Three contributions to topology, algebraic geometry and representation theory: homological finiteness of abelian covers, algebraic elliptic cohomology theory, and monodromy theorems in the elliptic setting
by Yaping Yang
B.S. in Mathematics, Zhejiang University M.S. in Mathematics, Northeastern University
A dissertation submitted to
The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy
April 18, 2014
Dissertation directed by
Valerio Toledano Laredo Professor of Mathematics Acknowledgments
I would like to express my deepest gratitude to my advisor, Prof. Valerio Toledano Laredo, and unofficial advisors, Prof. Alex Suciu and Prof. Marc Levine. During the past 5 years, I have received tremendous support and encouragement from them. They showed me the road and helped to get me started on Mathematics. They were always available for my questions and were very positive, patient and gave generously of their time and knowledge. I wrote my first paper with Prof. Alex Suciu, and gave my first talk under the help of Prof. Alex
Suciu. I am still benefit from his way of thinking and his attitude to mathematics. I am very grateful to Prof. Valerio Toledano Laredo for his enormous help, who treats everything seriously to make it perfect and always reminds me the seminars, workshops, talks, deadlines and etc. He has done me a great favor (even spent more time than me) in the process of
finding me a postdoctoral position.
I would like to thank Prof. Pavel Etingof and Prof. Ivan Losev for being members of the committee. Their enthusiasm and passion for mathematics always inspires me. Their responsible, humor and pizza make the seminar enjoyable and memorable. I am particularly grateful to Prof. Pavel Etingof for encouragement and insights with admiration, who provides valuable advices to the work and generously shares his ideas, and always ready to help.
I would also like to thank my professors at Northeastern University, Prof. Venkatraman
Lakshmibai, Prof. Alina Marian, Prof. Jonathan Weitsman, Prof. Ben Webster, Prof. Jerzy
Weyman and Prof. Andrei Zelevinsky for offering excellent courses and seminars over the years, for their guidance and help.
ii In addition, I would like to give many thanks to my friends and colleagues, Gufang,
Andrea, Sachin, Salvo, Shih-Wei, Yinbang, Andy, Kavita, Federico, Thomas, Jason, Andras,
Nate, Jeremy, Barbara, Simone, Adina, Chen, Brian, Rueven, Floran, Undine, Nicholas,
Ryan and Ryan, Gouri, Anupam, Jose, Rahul, Liang, He, Ruoran, Huijun, Sasha, Saif,
Andrew, Peng, Yong, Thi, Toan, Lei, Annabelle, Tian, Jiuzu, Yi, John, Lee-Peng, and many others. Many years with them, full of happiness and joy. Especially, thanks Gufang and
Sasha who are the potentially speakers whenever I couldn’t find enough speakers of the graduate student seminar, thanks Andrea and Hanai for sharing their wonderful templates, thanks Shih-Wei for liking every post on my facebook.
And finally, I would like to thank my family, my parents, my sister, my brother in-law, my lovely niece for their love and support. I am particularly grateful to my fiance, and collaborator Gufang, for cooking up good math and delicious food for us over the past 5 years.
Yaping Yang
Northeastern University
April 182014
iii Abstract of Dissertation
My doctoral work consists of three projects.
The first project is joint work with A. Suciu and G. Zhao described in more details in Chapter 2 and 3 of this dissertation. Chapter 2 is based on the paper [82], where we exploit the classical correspondence between finitely generated abelian groups and abelian complex algebraic reductive groups to study the intersection theory of translated subgroups in an abelian complex algebraic reductive group, with special emphasis on intersections of
(torsion) translated subtori in an algebraic torus.
Chapter 3 is based on the paper [83], we present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of r-planes in H1(X, Q). Inside this parameter space, there
i is a subset ΩA(X) consisting of all the covers with finite Betti numbers up to degree i. Building on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth, quasi-projective varieties, the generalized Dwyer–Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts.
iv The second project is joint work with M. Levine and G. Zhao described in more details
in Chapter 4 of this dissertation, which is based on the paper [91]. We define the algebraic
elliptic cohomology theory coming from Krichever’s elliptic genus as an oriented cohomology
theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic
cobordism ring with rational coefficients, the ideal generated by differences of classical flops
coincides with the kernel of Krichever’s elliptic genus. This generalizes a theorem of B. Totaro
in the complex analytic setting.
The third project consists of two parts. Chapter 5 is joint work with N. Guay, where
we give a double loop presentation of the deformed double current algebras, which are de-
formations of the central extension of the double current algebras g[u, v], for a simple Lie algebra g. We prove some nice properties of the algebras using the double loop presentation.
Especially, we construct a central element of the deformed double current algebra.
Chapter 6 is joint work with V. Toledano Laredo. In [10], Calaque-Enriquez-Etingof constructed the universal KZB equation, which is a flat connection on the configuration space of n points on an elliptic curve. They show that its monodromy yields an isomorphism between the completions of the group algebra of the elliptic braid group of type An−1 and the holonomy algebra of coefficients of the KZB connection. We generalized this connection and the corresponding formality result to an arbitrary root system in [91]. We also gave two concrete incarnations of the connection: one valued in the rational Cherednik algebra of the corresponding Weyl group, the other in the double deformed current algebra D(g) of the corresponding Lie algebra g. The latter is a deformation of the double current algebra g[u, v] recently defined by Guay in [34, 35], and gives rise to an elliptic version of the Casimir connection.
v Table of Contents
Acknowledgments ii
Abstract of Dissertation iv
Table of Contents vi
Disclaimer xiv
Chapter 1 Introduction1
1.1 Brief Overview...... 1
1.2 Characteristic varieties and Betti numbers of abelian covers...... 2
1.2.1 Motivations...... 2
1.2.2 Results achieved...... 3
1.2.3 Organization of Chapter 3...... 4
1.3 The algebraic elliptic cohomology theory...... 5
1.3.1 Motivations...... 5
1.3.2 Results achieved...... 7
1.3.3 Future directions...... 9
1.3.4 Organization of Chapter 4...... 11
1.4 Universal KZB equations and the elliptic Casimir connection...... 12
1.4.1 Motivations...... 12
vi 1.4.2 Results achieved...... 16
1.4.3 Future directions...... 19
Chapter 2 Intersections of translated algebraic subtori 22
2.1 Finitely generated abelian groups and abelian reductive groups...... 22
2.1.1 Abelian reductive groups...... 22
2.1.2 The lattice of subgroups of a finitely generated abelian group..... 24
2.1.3 The lattice of algebraic subgroups of a complex algebraic torus.... 25
2.1.4 Counting algebraic subtori...... 27
2.2 Primitive lattices and connected subgroups...... 28
2.2.1 Primitive subgroups...... 28
2.2.2 The dual lattice...... 30
2.3 Categorical reformulation...... 33
2.3.1 Fibered categories...... 33
2.3.2 A lattice over AbRed ...... 34
2.3.3 An equivalence of fibered categories...... 35
2.4 Intersections of translated algebraic subgroups...... 35
2.4.1 Two morphisms...... 35
2.4.2 Translated algebraic subgroups...... 36
2.4.3 Some consequences...... 39
2.4.4 Abelian covers...... 40
2.5 Exponential interpretation...... 41
2.5.1 The exponential map...... 41
2.5.2 Exponential map and Pontrjagin duality...... 42
2.5.3 Exponentials and determinant groups...... 43
2.5.4 Some applications...... 44
2.6 Intersections of torsion-translated subtori...... 45
vii 2.6.1 Virtual belonging...... 45
2.6.2 Torsion-translated tori...... 47
Chapter 3 Homological finiteness of abelian covers 48
3.1 A parameter set for regular abelian covers...... 48
3.1.1 Regular covers...... 48
3.1.2 Functoriality properties...... 49
3.1.3 Regular abelian covers...... 50
3.1.4 Splitting the torsion-free part...... 51
3.1.5 A pullback diagram...... 53
3.2 Reinterpreting the parameter set for A-covers...... 54
3.2.1 Splittings...... 55
3.2.2 A fibered product...... 55
3.2.3 Further discussion...... 57
3.3 Dwyer–Fried sets and their generalizations...... 57
3.3.1 Generalized Dwyer–Fried sets...... 57
3.3.2 Naturality properties...... 59
3.3.3 Abelian versus free abelian covers...... 60
3.3.4 The comparison diagram...... 62
3.4 Pontryagin duality...... 63
3.4.1 A functorial correspondence...... 63
3.4.2 Primitive lattices and connected subgroups...... 64
3.4.3 Pulling back algebraic subgroups...... 65
3.4.4 Intersections of translated subgroups...... 65
3.5 An algebraic analogue of the exponential tangent cone...... 66
3.5.1 A collection of subgroups...... 67
3.5.2 The exponential map...... 68
viii 3.5.3 Exponential interpretation...... 69
3.6 The incidence correspondence for subgroups of H ...... 70
3.6.1 The sets σA(ξ)...... 70
3.6.2 The sets UA(ξ)...... 72
3.7 The incidence correspondence for subvarieties of Hb ...... 74
3.7.1 The sets UA(W )...... 74
3.7.2 The sets ΥA(W )...... 75
3.7.3 Translated subgroups...... 76
3.7.4 Deleted subgroups...... 77
3.7.5 Comparing the sets ΥA(W ) and ΥA(W )...... 78 3.8 Support varieties for homology modules...... 81
3.8.1 Support varieties...... 81
3.8.2 Homology modules...... 82
3.8.3 Finite supports...... 84
3.9 Characteristic varieties and generalized Dwyer–Fried sets...... 85
3.9.1 The equivariant chain complex...... 85
3.9.2 Characteristic varieties...... 86
3.9.3 The first characteristic variety of a group...... 87
3.9.4 A formula for the generalized Dwyer–Fried sets...... 88
3.9.5 An upper bound for the Ω-sets...... 89
3.10 Comparison with the classical Dwyer–Fried invariants...... 91
i 3.10.1 The Dwyer–Fried invariants Ωr(X)...... 91 3.10.2 The comparison diagram, revisited...... 91
3.10.3 Maximal abelian versus free abelian covers...... 93
3.11 The rank 1 case...... 94
i 3.11.1 A simplified formula for ΩA(X)...... 94
ix 3.11.2 The singular set...... 95
3.11.3 A particular case...... 97
3.11.4 Another particular case...... 98
3.12 Translated tori in the characteristic varieties...... 99
3.12.1 A refined formula for the Ω-sets...... 99
3.12.2 Another formula for the Ω-sets...... 100
3.12.3 Toric complexes...... 101
3.12.4 When the translation order is coprime to |Tors(A)| ...... 102
3.12.5 When the translation order divides |Tors(A)| ...... 103
3.13 Quasi-projective varieties...... 105
3.13.1 Characteristic varieties...... 105
3.13.2 Dwyer–Fried sets...... 106
3.13.3 Brieskorn manifolds...... 107
3.13.4 The Catanese–Ciliberto–Mendes Lopes surface...... 109
Chapter 4 Algebraic elliptic cohomology theory and flops 111
4.1 Oriented cohomology...... 111
4.1.1 Oriented cohomology theories and algebraic cobordism...... 111
4.1.2 Motivic oriented cohomology theories...... 115
4.1.3 Specialization of the formal group law...... 117
4.1.4 Landweber exactness...... 119
4.1.5 Exponential and logarithm...... 120
4.1.6 Twisting a cohomology theory...... 121
4.1.7 Universality...... 124
4.2 The algebraic elliptic cohomology theory...... 129
4.2.1 The elliptic formal group law...... 129
4.3 Flops in the cobordism ring...... 134
x 4.3.1 The double point formula...... 134
4.3.2 Flops in the cobordism ring...... 137
4.3.3 Flops in the elliptic cohomology ring...... 140
4.4 The algebraic elliptic cohomology ring with rational coefficients...... 143
4.5 Birational symplectic varieties...... 145
4.5.1 Specialization in algebraic cobordism theory...... 145
4.5.2 Cobordism ring of birational symplectic varieties...... 147
4.6 Appendix: The ideal generated by differences of flops...... 150
Chapter 5 Deformed double current algebras 155
5.1 Deformed double current algebras...... 155
5.2 The map ϕ is a homomorphism of algebras...... 157
5.3 The map ϕ is an isomorphism of algebras...... 161
5.4 Some properties of the deformed double current algebras...... 170
5.5 A central element of D(g)...... 174
5.6 Deformed double current algebras with two parameters...... 179
Chapter 6 The elliptic Casimir connection 187
6.1 The generalized holonomy Lie algebra Aell ...... 187
6.2 The universal connection for arbitrary root systems...... 188
6.2.1 Principal bundles...... 188
6.2.2 Some facts about theta functions...... 189
6.2.3 General form...... 190
6.3 Flatness of the universal connection, part 1...... 193
6.4 Flatness of the universal connection, part 2...... 195
6.4.1 Case-A2...... 198
6.4.2 Case-B2...... 199
xi 6.4.3 Case-G2...... 200
6.5 Equivariance under Weyl group action...... 201
6.6 Degeneration of the elliptic connection...... 202
6.6.1 The trigonometric connection...... 202
6.6.2...... 204
6.6.3...... 205
6.7 Generalized Formality Isomorphism...... 207
6.7.1 The double affine braid group...... 208
6.7.2...... 209
6.8 Some set S ...... 217
Φ 6.9 Derivation of the Lie algebra Aell ...... 218
6.9.1 δ2m preserve the commutating relation of [y, x]...... 219 ˜ P P ˜ 6.9.2 Check [δ2m(tα), β∈Ψ+ tβ] + [tα, β∈Ψ+ δ2m(tβ)] = 0...... 220 ˜ ˜ 6.9.3 Check that [y(u), δ2m(tα)] + [δ2m(y(u)), tα]=0, if (α, u) = 0...... 221 ˜ ˜ 6.9.4 Check [y(u), δ2m(y(v))] + [δ2m(y(u)), y(v)] = 0...... 226
6.10 Bundle on the moduli space M1,n ...... 228
6.11 Flat connection on the moduli space M1,n ...... 230
6.11.1 Check that dA = 0...... 231
6.11.2 Check that A ∧ A = 0...... 232
6.11.3 Case A2 ...... 237
6.11.4 Case B2 ...... 238
6.11.5 Case G2 ...... 238
6.12 Simply connected type and adjoint type...... 241
6.13 The Homomorphism from Aell to rational Cherednik algebras...... 243
6.13.1 The rational Cherednik algebra...... 243
6.13.2 The homomorphism from Aell to the rational Cherednik algebra... 244
xii 6.13.3 The extension of the homomorphism...... 245
6.14 The Homomorphism from Aell to the deformed double current algebras... 250
6.14.1 The deformed double current algebras...... 250
6.14.2 The homomorphism to the deformed double current algebras..... 252
6.15 The extension of the elliptic Casimir connection...... 254
6.15.1...... 254
6.15.2...... 255
6.15.3...... 256
6.15.4...... 257
6.16 Proof of Conjecture 6.15.2 in special cases...... 261
Bibliography 267
xiii Disclaimer
I hereby declare that the work in this thesis is that of the candidate alone, except where indicated in the text, and as described below.
Chapter 2, Chapter 3 are based on a joint work with Dr. Alexandru I. Suciu and G- ufang Zhao, published as [82] Intersections of translated algebraic subtori. J. Pure Appl.
Algebra 217 (2013), no. 3, 481-494. [83] Homological finiteness of abelian covers. Accepted for publication by Annali della Scuola Normale Superiore di Pisa, 42 pages, available at arXiv:1204.4873. available at arXiv:1109.1023.
Chapter 4 is based on a joint work with Dr. Marc N. Levine and Gufang Zhao, published as [56] Algebraic Elliptic cohomology theory and flops I, 33 pages, available at arXiv:1311.2159.
Chapter 5 is based on a joint work with Dr. Nicolas Guay, published as [36] On deformed double current algebras, preprint.
Chapter 6 is based on a joint work with Dr. Valerio Toledano Laredo, published as [91]
Universal KZB equations for arbitrary root systems and the elliptic Casimir connection, 45 pages, preprint.
xiv Chapter 1
Introduction
1.1 Brief Overview
My doctoral work consists of three parts described in more details in Chapter 2, 3, 4, 5, 6
respectively of this dissertation.
The first project is joint work with A. Suciu and G. Zhao. In [70, 81], S. Papadima
and A. Suciu developed a method for determining when a regular free abelian cover of a
CW-complex X has finite Betti numbers. They showed that it can be tested through the characteristic variety of X, which are the jump loci for the cohomology with coefficients in
rank 1 local systems on X. We generalized this method to the set of regular abelian covers
of X in [82, 83].
The second project is joint work with M. Levine and G. Zhao. Totaro in [84] showed that
the kernel of the elliptic genus from the complex cobordism ring to the elliptic cohomology
ring is generated by difference of classical flops. In [56], we generalized Totaro’s theorem to
the algebraic setting. Our proof can be applied to a field with arbitrary characteristic.
The third project is joint work with V. Toledano Laredo. In [10], Calaque-Enriquez-
Etingof constructed the universal KZB equation, which is a flat connection on the con-
figuration space of n points on an elliptic curve. They show that its monodromy yields an
1 isomorphism between the completions of the group algebra of the elliptic braid group of type
An−1 and the holonomy algebra of coefficients of the KZB connection. We generalized this connection and the corresponding formality result to an arbitrary root system in [91]. We also gave two concrete incarnations of the connection: one valued in the rational Cherednik algebra of the corresponding Weyl group, the other in the double deformed current algebra
D(g) of the corresponding Lie algebra g. The latter is a deformation of the double current algebra g[u, v] recently defined by Guay in [34, 35], and gives rise to an elliptic version of the Casimir connection.
1.2 Characteristic varieties and Betti numbers of a-
belian covers
1.2.1 Motivations
Let X be a connected finite CW-complex. Let G = π1(X, x0) be the fundamental group.
Consider the regular covers of X, with group of deck transformations A. In the case when A ∼=
r n Z , the parameter set for regular, A-covers can be identified with Grr(Q ), the Grassmannian 1 ∼ n of r-planes in H (X, Q) = Q .
i In a foundational paper [18], Dwyer and Fried isolated the subsets Ωr(X) of the Grass-
n mannian Grr(Q ), consisting of those covers for which the Betti numbers up to degree i are finite.
i The Dwyer–Fried sets Ωr(X) have been studied in depth in [70, 81], using the character- istic varieties of X. These varieties, V i(X), are Zariski closed subsets of the character group
∗ Hb = Hom(H, C ), where H = H1(X, Z); they consist of those rank 1 local systems on X for which the corresponding cohomology groups do not vanish, for some degree less or equal to i.
2 Theorem 1.2.1 (Papadima–Suciu [70, 81]). Let X be a connected, finite CW-complex, and
let H = H1(X, Z). Then,
i n i Ωr(X) = {P ∈ Grr(Q ) | exp(P ⊗ C) ∩ W (X) is finite}, where exp(P ⊗ C) lies inside the identity component Hb 0 of the character group Hom(H, C∗) and W i(X) = V i(X) ∩ Hb 0 is the Alexander variety of X.
The above theorem gives a method for deciding when a regular free abelian cover of X has finite Betti numbers. In particular, when the Alexander variety W i(X) of X is finite, all regular, free abelian covers of X have finite Betti numbers.
A natural question is the following.
Question 1.2.2. For any quotient group A of π1(X, x0), is there a similar description of the set of regular A-covers of X for which the Betti numbers are finite?
1.2.2 Results achieved
In the joint papers [82, 83] with Alex Suciu, and Gufang Zhao, we answered the Question
1.2.2 in the case when A is an abelian group.
The regular covers of X with group of deck transformations isomorphic to A can be parameterized by the set
Γ(G, A) = Epi(G, A)/ Aut(A),
where Epi(G, A) is the set of all epimorphisms from G to A and Aut(A) is the group of
automorphisms of A, acting on Epi(G, A) by composition. For an epimorphism ν : G A, we write its class in Γ(G, A) by [ν], and the corresponding cover by Xν → X.
We first identify this set Γ(G, A) with a (set-theoretical) twisted product over a rational
Grassmannian, whose fiber is a concrete finite set. We define the generalized Dwyer–Fried
i invariants of X to be the subsets ΩA(X) of Γ(G, A) consisting of those regular A-covers having finite Betti numbers up to degree i. In the case when A is a finitely generated (not necessarily
3 torsion-free) abelian group, we establish a similar formula as Theorem 1.2.1, computing the
i invariants ΩA(X), viewed now as subsets of Γ(G, A), in terms of the characteristic varieties of X.
Theorem 1.2.3 (Suciu–Yang–Zhao [83]). Let X be a connected, finite CW-complex, and let
H = H1(X, Z). Suppose ν : H A is an epimorphism to an abelian group A. Then
i i ΩA(X) = {[ν] ∈ Γ(H,A) | im(ˆν) ∩ V (X) is finite}, where νˆ: Ab → Hb is the induced morphism between character groups.
As shown by Arapura [3], the characteristic varieties of a connected, smooth, quasi- projective variety X consist of translated subtori of the character torus Hb. Understanding the way translated subtori intersect gives valuable information on the Betti numbers of regular, abelian covers of X.
In [82], building on the approach taken by E. Hironaka in [38], we use Pontrjagin duality of algebraic subgroups of Hb and subgroups of H to study intersections of translated subtori in a complex algebraic torus. This allows us to decide whether a finite collection of translated subgroups intersect non-trivially, and, if so, what the dimension of their intersection is.
Use the method developed in [82], we are able to compare generalized Dwyer–Fried invari-
i i ants ΩA(X) of X with their classical counterparts Ωr(X). Topologically, this comparison gives a criterion whether the finiteness of the Betti numbers of an A-cover can be test- ed through the corresponding A/ Tors(A)-cover. Some nice classes of spaces to which our theory applies is that of toric complexes and quasi-projective varieties.
1.2.3 Organization of Chapter 3
Chapter 3 is organized as follows:
In §3.1 and §3.2, we describe the structure of the parameter set Γ(H,A) for regular A- covers of a finite, connected CW-complex X with H1(X, Z) = H, while in §3.3 we define the
4 i generalized Dwyer–Fried invariants ΩA(X), and study their basic properties. In §3.4, we review the Pontryagin correspondence between subgroups of H and algebraic
subgroups of the character group Hb, while in §3.5 we associate to each subvariety W ⊂ Hb a family of subgroups of H generalizing the exponential tangent cone construction. In §3.6
and §3.7, we introduce and study several subsets of the parameter set Γ(H,A), which may
be viewed as analogues of the special Schubert varieties and the incidence varieties from
classical algebraic geometry.
In §3.8 we revisit the Dwyer–Fried theory in the more general context of (not necessarily
i torsion-free) abelian covers, while in §3.9 we show how to determine the sets ΩA(X) in terms of the jump loci for homology in rank 1 local systems on X. In §3.10, we compare the
Dwyer–Fried invariants Ωi (X) with their classical counterparts, Ωi (X), while in §3.11 we A A discuss in more detail these invariants in the case when rank A = 1.
Finally, in §3.12 we study the situation when all irreducible components of the charac-
teristic varieties of X are (possibly translated) algebraic subgroups of the character group, while in §3.13 we consider the particular case when X is a smooth, quasi-projective variety.
1.3 The algebraic elliptic cohomology theory
1.3.1 Motivations
In the category of differentiable manifolds, the cohomology theories have been studied by topologist for a long time, e.g., the singular cohomology groups, topological K-theory and etc.
The analog cohomology theories in algebraic geometry are also important tools to under- stand algebraic varieties. Let k be a field and Smk be the category of smooth, quasi-projective schemes over k. Let R∗ denote the category of commutative, graded rings with unit. One
∗ op ∗ can define the oriented cohomology theory on Smk to be an additive functor A : Smk → R
5 with smooth pull-backs and proper push-forwards, which satisfy some formal Axioms. See
[54] for details. The fundamental insight of Quillen is that it is not true in general that one has the formula
c1(L ⊗ M) = c1(L) + c1(M), for line bundles L and M over the same base X.
In general, the power series
FA(c1(L), c1(M)) := c1(L ⊗ M) defines a formal group law on the ring A∗(k). The Chow ring X 7→ CH∗(X) is a basic
∗ example of an oriented cohomology theory on Smk. The formal group law on CH (k) = Z is
0 0 the additive formal group law Fa(u, v) = u + v. The Grothendieck K functor X 7→ K (X) is another fundamental example of an oriented cohomology theory. The formal group law
∗ −1 on K (k) = Z[β, β ] is the multiplicative formal group law Fm(u, v) = u + v − βuv. Assume k has characteristic zero. Levine and Morel construct the universal oriented cohomology theory, whose formal group law is the universal formal group law FΩ on the
Lazard ring Laz := Z[p1, p2, ··· ].
Theorem 1.3.1 (Levine and Morel). Assume k has characteristic zero. Then, there exists a universal oriented cohomology theory on Smk, denoted by
X 7→ Ω∗(X),
∗ which called the algebraic cobordism. Thus, given an oriented cohomology theory A on Smk, there is a unique morphism Ω∗ → A∗ of oriented cohomology theories.
Let the power series λ(x) ∈ A∗({pt}) ⊗ Q[x] be the exponential of the formal group law F , that is,
F (u, v) = λ(λ−1(u) + λ−1(v)).
6 The map φ :Ω∗({pt}) → A∗({pt}) called the Hirzebruch genus, is given by the Hirzebruch
x characteristic power series Q(x) = λ(x) , more explicitly, the map φ is defined to be Z n Y xi φ(X) := ( ), λ(x ) X i=1 i
where x1, ··· , xn are the Chern roots of X. In the case the A∗ is the K-theory. The Hirzebruch genus specializes to the classical
Todd genus with characteristic power series
x βx Q(x) = = . λ(x) 1 − e−βx
Let
∗ 2πiz 2iπτ φE :Ω → Q((e ))[[e , k]] be the elliptic genus studied by Krichever in [49]. The Krichever elliptic genus has the rigidity property, see [49] and [43]. Use this rigidity property, H¨ohnand Totaro studied the kernel of the genus φE, see [43] and [84].
Theorem 1.3.2 (Totaro). Let I be the ideal in the complex cobordism ring MU ∗⊗Q, which is additively generated by differences X1 −X2, where X1 and X2 are smooth projective varieties related by a classical flop. Then the complex elliptic genus, viewed as a ring homomorphism
∗ φE : MU ⊗ Q → Q[x1, x2, x3, x4], is surjective with kernel equal to I.
It is worth mentioning that in Totaro’s work, the proof depends on the category of weakly complex manifolds, and topological constructions which apparently does not lend themselves to a field of positive characteristic.
1.3.2 Results achieved
In the joint paper [56] with Levine and Zhao, we study the algebraic version of the elliptic cohomology theory over an arbitrary field k. Based on a theorem of Landweber, any formal
7 group law gives rise to a P1-spectrum, if certain flatness assumption called Landweber ex- actness is satisfied. The elliptic formal group law corresponding to the Krichever’s elliptic genus is defined on the ring Z[a1, a2, a3, a4], with explicit descriptions of the four elements ai. We show that this formal group law is Landweber exact if we enlarge the coefficient ring by
−1 taking Ell[1/2] = Z[1/2][a1, a2, a3, a4][∆ ], where ∆ is the discriminant. As a consequence,
∗ ∗ the elliptic cohomology theory can be defined as Ell (X) := Ω (X) ⊗Laz Ell[1/2]. We extended Totaro’s Theorem 1.3.2 to the algebraic setting, which also works when the base field k has positive characteristic. In the case when k has positive characteristic, though the algebraic cobordism theory Ω∗ is not constructed. There is another theory called the
∗,∗ algebraic cobordism spectra MGL in the motivic stable homotopy category of P1-spectra
2n,n of Morel and Voevodsky (see [63]), whose geometric part ⊕n MGL (X) coincides with the algebraic cobordism Ω∗(X) when the base field has characteristic zero. The geometric part
2n,n ∗ ⊕n MGL ({pt}) is denoted by Ω ({pt}). The following is the main theorem of the project we obtained so far.
Theorem 1.3.3 (Levine–Yang–Zhao [56]). The kernel of the algebraic elliptic genus
∗ ∗ φQ :Ω ({pt}) ⊗ Q → Ell ({pt}) ⊗ Q is generated by the difference of classical flops, and its image is the free polynomial ring
Q[a1, a2, a3, a4].
The main tools we use to show the above theorem are
• the double point relations formula of Ω∗ found by Levine and Pandharipande, see [55].
• the Quillen’s push-forward formula, see [77].
For F ⊆ X be a subscheme of X, the double point relation yields the following blow-up formula in Ω∗:
X = BlF X + P(NF X ⊕ O) − P(ONF X (1) ⊕ O),
8 where BlF X is blow-up of X along F , and NF X is the normal bundle of F . This allows us to express the difference of flops X1 − X2 in terms of linear combinations of projective spaces. Then, we pushforward those projective spaces to Ell∗({pt}) using Quillen’s formula, which yields zero due to a classical identity of sigma-functions. Thus, hX1 − X2i ⊂ ker φQ.
1.3.3 Future directions
In this section, we describe future directions of the ongoing project of the author joint with
Levine and Zhao.
One may wonder whether the Theorem 1.3.3 holds over then coefficient ring Z without tensoring Q. In the category of complex manifolds, as pointed out by Totaro that it is more natural to work with the SU-cobordism ring MSU ∗ rather than the cobordism ring
MU ∗, for example because the image of MU ∗ under the complex elliptic genus is not finitely generated, although after tensoring with Q it becomes the polynomial ring Q[x1, x2, x3, x4]. As shown by Totaro in [84],
∗ ∼ MSU ⊗ Z[1/2]/SU-flops = Z[1/2][x2, x3, x4]
In algebraic setting, for an integral questions such as this, the analogous of SU-cobordism is the algebraic cobordism spectra MSL∗,∗, see [74] for the definition. To extend the Theorem
1.3.3 to the integral setting, we need to know the coefficient ring of MSL∗,∗({pt}), at least
2n,n the geometric part MSL({pt}) := ⊕n MSL ({pt}). Based on the result of complex SU- cobordism ring MSU ∗, see [84], we make the following conjecture.
Conjecture 1.3.4. The coefficient ring of the MSL-cobordism ring
∼ MSL({pt}) ⊗ Z[1/2] = Z[1/2][x2, x3, x4,... ],
9 where the polynomial generators xi are determined as follows. For X ∈ MGL({pt})⊗Z[1/2] is a polynomial generator of MSL({pt}) ⊗ Z[1/2] in degree n if and only if ±p(a power of 2), if n is a power of an odd prime p; sn(X) = ±p(a power of 2), if n + 1 is a power of an odd prime p; ±(a power of 2), otherwise. where sn(X) is the Chern number of the tangent bundle of X, that is,
n n n s (X) := hx1 + ··· + xn, [X]i, with xi being the Chern roots of the tangent bundle of X.
Since the SU-cobordism ring is obtained by Novikov using the Adams spectral sequences, to prove the Conjecture 1.3.4, it would be helpful to compare the classical Adams spectal sequence with the Motivic Adams spectral sequences via the Betti and ´etalerealizations.
Conjecture 1.3.4 implies the following conjecture.
Conjecture 1.3.5. The kernel of the algebraic elliptic genus on MSL∗({pt}) ⊗ Z[1/2] is equal to the ideal of SL-flops. The quotient ring is a polynomial ring Z[1/2][a2, a3, a4].
Nakajima quiver varieties form a large class of complex algebraic varieties arising from repre- sentation theory. The cotangent bundle of flag varieties, Hilbert scheme of points on surfaces are special cases of the Nakajima quiver varieties.
In [64], Nakajima used these varieties to provide a geometric construction of the irre- ducible integrable representations of the Kac-Moody Lie algebras g. In other words, there is an action of the Kac-Moody Lie algebras g on the top degree of the Borel-Moore homology of the Nakajima quiver variety associated to the Dynkin quiver of g and this action gives all the irreducible integrable representations of g.
In [94], the action of the Yangians on the equivariant Borel-Moore homology of quiver va- rieties is established by Varagnolo. In [65], Nakajima constructed the action of the quantum loop algebra on the equivariant K-theory of quiver varieties. Those actions give geometric
10 realizations of the algebras Yangians and quantum loop algebra respectively. As a corol-
lary, there is a set-theoretic map from the category of finite dimensional representations of
Yangians Repfd(Y~(g)) to the category of finite dimensional representations of quantum loop algebra Repfd(U~(Lg)). Motivated by this work, Gautam and Toledano Laredo [30] constructed a functor from
Repfd(Y~(g)) to Repfd(U~(Lg)), when ~ is a formal parameter, and showed it’s an equivalence of appropriate subcategories. In [29], Gautam and Toledano Laredo constructed a functor from Repfd(Y~(g)) to Repfd(U~(Lg)) for numerical parameter ~, using monodromy of abelian difference equations.
Guided by the same principal, one may ask the following question.
Question 1.3.6. Is there a quantum algebra action on the equivariant elliptic cohomology of the Nakajima quiver varieties?
It is expected that the elliptic quantum groups act on the equivariant elliptic cohomol- ogy of quiver varieties, as suggested to the author by Toledano Laredo. While the elliptic quantum groups have not been well-studied yet. The twisted constructions of the equiv- ariant elliptic cohomology theory (that is, the equivariant elliptic cohomology theory is the equivariant Chow theory tensoring with the elliptic cohomology ring) constructed in [54] is supposed to be useful to obtain such an action. The product of the equivariant elliptic cohomology of the quiver varieties can be computed using the twisted constructions of the equivariant elliptic cohomology theory.
1.3.4 Organization of Chapter 4
Chapter 4 is organized as follows: In §4.1 we recall some foundational material concerning oriented cohomology and motivic oriented cohomology. We describe two methods of con- struction oriented cohomology theories from a given formal group law: specialization and twisting, and recall the construction of a motivic oriented cohomology theory through spe-
11 cialization to a Landweber exact formal group law. We use the twisting construction to
construct the universal oriented cohomology theory on Smk for k a field of positive chara- teristic, after restriction to theories with coefficient ring a Q-algebra; for k of characteristic zero, one already has the universal theory among all oriented theories, namely, Ω∗. In §4.2 we apply these results to give our construction of elliptic cohomology as an oriented coho- mology theory on Smk for k an arbitrary perfect field, and we prove our main result on the existence of a motivic oriented theory representing elliptic cohomology (Theorem ??). In
§4.3 we introduce the double-point cobordism and use this theory to give an explicit de- scription of the difference of two flops in MGL∗-theory. We then apply this formula to show that the difference of two flops vanishes in elliptic cohomology. In §4.4 we use H¨ohn’sand
Totaro’s algebraic computations to prove our main result (Theorem ??) on rational elliptic
∗ cohomology and its relation to MGLQ (we recall Totaro’s computations in an appendix). We conclude in §4.5 with another application of algebraic cobordism, showing that birational smooth projective symplectic varieties that arise from different specializations of generically isomorphic families have the same class in algebraic cobordism, and in particular, have the same Chern numbers (see proposition 4.5.2 and corollary 4.5.5).
1.4 Universal KZB equations and the elliptic Casimir
connection
1.4.1 Motivations
The KZ connection
Around 1990, T. Kohno and V. G. Drinfeld proved the Kohno-Drinfeld Theorem, see [48]
[19]. Roughly speaking, the theorem states that quantum groups can be used to describe monodromy of certain first order Fuchsian PDEs known as Knizhnik- Zamolodchikov (KZ)
12 equations.
To be more precise, given a simple Lie algebra g, a representation V of g and a positive
integer n, the KZ equations are the following system of PDEs:
dΦ X Ωij = ~ Φ, (1.4.1) dzi zi − zj i6=j where
1. Φ is a function on the configuration space of n-ordered points on C (denoted by C(C, n)) with values in V ⊗n.
2.Ω ij ∈ g ⊗ g is the Casimir operator
3. ~ is a complex deformation parameter.
This system is integrable and invariant under the natural symmetric group Sn action and hence defines a one-parameter family of monodromy representations of Artin’s braid group
Bn = π1(C(C, n)/Sn). The Kohno-Drinfeld theorem asserts that this representation is equivalent to the R-matrix representation of Bn arising from the quantum group U~(g).
The rational Casimir connection
In subsequent years, J. Millson and V. Toledano Laredo [62, 87], and independently C. De
Concini(unpublished) and G. Felder et al [26] constructed another flat connection ∇C , the Casimir connection of a simple Lie algebra g, which is described as follows.
Let h be a Cartan subalgebra of g and let hreg be the complement of the root hyperplanes
in h. For a finite-dimensional g module V , the Casimir connection ∇C is a connection on the trivial vector bundle V of the following form:
X dα ∇ := d − C , C ~ α α α>0
where
13 1. the summation is over a set of chosen positive roots of g.
2. Cα ∈ g ⊗ g is the Casimir operator of the sl2 subalgebra of g corresponding to the root α.
This connection is flat and equivariant with respect to the Weyl group W , thus, gives rise
to a one-parameter family of mondromy representations of the generalized braid group Bg :=
π1(hreg/W ). In this setting a Kohno-Drinfeld type theorem was obtained by V. Toledano Laredo
[85, 87, 88, 89, 90], stating that the monodromy of the Casimir connection is described by
the quantum Weyl group operators of the quantum group U~(g).
The trigonometric Casimir connection
V. Toledano Laredo constructed a trigonometric extension of ∇C in [86] recently, now known
as the trigonometric Casimir connection. Let Hreg be the complement of the root hypertori of a maximal torus H of the simply connected Lie group G with Lie algebra g.
This trigonometric Casimir connection ∇trig,C is a connection on the trivial vector bundle
Hreg ×V , where the fiber V is a finite-dimensional representation of the Yangian Y~(g), which
i ∗ is a deformation of U(g[s]). Let {u } and {ui} be the dual basis of h and h ,
X dα ∇ = d − ~ κ − A(ui)du , trig,C 2 eα − 1 α i α∈Φ+ is flat and W -equivariant, where
i • A(u )dui is a translation invariant 1-form on H;
• A : h → Y~(g) is a linear map such that A(u) ≡ u ⊗ s mod ~.
The connection ∇trig,C is flat and W -equivariant and its monodromy yields a one parameter family of monodromy representations of the affine braid group
Aff Bg := π1(Hreg/W ).
14 By analogy with the monodromy theorem of rational Casimir connection [87], Toledano
Laredo formulated the following [86]
Conjecture 1.4.1 (Toledano Laredo). The monodromy of the trigonometric Casimir con-
nection is described by the quantum Weyl group operators of the quantum loop algebra
U~(Lg).
Partial results of the Conjecture 1.4.1 was proved by Gautam and Toledano Laredo in
[29].
The elliptic Casimir connection
My joint project with V. Toledano Laredo [91] is aimed at addressing the following
Problem 1.4.2. Construct an elliptic analogue of the Casimir connection, and describe its
monodromy in terms of quantum groups.
The universal KZB connection
In type An, elliptic connections were constructed by Calaque-Enriquez-Etingof [10] in the following context. Let C(X, n) be the configuration space of n-distinct points on a Riemann
surface X. Bezrukavnikov showed that there is an isomorphism between the completion of
the group algebra of π1(C(X, n)) and that of the universal enveloping algebra of an explicit
Lie algebra tg,n, where g is the genus of the Riemann surface X. We call this isomorphism Bezrukavnikov’s isomorphism.
In the case when X = E is an elliptic curve, Bezrukavnikov’s isomorphism is construct- ed explicitly by Calaque-Enriquez-Etingof using the monodromy of a flat connection on a suitable principal bundle over the configuration space C(E, n) in [10]. The flat connection,
known as the universal KZB connection, valued in the Lie algebra t1,n, is given by:
n n X X X ∇KZB = d − ( k(zi − zj, ad xi|τ)(tij))dzi + yidzi, (1.4.2) i=1 {j|j6=i} i=1
15 where:
n 1. zi’s are coordinates of C ,
2. the elements tij, xi, and yi are generators of t1,n,
3. the function k(z, x|τ) is expressed as a ratio of theta functions. k(z, x|τ) has a simple
pole at z = 0 with residue 1, but no pole at x = 0.
Calaque-Enriquez-Etingof showed that the monodromy of ∇KZB induces an isomorphism from the completion of the group algebra of π1(C(E, n)) to the completion of the universal enveloping algebra of t1,n. A similar construction of Bezrukavnikov’s isomorphism for Riemann surfaces of higher genus case using the monodromy of flat connections was done by Enriquez in [22].
1.4.2 Results achieved
The universal KZB-connection
In joint work with Toledano Laredo, we generalized the universal KZB-connection and the corresponding formality result to the root system associated to an arbitrary simple Lie algebra g.
∨ reg Let h ⊂ g be a Cartan subalgebra, P ⊂ h the corresponding coweight lattice, and Hτ the complement of the root hyperplanes in h/(P ∨ + τP ∨), where τ is a complex number
reg with positive imaginary part. Note that Hτ is a complement of a divisor in the rth power
reg of the elliptic curve Eτ = C/(Z + τZ), where r = dim h. By definition, Hτ is the elliptic configuration space.
Let A be an algebra endowed with the following data:
• a set of elements {tα}α∈Φ ⊂ A such that t−α = tα,
• a linear map x : h → A,
16 • a linear map y : h → A.
reg Consider the A-valued connection on Hτ given by n X xα∨ X ∇ = d − k(α, ad( )|τ)(t )dα + y(ui)du , (1.4.3) τ 2 α i α>0 i=1 where the summation is over a chosen system of positive roots, and α∨ ∈ h are the coroots of g.
In [91], we give a criteria, in terms of relations of the algebra A, of the flatness and W - equivariance of the connection ∇τ . This motives the definition of the holonomy Lie algebra
A by imposing those relations which are satisfied to make the connection ∇τ flat.
Theorem 1.4.3 (Toledano Laredo–Yang [91]). The connection ∇τ,n in (6.2.3) is flat if and only if the following relations holds:
1. For any rank 2 root subsystem Ψ ⊂ Φ, and α ∈ Ψ,
X [tα, tβ] = 0; β∈Ψ+
2. [x(u), x(v)] = 0, [y(u), y(v)] = 0, for any u, v ∈ h;
3. [y(u), x(v)] = Σγ∈Φ+ hv, γihu, γitγ.
4. [tα, x(u)] = 0, [tα, y(u)] = 0, if hα, ui = 0.
The connection ∇τ,n in (6.2.3) is W -equvariant, if and only if:
1. si(tα) = tsiα;
2. si(x(u)) = x(siu);
3. si(y(v)) = y(siv)
We extended the formality result in this case.
Theorem 1.4.4 (Toledano Laredo–Yang [91]). The induced map µ : C[π\1(Hreg)] → Ab is an isomorphism of Hopf algebras.
17 Rational Cherednik algebras
We give in [91] a concrete incarnation ∇RCA of the connection ∇τ with values in the ra- tional Cherednik algebra of W . The connection ∇RCA is flat and W -equivariant. Its mon- odromy yields a one parameter family of monodromy representations of the elliptic braid group π1(Hreg/W ), which factors through the double affine Hecke algebra of W [14]. Thus, the monodromy induces a functor from the category of finite-dimensional representations of rational Cherednik algebra to the category of finite-dimensional representations of the double affine Hecke algebra.
The elliptic Casimir connection
In [91], we also give another concrete incarnation of the universal connection ∇τ related to the Lie algebra g. Thus, solving the first part of Problem 1.4.2. This elliptic Casimir connection takes values in the deformed double current algebra D(g), a deformation of the universal central extension of the double current algebra g[u, v] recently introduced by Guay[34, 35].
The construction of a map from the algebra of coefficients of ∇τ to D(g) relies crucially on a double loop presentation of D(g). Such a presentation was obtained by Guay for sln in [34] by relying on Schur-Weyl duality but was otherwise unknown. In work in progress with
Guay [36], I obtained the following double loop presentation of the deformed double current algebra D(g).
Theorem 1.4.5 (Guay–Yang [36]). The algebra D(g) is generated by elements X,K(X),Q(X),P (X) such that
• K(X),X ∈ g generate a subalgebra which is an image of g⊗C C[u] with X ⊗u 7→ K(X);
• Q(X),X ∈ g generate a subalgebra which is an image of g⊗C C[v] with X ⊗v 7→ Q(X);
• P (X) is linear in X, and for any X,X0 ∈ g, [P (X),X0] = P [X,X0].
18 and the following relation holds for all root vectors Xβ1 ,Xβ2 ∈ g with β1 6= −β2:
(β1, β2) 1 X [K(X ),Q(X )] = P ([X ,X ]) − S(X ,X ) + S([X ,X ], [X ,X ]), β1 β2 β1 β2 4 β1 β2 4 β1 α −α β2 α∈∆ where S(a1, a2) = a1a2 + a2a1 ∈ U(g).
Based on the double loop presentation of the deformed double current algebra Dλ(g), we define the elliptic Casimir connection as follows:
Theorem 1.4.6 ([91]). The elliptic Casimir connection valued in the deformed double cur- rent algebra Dλ(g) n λ X Q(α∨) X ∇ = d − k(α, ad( )|τ)(C )dα + K(ui)du , Ell,C 2 2 α i α∈Φ+ i=1
is a flat and W -equivariant, where Cα ∈ U(g) is the Casimir operator.
1.4.3 Future directions
Extension to the Moduli spaces
The universal KZB-connection ∇KZB (1.4.2) and ∇τ (6.2.3) both depend on the modular
parameter τ. As shown in [10], the KZB-connection ∇KZB (1.4.2) can be extended to the moduli space of elliptic curves with n marked points. In [91], we extended this result to the
universal connection ∇τ (6.2.3). With regards to the two concrete incarnations described in § 1.4.2 and § 1.4.2, we also
show in [91] that the connection ∇RCA with values in the rational Cherednik algebra can be extended to the moduli space of elliptic curves.
Problem 1.4.7. Extend the elliptic Casimir connection ∇Ell,C to the moduli space of elliptic curves with n marked points with values in the deformed double current algebra.
Deformed double current algebras
The deformed double current algebras is yet to be fully understood. One open problem is
to show that it satisfies the PBW property.
19 The monodromy of the elliptic Casimir connection
By analogy with the Kohno-Drinfeld theorem [48, 19], and the Conjecture 1.4.1. We state
the following
Conjecture 1.4.8. The monodromy of the elliptic Casimir connection ∇Ell,C is described
tor by the quantum Weyl group operators of the quantum toroidal algebra U~(g ).
One problem in addressing Conjecture 1.4.8 is to find a correct category of representations
of D(g) on which to take monodromy. These should have in particular finite dimensional
weight spaces under h so that the monodromy is well-defined and be invariant under some
action of SL(2, Z) so that the connection has a chance to be extended in the modular direction.
In [31], Gautam and Toledano Laredo constructed an explicit equivalence of categories
between finite dimensional representations of the Yangian Y~(g) and finite dimensional rep-
resentations of the quantum loop algebra U~(Lg), when g is a finite dimensional simple Lie algebra. In the case when g is a Kac-Moody Lie algebra, the functor in [31] establishes an equivalence between subcategories of the two categories Oint(Y~(bg)) of the affine Yangian and tor Oint(U~(g )) of the quantum toroidal algebra.
Problem 1.4.9. Construct a functor from a suitable category of representations of the affine
Yangian Y~(bg) to a suitable category of the deformed double current algebra D~(g).
This should lead us to the correct category of representations of the deformed double current algebra and the quantum toroidal algebra in Problem 1.4.8.
The (glk, gln) duality
There is a (glk, gln) duality between the rational KZ-connection ∇KZ for g = glk and the rational Casimir connection ∇C for g = gln found by Toledano Laredo [87]. Such a duality is a powerful tool to study the monodromy of the Casimir connection. Using this duality,
20 Toledano Laredo reduces the proof of the monodromy theorem of Casimir connections for sln
to the case of the Kohno-Drinfeld theorem for g = slk, see [87] for details. Such a (glk, gln) duality also exists between the trigonometric KZ-connection ∇trig,KZ for g = glk and the trigonometric Casimir connection ∇trig,C for g = gln [29]. Guided by the same principle, we state the following
Problem 1.4.10. Establish a (glk, gln) duality between the universal KZB connection for g = glk and the elliptic Casimir connection for g = gln.
21 Chapter 2
Intersections of translated algebraic subtori
2.1 Finitely generated abelian groups and abelian re-
ductive groups
In this section, we describe an order-reversing isomorphism between the lattice of subgroups of a finitely generated abelian group H and the lattice of algebraic subgroups of the corre- sponding abelian, reductive, complex algebraic group T .
2.1.1 Abelian reductive groups
We start by recalling a well-known equivalence between two categories: that of finitely generated abelian groups, AbFgGp, and that of abelian, reductive, complex algebraic groups,
AbRed. For a somewhat similar approach, see also [38] and [61].
Let C∗ be the multiplicative group of units in the field C of complex numbers. Given a group G, let Gb = Hom(G, C∗) be the group of complex-valued characters of G, with pointwise multiplication inherited from C∗, and identity the character taking constant value 1 ∈ C∗
22 for all g ∈ G. If the group G is finitely generated, then Gb is an abelian, complex reductive algebraic group.
Note that Gb ∼= Hb, where H is the maximal abelian quotient of G. If H is torsion-free, say, H = Zr, then Hb can be identified with the complex algebraic torus (C∗)r. If A is a finite abelian group, then Ab is, in fact, isomorphic to A. ˆ Given a homomorphism φ: G1 → G2, let φ: Gb2 → Gb1 be the induced morphism between character groups, given by φˆ(ρ) = ρ ◦ φ. Since C∗ is a divisible abelian group, the functor
H ; Hb = Hom(H, C∗) is exact. Now let T be an abelian, complex algebraic reductive group. We can then associate to T
ˇ ∗ its weight group, T = Homalg(T, C ), where the hom set is taken in the category of algebraic groups. It turns out that Tˇ is a finitely generated abelian group, which can be described concretely, as follows.
According to the classification of abelian reductive groups over C (cf. [80]), the identity component of T is an algebraic torus, i.e., it is of the form (C∗)r for some integer r ≥ 0. Furthermore, this identity component has to be a normal subgroup. Thus, the algebraic group T is isomorphic to (C∗)r × A, for some finite abelian group A. The coordinate ring O[T ] decomposes as
∗ r ∼ ∗ r ∼ r O[(C ) × A] = O[(C ) ] ⊗ O[A] = C[Z ] ⊗ C[Ab], (2.1.1) where C[G] denotes the group ring of a group G. Let O[T ]∗ be the group of units in the coordinate ring of T . By (2.1.1), this group is isomorphic to C∗ × Zr × A, where C∗
corresponds to the non-zero constant functions. Taking the quotient by this C∗ factor, we get isomorphisms
∼ ∗ ∗ ∼ r Tˇ = O[T ] /C = Z × A. (2.1.2)
ˇ ˇ ˇ ∗ Clearly, maxSpec (C[T ]) = Homalg(C[T ], C) = Homgroup(T, C ) = T .
Now let f : T1 → T2 be a morphism in AbRed. Then the induced morphism on coordinate
∗ ∗ ∗ ∗ rings, f : O[T2] → O[T1], restricts to a group homomorphism, f : O[T2] → O[T1] , which
23 takes constants to constants. Under the identification from (2.1.2), f ∗ induces a homomor- ˇ ˇ ˇ phism f : T2 → T1 between the corresponding weight groups.
The following proposition is now easy to check.
Proposition 2.1.1. The functors H ; Hb and T ; Tˇ establish a contravariant equivalence between the category of finitely generated abelian groups and the category of abelian reductive
groups over C.
Recall now that the functor H ; Hb is exact. Hence, the functor T ; Tˇ is also exact.
Remark 2.1.2. The above functors behave well with respect to (finite) direct products. For instance, let α: A → C and β : B → C be two homomorphisms between finitely generated abelian groups, and consider the homomorphism δ : A × B → C defined by δ(a, b) = α(a) +
β(b). The morphism δˆ: Cb → A\× B = Ab × Bb is then given by δˆ(f) = (ˆα(f), βˆ(f)).
2.1.2 The lattice of subgroups of a finitely generated abelian group
Recall that a poset (L, ≤) is a lattice if every pair of elements has a least upper bound and a greatest lower bound. Define operations ∨ and ∧ on L (called join and meet, respectively) by x ∨ y = sup{x, y} and x ∧ y = inf{x, y}.
The lattice L is called modular if, whenever x < z, then x ∨ (y ∧ z) = (x ∨ y) ∧ z, for all y ∈ L. The lattice L is called distributive if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and
x∧(y ∨z) = (x∧y)∨(x∧z), for all x, y, z ∈ L. A lattice is modular if and only if it does not
contain the pentagon as a sublattice, whereas a modular lattice is distributive if and only if
it does not contain the diamond as a sublattice.
Finally, L is said to be a ranked lattice if there is a function r : L → Z such that r is constant on all minimal elements, r is monotone (if ξ1 ≤ ξ2 , then r(ξ1) ≤ r(ξ2)), and r preserves covering relations (if ξ1 ≤ ξ2, but there is no ξ such that ξ1 < ξ < ξ2, then
r(ξ2) = r(ξ1) + 1).
24 Given a group G, the set of subgroups of G forms a lattice, L(G), with order relation given by inclusion. The join of two subgroups, γ1 and γ2, is the subgroup generated by γ1 and γ2, and their meet is the intersection γ1 ∩ γ2. There is unique minimal element—the trivial subgroup, and a unique maximal element—the group G itself. The lattice L(G) is distributive if and only if G is locally cyclic (i.e., every finitely generated subgroup is cyclic).
Similarly, one may define the lattice of normal subgroups of G; this lattice is always modular.
We refer to [79] for more on all this.
Now let H be a finitely generated abelian group, and let L(H) be its lattice of subgroups.
In this case, the join of two subgroup, ξ1 and ξ2, equals the sum ξ1 + ξ2. By the above, the lattice L(H) is always modular, but it is not a distributive lattice, unless H is cyclic.
Furthermore, L(H) is a ranked lattice, with rank function ξ 7→ rank(ξ) = dimQ(ξ ⊗ Q) enjoying the following property: rank(ξ1) + rank(ξ2) = rank(ξ1 ∧ ξ2) + rank(ξ1 ∨ ξ2).
2.1.3 The lattice of algebraic subgroups of a complex algebraic
torus
Now let T be a complex abelian reductive group, and let (Lalg(T ), ≤) be the poset of algebraic subgroups of T , ordered by inclusion. It is readily seen that Lalg(T ) is a ranked modular lattice. For any algebraic subgroups P1 and P2 of T , the join P1 ∨P2 = P1 ·P2 is the algebraic subgroup generated by the two subgroups P1 and P2, while the meet P1 ∧ P2 = P1 ∩ P2 is the intersection of the two subgroups. Furthermore, the rank of a subgroup P is its dimension.
The next theorem shows that the natural correspondence from Proposition 2.1.1 is lattice- preserving. Recall that, if H is a finitely generated abelian group, the character group
∗ Hb = Homgroup(H, C ) is an abelian reductive group over C, and conversely, if T is an abelian ˇ ∗ reductive group, the weight group T = Homalg(T, C ) is a finitely generated abelian group.
Theorem 2.1.3. Suppose H ∼= Tˇ, or equivalently, T ∼= Hb. There is then an order-reversing isomorphism between the lattice of subgroups of H and the lattice of algebraic subgroups of
25 T .
Proof. For any subgroup ξ ≤ H, let
V (ξ) = maxSpec(C[H/ξ]) (2.1.3)
∼ be the set of closed points of Spec(C[H/ξ]). Clearly, the variety V (ξ) embeds into maxSpec(C[H]) = T as an algebraic subgroup. This subgroup can be naturally identified with the group of
characters of H/ξ, that is, V (ξ) ∼= H/ξd , or equivalently,
ξb∼= T/V (ξ). (2.1.4)
For any algebraic subset W ⊂ T , define
∗ ∗ (W ) = ker (Homalg(T, C ) Homalg(W, C )). (2.1.5)
∗ r ∗ Write T = (C ) × Zk1 × · · · × Zks , where Zki embeds in C as the subgroup of ki-th roots of
unity. Using the standard coordinates of (C∗)r+s, we can identify (W ) with the subgroup {λ ∈ H : tλ − 1 vanishes on W }.
The proof of the theorem is completed by the next three lemmas.
Lemma 2.1.4. If ξ is a subgroup of H, then (V (ξ)) = ξ.
Proof. The inclusion (V (ξ)) ⊇ ξ is clear. Now suppose λ ∈ H \ ξ. Then we may define a character ρ ∈ Hb such that ρ(λ) 6= 1, but ρ(ξ) = 1. Evidently, λ 6∈ (V (ξ)), and we are done.
Given two algebraic subgroups, P and Q of T , let Homalg(P,Q) be the set of morphisms from P to Q which preserve both the algebraic and multiplicative structure.
Lemma 2.1.5. Any algebraic subgroup P of T is of the form V (ξ), for some subgroup ξ ⊆ H.
∗ Proof. Let ι: P,→ T be the inclusion map. Applying the functor Homalg(−, C ), we obtain
∗ ∗ ∗ an epimorphism ι : H Homalg(P, C ). Set ξ = ker(ι ); then P = V (ξ).
26 The above two lemmas (which generalize Lemmas 3.1 and 3.2 from [38]), show that
we have a natural correspondence between algebraic subgroups of T and subgroups of H.
This correspondence preserves the lattice structure on both sides. That is, if ξ1 ≤ ξ2, then
V (ξ1) ⊇ V (ξ2), and similarly, if P1 ⊆ P2, then (P1) ≥ (P2).
Lemma 2.1.6. The natural correspondence between algebraic subgroups of T and subgroups
of H is an order-reversing lattice isomorphism. In particular,
V (ξ1 + ξ2) = V (ξ1) ∩ V (ξ2) and V (ξ1 ∩ ξ2) = V (ξ1) · V (ξ2).
Proof. The first claim follows from the two lemmas above. The two equalities are conse-
quences of this.
The above correspondence reverses ranks, i.e., rank(ξ) = codim V (ξ) and corank(ξ) =
dim V (ξ).
2.1.4 Counting algebraic subtori
As a quick application, we obtain a counting formula for the number of algebraic subgroups
of an r-dimensional complex algebraic torus, having precisely k connected components, and
a fixed, n-dimensional subtorus as the identity component. It is convenient to restate such
a problem in terms of a zeta function.
Definition 2.1.7. Let T be an abelian reductive group, and let T0 be a fixed connected algebraic subgroup. Define the zeta function of this pair as ∞ X ak(T,T0) ζ(T,T , s) = , 0 ks k=1
where ak(T,T0) is the number of algebraic subgroups W ≤ T with identity component equal
to T0 and such that |W/T0| = k.
This definition is modeled on that of the zeta function of a finitely generated group G,
P∞ −s which is given by ζ(G, s) = k=1 ak(G)k , where ak(G) is the number of index-k subgroups of G, see for instance [78].
27 ∼ ∗ r ∼ ∗ n Corollary 2.1.8. Suppose T = (C ) and T0 = (C ) , for some 0 ≤ n ≤ r. Then P∞ −s ζ(T,T0, s) = ζ(s)ζ(s − 1) ··· ζ(s − r + n + 1), where ζ(s) = k=1 k is the usual Riemann zeta function.
Proof. First assume n = 0, so that T0 = {1}. By Theorem 2.3.3, ak(T, {1}) is the number of
algebraic subgroups W ≤ T of the form W = V (ξ), where ξ ≤ Zr and [Zr : ξ] = k. Clearly,
r r this number equals ak(Z ), and so ζ(T, {1}, s) = ζ(Z , s). By a classical result of Bushnell
and Reiner (see [78]), we have that ζ(Zr, s) = ζ(s)ζ(s − 1) ··· ζ(s − r + 1). This proves the claim for n = 0.
For n > 0, we simply take the quotient of the group T by the fixed subtorus T0, to get
r−n ζ(T,T0, s) = ζ(T/T0, {1}, s) = ζ(Z , s). This ends the proof.
2.2 Primitive lattices and connected subgroups
In this section, we show that the correspondence between L(H) and Lalg(T ) restricts to a correspondence between the primitive subgroups of H and the connected algebraic subgroups of T . We also explore the relationship between the connected components of V (ξ) and the
determinant group, ξ/ξ, of a subgroup ξ ≤ H.
2.2.1 Primitive subgroups
As before, let H be a finitely generated abelian group. We say that a subgroup ξ ≤ H is primitive if there is no other subgroup ξ0 ≤ H with ξ < ξ0 and [ξ0 : ξ] < ∞. In
particular, a primitive subgroup must contain the torsion subgroup, Tors(H) = {λ ∈ H |
∃ n ∈ N such that nλ = 0}. The intersection of two primitive subgroups is again a primitive subgroup, but the sum
of two primitive subgroups need not be primitive (see, for instance, the proof of Corollary
2.5.6). Thus, the set of primitive subgroups of H is not necessarily a sublattice of L(H).
28 When H is free abelian, then all subgroups ξ ≤ H are also free abelian. In this case, ξ is
primitive if and only if it has a basis that can be extended to a basis of H, or, equivalently,
H/ξ is torsion-free. It is customary to call such a subgroup a primitive lattice.
Returning to the general situation, let H be a finitely generated abelian group. Given
an arbitrary subgroup ξ ≤ H, define its primitive closure, ξ, to be the smallest primitive
subgroup of H containing ξ. Clearly,
ξ = {λ ∈ H : ∃ n ∈ N such that nλ ∈ ξ}. (2.2.1)
Note that H/ξ is torsion-free, and thus we have a split exact sequence,
} 0 / ξ / H / H/ξ / 0 . (2.2.2)
By definition, ξ is a finite-index subgroup of ξ; in particular, rank(ξ) = rank(ξ). We call
the quotient group, ξ/ξ, the determinant group of ξ. We have an exact sequence,
0 / H/ξ / H/ξ / ξ/ξ / 0 , (2.2.3)
with ξ/ξ finite. The inclusion ξ ,→ H induces a monomorphism ξ/ξ ,→ H/ξ, which yields a splitting for the above sequence, showing that ξ/ξ ∼= Tors(H/ξ). Since the group ξ/ξ is
finite, it is isomorphic to its character group, ξ/ξc , which in turn can be viewed as a (finite) subgroup of Hb = T . Using an approach similar to the one from [38, Lemma 3.3], we sharpen and generalize that result, as follows.
Lemma 2.2.1. For every subgroup ξ ≤ H, we have an isomorphism of algebraic groups,
V (ξ) ∼= ξ/ξc · V (ξ). (2.2.4)
Moreover,
1. V (ξ) = S ρV (ξ) is the decomposition of V (ξ) into irreducible components, with ρ∈ξ/ξd V (ξ) as the component of the identity.
29 2. V (ξ)/V (ξ) ∼= ξ/ξc . In particular, if rank ξ = dim H, then V (ξ) ∼= ξ/ξc .
Proof. As noted in §2.1.1, the functor Hom(−, C∗) is exact. Applying this functor to sequence (3.4.4), we obtain an exact sequence in AbRed,
0 / ξ/ξc / H/ξd / H/dξ / 0 . (2.2.5)
V (ξ) V (ξ)
Since sequence (3.4.4) is split, sequence (2.2.5) is also split, and thus we get decomposition
(3.4.5).
Now recall that H/ξ is torsion-free; thus, V (ξ) = maxSpec(C[H/ξ]) is a connected alge- braic subgroup of T . Claims (1) and (2) readily follow.
Example 2.2.2. Let H = Z and identify Hb = C∗. If ξ = 2Z, then V (ξ) = {±1} ⊂ C∗, whereas ξ = H and V (ξ) = {1} ⊂ C∗.
Clearly, the subgroup ξ is primitive if and only if ξ = ξ. Thus, ξ is primitive if and only if the variety V (ξ) is connected. Putting things together, we obtain the following corollary to Theorem 2.1.3 and Lemma 2.2.1.
Corollary 2.2.3. Let H be a finitely generated abelian group, and let T = Hb. The natural correspondence between L(H) and Lalg(T ) restricts to a correspondence between the primitive subgroups of H and the connected algebraic subgroups of T .
2.2.2 The dual lattice
Given an abelian group A, let A∨ = Hom(A, Z) be the dual group. Clearly, if H is a finitely generated abelian group, then H∨ is torsion-free, with rank H∨ = rank H.
Now suppose ξ ≤ H is a subgroup. By passing to duals, the projection map π : H H/ξ yields a monomorphism π∨ :(H/ξ)∨ ,→ H∨. Thus, (H/ξ)∨ can be viewed in a natural way as a subgroup of H∨. In fact, more is true.
30 Lemma 2.2.4. Let H be a finitely generated abelian group, and let ξ ≤ H be a subgroup.
Then (H/ξ)∨ is a primitive lattice in H∨.
Proof. Dualizing the short exact sequence 0 → ξ → H −→π H/ξ → 0, we obtain a long exact
sequence,
π∨ 0 / (H/ξ)∨ / H∨ / ξ∨ / Ext(H/ξ, Z) / 0 . (2.2.6)
Upon identifying Ext(H/ξ, Z) = Tors(H/ξ) = ξ/ξ and setting K = coker(π∨), the above sequence splits into two short exact sequences, 0 → (H/ξ)∨ → H∨ → K → 0 and
0 / K / ξ∨ / ξ/ξ / 0 . (2.2.7)
Now, since K is a subgroup of ξ∨, is must be torsion free. Thus, (H/ξ)∨ is a primitive
lattice in H∨.
Given two primitive subgroups ξ1, ξ2 ≤ H, their sum, ξ1 + ξ2, may not be a primitive
∨ ∨ subgroup of H. Likewise, although both (H/ξ1) and (H/ξ2) are primitive subgroups of H∨, their sum may not be primitive. Nevertheless, the following lemma shows that the
respective determinant groups are the same.
Proposition 2.2.5. Let H be a finitely generated abelian group, and let ξ1 and ξ2 be primitive
∨ ∨ subgroups of H, with ξ1 ∩ ξ2 finite. Set ξ = (H/ξ1) + (H/ξ2) , and let ξ be the primitive closure of ξ in H∨. Then ∼ ξ/ξ = ξ1 + ξ2/ξ1 + ξ2.
Proof. Replacing H by H/ Tors(H) and ξi by ξi/ Tors(H) if necessary, we may assume that H
is free abelian, and ξ1 and ξ2 are primitive lattices with ξ1 ∩ξ2 = {0}. Furthermore, choosing
a splitting of H∨/ξ ,→ H∨ if necessary, we may assume that ξ ∼= H∨, or equivalently,
ξ1 + ξ2 = H. Write s = rank ξ1 and t = rank ξ2. Then s + t = n, where n = rank H.
Choose a basis {e1, . . . , es} for ξ1. Since ξ1 is a primitive lattice in H, we may extend this basis to a basis {e1, . . . , es, f1, . . . , ft} for H. Picking a suitable basis for ξ2, we may assume
31 that the inclusion ι: ξ1 + ξ2 ,→ H is given by a matrix of the form
I C s , (2.2.8) 0 D
where Is is the s × s identity matrix and D = diag(d1, . . . , dt) is a diagonal matrix with positive diagonal entries d1, . . . , dt such that 1 = d1 = ··· = dm−1 and 1 6= dt | dt−1 | · · · | dm,
for some 1 ≤ m ≤ t + 1. Then ξ1 + ξ2/ξ1 + ξ2 = Z/dmZ ⊕ · · · ⊕ Z/dtZ. C Notice that the columns of the matrix form a basis for ξ2. Since ξ2 is a primitive D lattice in H, the last t − m columns of C must have a minor of size t − m equal to ±1.
Without loss of generality, we may assume that the corresponding rows are also the last
t − m ones.
The canonical projection π : H H/ξ1 + H/ξ2 is given by a matrix of the form X Y . (2.2.9) 0 It
Using row and column operations, the matrix X can be brought to the diagonal form
diag(x1, . . . , xs), where xi are positive integers with 1 = x1 = ··· = xa−1 and 1 6= xs | xs−1 |
· · · | xa. Moreover, the submatrix of Y involving the last s−a+1 rows and columns is invert-
∨ ∨ ∨ ∨ ible. Taking the dual basis of H = Hom(H, Z), the inclusion ξ = (H/ξ1) + (H/ξ2) ,→ H
XT 0 is given by the matrix T . It follows that ξ/ξ = /xa ⊕ · · · ⊕ /xs . Y It Z Z Z Z
Evidently, the restriction of π ◦ ι to ξ2 is the zero map; hence, XC = −YD. For a fixed
integer k ≤ s, set δk := dtdt−1 ··· dk−s+t and yk := xsxs−1 ··· xk. Clearly, δk is the gcd of all minors of size s−k +1 of the submatrix in YD involving the last s−a+1 rows and columns.
Hence, δk is also the gcd of all minors of size s − k + 1 of the corresponding submatrix of
XC. Thus, δk = yk. Hence, dt = xs, . . . , dm = xs−t+m, and 1 = dm−1 = xs−t+m−1, which implies s − t + m = a. This yields the desired conclusion.
32 2.3 Categorical reformulation
In this section, we reformulate Theorem 2.1.3 using the language of categories. In order to
simultaneously consider the category of all finitely generated abelian groups, and the lattice
structure for all the subgroups of a fixed abelian group, we need the language of fibered
categories from [33, §5.1], which we briefly recall here.
2.3.1 Fibered categories
Recall that a poset (L, ≤) can be seen as a small category, with objects the same as the
elements of L, and with an arrow p → p0 in the category L if and only if p ≤ p0.
Let E be a category. We denote by Ob E the objects of E, and by Mor E its morphisms.
A category over E is a category F, together with a functor Φ: F → E. For T ∈ Ob E, we
denote by F(T ) the subcategory of F consisting of objects ξ with Φ(ξ) = T , and morphisms
f with Φ(f) = idT .
Definition 2.3.1. Let Φ: F → E be a category over E. Let f : v → u be a morphism in F,
and set (F : V → U) = (Φ(f : v → u)). Then f is said to be a Cartesian morphism if for
any h: v0 → u with Φ(h: v0 → u) = F , there exists a unique h0 : v0 → v in Mor F(F ) such
that h = f ◦ h0.
The above definition is summarized in the following diagram:
v0 (2.3.1) h h0 f v / u F V / U Definition 2.3.2. We say that Φ: F → E is a fibered category if for any morphism F : V → U in E, and any object u ∈ Ob(F(U)), there exists a Cartesian morphism f : v → u, with
Φ(f) = F . Moreover, the composition of Cartesian morphisms is required to be a Cartesian morphism.
33 We say F is a lattice over E (or less succinctly, a category fibered in lattices over E) if F
is a fibered category, and every F(T ) is a lattice.
2.3.2 A lattice over AbRed
We now construct a category fibered in lattices over the category of abelian complex algebraic
reductive groups, AbRed. Let SubAbRed be the category with objects
Ob SubAbRed = {i: W,→ P | i is a closed immersion of algebraic subgroups}, (2.3.2)
and morphisms between i: W → P and i0 : W 0 → P 0 the set of pairs
{(f, g) | f : W → W 0 and g : P → P 0 such that i0 ◦ f = g ◦ i}. (2.3.3)
Projection to the target,
i W / P 7→ P (2.3.4)
f g g i0 W 0 / P 0 P 0 defines a functor from SubAbRed to AbRed. One can easily check that this functor is a cate- gory fibered in lattices over AbRed, with Cartesian morphisms obtained by taking preimages of subtori. More precisely, suppose F : W → P is a morphism in AbRed, and θ ,→ P is an object in Ob SubAbRed; the corresponding Cartesian morphism is then
−1 F F (θ) / θ (2.3.5) _ _ i i F W / P
A similar construction works for AbFgGp, the category of finitely generated abelian group- s. That is, we can construct a category SubAbGp fibered in lattices over AbFgGp by taking injective morphisms of finitely generated abelian groups.
34 2.3.3 An equivalence of fibered categories
We now construct an explicit (contravariant) equivalence between these two fibered categories considered above. For any inclusion of subgroups η ,→ ξ, let
V (η ,→ ξ) := (maxSpec(C[ξ/η]) ,→ maxSpec(C[ξ])). (2.3.6)
This algebraic subgroup is naturally identified with the subgroup of characters of ξ/η, i.e., the closed embedding of subtori Hom(ξ/η, C∗) → Hom(ξ, C∗). Finally, for any algebraic subgroups W,→ P , let
∗ λ ∗ (W,→ P ) := ({λ ∈ Hom(P, C ): t − 1 vanishes on W } ⊆ Hom(P, C )). (2.3.7)
For a fixed algebraic group T in AbRed, with character group H = Tˇ, the subgroup
(W,→ T ) of H coincides with the subgroup (W ) defined in (2.1.5), and the algebraic subgroup V (ξ ,→ H) of T coincides with the algebraic subgroup V (ξ) defined in (2.1.3).
Tracing through the definitions, we obtain the following result, which reformulates The- orem 2.1.3 in this setting.
Theorem 2.3.3. The two fibered categories SubAbRed and SubAbGp are equivalent, with
(contravariant) equivalences given by the functors V and defined above.
2.4 Intersections of translated algebraic subgroups
In Sections 2.1 and 2.2, we only considered intersections of algebraic subgroups. In this section, we consider the more general situation where translated subgroups intersect.
2.4.1 Two morphisms
As usual, let T be a complex abelian reductive group, and let H = Tˇ be the weight group corresponding to T = Hb.
35 Let ξ1, . . . , ξk be subgroups of H. Set ξ = ξ1 + ··· + ξk, and let σ : ξ1 × · · · × ξk → ξ be
the homomorphism given by (λ1, . . . , λk) 7→ λ1 + ··· + λk. Consider the induced morphism on character groups,
σˆ : ξb / ξb1 × · · · × ξbk . (2.4.1)
Using Remark 2.1.2, the next lemma is readily verified.
∼ ∼ T Lemma 2.4.1. Under the isomorphisms ξbi = T/V (ξi) and ξb = T/V (ξ) = T/( i V (ξi)), the T morphism σˆ gets identified with the morphism δ : T/( i V (ξi)) → T/V (ξ1) × · · · × T/V (ξk) induced by the diagonal map ∆: T → T k.
Next, let γ : ξ1 ×· · ·×ξk → H ×· · ·×H be the product of the inclusion maps γi : ξi ,→ H, and consider the induced homomorphism on character groups,
γˆ : Hb × · · · × Hb / ξb1 × · · · × ξbk . (2.4.2)
The next lemma is immediate.
∼ Lemma 2.4.2. Under the isomorphisms ξbi = T/V (ξi), the morphism γˆ gets identified with
k the projection map π : T → T/V (ξ1) × · · · × T/V (ξk).
2.4.2 Translated algebraic subgroups
Given an algebraic subgroup P ⊆ T , and an element η ∈ T , denote by ηP the translate of
P by η. In particular, if C is an algebraic subtorus of T , then ηC is a translated subtorus.
If η is a torsion element of T , we denote its order by ord(η). Finally, if A is a finite group,
denote by c(A) the largest order of any element in A.
We are now in a position to state and prove the main result of this section (Theorem ?? ˇ from the Introduction). As before, let ξ1, . . . , ξk be subgroups of H = T . Let η1, . . . , ηk be
elements in T , and consider the translated subgroups Q1,...,Qk of T defined by
Qi = ηiV (ξi). (2.4.3)
36 Clearly, each Qi is a subvariety of T , but, unless ηi ∈ V (ξi), it is not an algebraic subgroup.
k Theorem 2.4.3. Set ξ = ξ1 + ··· + ξk and η = (η1, . . . , ηk) ∈ T . Then ∅ if γˆ(η) ∈/ im(ˆσ), Q1 ∩ · · · ∩ Qk = (2.4.4) ρV (ξ) otherwise, where ρ is any element in the intersection Q = Q1 ∩· · ·∩Qk. Furthermore, if the intersection is non-empty, then
1. The variety Q decomposes into irreducible components as Q = S ρτV (ξ), and τ∈ξ/ξd dim(Q) = dim(T ) − rank(ξ).
2. If η has finite order, then ρ can be chosen to have finite order, too. Moreover, ord(η) |
ord(ρ) | ord(η) · c(ξ/ξ).
Proof. We have:
Q1 ∩ · · · ∩ Qk 6= ∅ ⇐⇒ η1a1 = ··· = ηkak, for some ai ∈ V (ξi)
⇐⇒ γˆ1(η1) = ··· =γ ˆk(ηk) by Lemma 2.4.2
⇐⇒ γˆ(η) ∈ im(ˆσ) by Lemma 2.4.1.
Now suppose Q1 ∩ · · · ∩ Qk 6= ∅. For any ρ ∈ Q1 ∩ · · · ∩ Qk, and any 1 ≤ i ≤ k, there is
−1 a ρi ∈ V (ξi) such that ρ = ηiρi; thus, ρ ηi ∈ V (ξi). Therefore,
−1 −1 ρ (Q1 ∩ · · · ∩ Qk) = ρ (η1V (ξ1) ∩ · · · ∩ ηkV (ξk))
−1 −1 = ρ (η1V (ξ1)) ∩ · · · ∩ ρ (ηkV (ξk))
= V (ξ1) ∩ · · · ∩ V (ξk)
= V (ξ1 + ··· + ξk).
Hence, Q1 ∩ · · · ∩ Qk = ρV (ξ).
37 Finally, suppose η has finite order. Letρ ¯ be an element in T/V (ξ) such thatσ ˆ(¯ρ) =γ ˆ(η).
Note that ord(¯ρ) = ord(η). Using the exact sequence (2.2.5) and the third isomorphism theorem for groups, we get a short exact sequence,
q 0 / ξ/ξc / T/V (ξ) / T/V (ξ) / 0 . (2.4.5)
∗ Applying the Homgroup(−, C ) functor to the split exact (2.2.2), we get a split exact sequence,
s | 0 / V (ξ) / T / T/V (ξ) / 0 . (2.4.6)
Now pick an elementρ ˜ ∈ q−1(¯ρ). We then have q ρ˜ord(η) =ρ ¯ord(η) = 1, which implies
thatρ ˜ord(η) ∈ ξ/ξc . Hence,ρ ˜ has finite order in T/V (ξ), and, moreover, ord(η) | ord(˜ρ) |
ord(˜ρ) · c(ξ/ξ). Setting ρ = s(˜ρ) gives the desired translation factor.
When k = 2, the theorem takes a slightly simpler form.
Corollary 2.4.4. Let ξ1 and ξ2 be two subgroups of H, and let η1 and η2 be two elements in
T = Hb. Then
−1 1. The variety Q = η1V (ξ1) ∩ η2V (ξ2) is non-empty if and only if η1η2 belongs to the
subgroup V (ξ1) · V (ξ2).
2. If the above condition is satisfied, then dim Q = rank H − rank(ξ1 + ξ2).
In the special case when H = Zr and T = (C∗)r, Theorem 2.4.3 allows us to recover Proposition 3.6 from [38].
r Corollary 2.4.5 (Hironaka [38]). Let ξ1, . . . , ξk be subgroups of Z , let η = (η1, . . . , ηk) be
∗ rk an element in (C ) , and set Qi = ηiV (ξi). Then
Q1 ∩ · · · ∩ Qk 6= ∅ ⇐⇒ γˆ(η) ∈ im(ˆσ). (2.4.7)
Moreover, for any connected component Q of Q1 ∩ · · · ∩ Qk, we have:
38 1. Q = ρV (ξ), for some ρ ∈ (C∗)r.
2. dim(Q) = r − rank(ξ).
3. If η has finite order, then ord(η) | ord(ρ) | ord(η) · c(ξ/ξ).
2.4.3 Some consequences
We now derive a number of corollaries to Theorem 2.4.3. Fix a complex abelian reductive
group T . To start with, we give a general description of the intersection of two arbitrary unions of translated subgroups.
S 0 S 0 0 Corollary 2.4.6. Let W = i ηiV (ξi) and W = j ηjV (ξj) be two unions of translated subgroups of T . Then
0 [ 0 0 W ∩ W = ηiV (ξi) ∩ ηjV (ξj), (2.4.8) i,j 0 0 where ηiV (ξi) ∩ ηjV (ξj) is either empty (which this occurs precisely when γˆi,j(ηi, ηj) does not
0 belong to im(ˆσi,j)), or equals ηi,jV (ξi + ξj), for some ηi,j ∈ T .
Corollary 2.4.7. With notation as above, W ∩ W 0 is finite if and only if W ∩ W 0 = ∅ or
0 rank(ξi + ξj) = dim(T ), for all i, j.
Corollary 2.4.8. Let W and W 0 be two unions of (torsion-) translated subgroups of T . Then
W ∩ W 0 is again a union of (torsion-) translated subgroups of T .
The next two corollaries give a comparison between the intersections of various translates of two fixed algebraic subgroups of T . Both of these results will be useful in another paper
[83].
Corollary 2.4.9. Let T1 and T2 be two algebraic subgroups in T . Suppose α, β, η are elements
in T , such that αT1 ∩ ηT2 6= ∅ and βT1 ∩ ηT2 6= ∅. Then
dim (αT1 ∩ ηT2) = dim (βT1 ∩ ηT2). (2.4.9)
39 Proof. Set ξ = (T1)+(T2). From Theorem 2.4.3, we find that both αT1 ∩ηT2 and βT1 ∩ηT2 have dimension equal to the corank of ξ. This ends the proof.
Corollary 2.4.10. Let T1 and T2 be two algebraic subgroups in T . Suppose α1 and α2 are torsion elements in T , of coprime order. Then
T1 ∩ α2T2 = ∅ =⇒ α1T1 ∩ α2T2 = ∅. (2.4.10)
ˇ Proof. Set H = T , ξi = (Ti), and ξ = ξ1 + ξ2. By Theorem 2.4.3, the condition that
T1 ∩ α2T2 = ∅ impliesγ ˆ(1, α2) ∈/ im(ˆσ), where σ : ξ1 × ξ2 ξ is the sum homomorphism, and γ : ξ1 × ξ2 ,→ H × H is the inclusion map.
Suppose α1T1 ∩ α2T2 6= ∅. Then, from Theorem 2.4.3 again, we know thatγ ˆ(α1, α2) ∈
n im(ˆσ); thus, (ˆγ(α1, α2)) ∈ im(ˆσ), for any integer n. From our hypothesis, the orders of α1 and α2 are coprime; thus, there exist integers p and q such that p ord(α1) + q ord(α2) = 1. Hence,
p ord(α1) 1−q ord(α2) p ord(α1) γˆ(1, α2) =γ ˆ(α1 , α2 ) = (ˆγ(α1, α2)) ∈ im(ˆσ),
a contradiction.
2.4.4 Abelian covers
In [83], we use Corollaries 2.4.9 and 2.4.10 to study the homological finiteness properties of
abelian covers. Let us briefly mention one of the results we obtain as a consequence.
Let X be a connected CW-complex with finite 1-skeleton. Let H = H1(X, Z) the first homology group. Since the space X has only finitely many 1-cells, H is a finitely generated
abelian group. The characteristic varieties of X are certain Zariski closed subsets Vi(X)
inside the character torus Hb = Hom(H, C∗). The question we study in [83] is the following:
ν ν Given a regular, free abelian cover X → X, with dimQ Hi(X , Q) < ∞ for all i ≤ k, which regular, finite abelian covers of Xν have the same homological finiteness property?
40 Proposition 2.4.11 ([83]). Let A be a finite abelian group, of order e. Suppose the charac-
i S teristic variety V (X) decomposes as j ρjTj, with each Tj an algebraic subgroup of Hb, and each ρj an element in Hb such that ρ¯j ∈ H/Tb j satisfies gcd(ord(ρ ¯j), e) = 1. Then, given any regular, free abelian cover Xν with finite Betti numbers up to some degree k ≥ 1, the regular
A-covers of Xν have the same finiteness property.
2.5 Exponential interpretation
In this section, we explore the relationship between the correspondence H ; T = Hb from §2.1 and the exponential map Lie(T ) → T .
2.5.1 The exponential map
Let T be a complex abelian reductive group. Denote by Lie(T ) the Lie algebra of T . The exponential map exp: Lie(T ) → T is an analytic map, whose image is T0, the identity component of T .
As usual, set H = Tˇ, and consider the lattice H = H∨ ∼= H/ Tors(H). We then can
∨ ∗ ∨ identify T0 = Hom(H , C ) and Lie(T ) = Hom(H , C). Under these identifications, the corestriction to the image of the exponential map can be written as
2π i z ∨ ∨ ∗ exp = Hom(−, e ): Hom(H , C) → Hom(H , C ), (2.5.1) where C → C∗, z 7→ ez is the usual complex exponential. Finally, upon identifying
∨ Hom(H , C) with H ⊗ C, we see that T0 = exp(H ⊗ C). The correspondence T ; H = (Tˇ)∨ sends an algebraic subgroup W inside T to the sublattice χ = (Wˇ )∨ inside H. Clearly, χ = Lie(W ) ∩ H is a primitive lattice; furthermore, exp(χ ⊗ C) = W0.
Now let χ1 and χ2 be two sublattices in H. Since the exponential map is a group
41 homomorphism, we have the following equality:
exp((χ1 + χ2) ⊗ C) = exp(χ1 ⊗ C) · exp(χ2 ⊗ C). (2.5.2)
On the other hand, the intersection of the two algebraic subgroups exp(χ1 ⊗C) and exp(χ2 ⊗
C) need not be connected, so it cannot be expressed solely in terms of the exponential map. Nevertheless, we will give a precise formula for this intersection in Theorem 2.5.3 below.
2.5.2 Exponential map and Pontrjagin duality
First, we need to study the relationship between the exponential map and the correspondence from Proposition 2.1.1.
Lemma 2.5.1. Let T be a complex abelian reductive group, and let H = (Tˇ)∨. Let χ ≤ H be a sublattice. We then have an equality of connected algebraic subgroups,
∨ V ((H/χ) ) = exp(χ ⊗ C), (2.5.3) inside T0 = exp(H ⊗ C).
Proof. Let π : H → H/χ be the canonical projection, and let K = coker(π∨). As in (2.2.7), we have an exact sequence, 0 → K → χ∨ → χ/χ → 0. Applying the functor Hom(−, C∗) to this sequence, we obtain a new short exact sequence,
0 / Hom(χ/χ, C∗) / Hom(χ∨, C∗) / Hom(K, C∗) / 0 . (2.5.4)
From the way the functor V was defined in (2.1.3), we have that Hom(K, C∗) = V ((H/χ)∨).
Composing with the map exp: C → C∗, we obtain the following commutative diagram:
exp Hom(χ∨, C) / Hom(χ∨, C∗) (2.5.5) =∼ exp Hom(K, ) / Hom(K, ∗) = V ((H/χ)∨) _ C C _
exp Hom(H∨, C) / Hom(H∨, C∗)
42 Identify now exp(χ ⊗ C) with the image of Hom(χ∨, C) in Hom(H∨, C∗). Clearly, this
∨ ∨ ∗ image coincides with the image of V ((H/χ) ) in T0 = Hom(H , C ), and so we are done.
Corollary 2.5.2. Let H be a finitely generated abelian group, and let ξ ≤ H be a subgroup.
Consider the sublattice χ = (H/ξ)∨ inside H = H∨. Then
V (ξ) = exp(χ ⊗ C). (2.5.6)
Proof. Note that ξ = (H/χ)∨, as subgroups of H/ Tors(H) = H∨. The desired equality follows at once from Lemma 2.5.1.
2.5.3 Exponentials and determinant groups
We are now in a position to state and prove the main result of this section (Theorem ??
from the Introduction).
Theorem 2.5.3. Let T be a complex abelian reductive group, and let χ1 and χ2 be two sublattices of H = Tˇ∨.
∨ ∨ ∨ ∨ ∨ 1. Set ξ = (H/χ1) + (H/χ2) ≤ H and χ = (H /ξ) ≤ H. Then
exp(χ1 ⊗ C) ∩ exp(χ2 ⊗ C) = ξ/ξc · exp(χ ⊗ C),
as algebraic subgroups of T0. Moreover, the identity component of both these groups is
V (ξ) = exp(χ ⊗ C).
2. Now suppose χ1 and χ2 are primitive sublattices of H, with χ1 ∩ χ2 = 0. We then have an isomorphism of finite abelian groups,
∼ exp(χ1 ⊗ C) ∩ exp(χ2 ⊗ C) = χ1 + χ2/χ1 + χ2.
43 Proof. To prove part (1), note that
∨ ∨ exp(χ1 ⊗ C) ∩ exp(χ2 ⊗ C) = V ((H/χ1) ) ∩ V ((H/χ2) ) by Lemma 2.5.1
∨ ∨ = V ((H/χ1) + (H/χ2) ) by Lemma 2.1.6
= ξ/ξc · V (ξ) by Lemma 2.2.1.
Finally, note that (H/χ)∨ = ξ; thus, V (ξ) = exp(χ ⊗ C), again by Lemma 2.5.1.
∨ To prove part (2), note that ξ = H , since we are assuming χ1 ∩ χ2 = 0. Hence,
V (ξ) = {1}. Since we are also assuming that the lattices χ1 and χ2 are primitive, Proposition ∼ 2.2.5 applies, giving that ξ/ξ = χ1 + χ2/χ1 + χ2. Using now part (1) finishes the proof.
The next corollary follows at once from Theorem 2.5.3.
Corollary 2.5.4. Let H be a finitely generated abelian group, and let ξ1 and ξ2 be two
∨ ∨ subgroups. Let H = H be the dual lattice, and χi = (H/ξi) the corresponding sublattices.
1. Set ξ = ξ1 + ξ2. Then:
exp(χ1 ⊗ C) ∩ exp(χ2 ⊗ C) = ξ/ξc · V (ξ).
2. Now suppose rank(ξ1 + ξ2) = rank(H). Then
∼ ∼ exp(χ1 ⊗ C) ∩ exp(χ2 ⊗ C) = χ1 + χ2/χ1 + χ2 = ξ/ξ.
2.5.4 Some applications
Let us now consider the case when T = (C∗)r. In this case, H = Zr, and the exponential map
r ∗ r 2π i z1 2π i zr (3.5.3) can be written in coordinates as exp: C → (C ) ,(z1, . . . , zr) 7→ (e , . . . , e ). Applying Theorem 2.5.3 to this situation, we recover Theorem 1.1 from [67].
r Corollary 2.5.5 (Nazir [67]). Suppose χ1 and χ2 are primitive lattices in Z , and χ1∩χ2 = 0.
Then exp(χ1 ⊗ C) ∩ exp(χ2 ⊗ C) is isomorphic to χ1 + χ2/χ1 + χ2.
44 As another application, let us show that intersections of subtori can be at least as com-
plicated as arbitrary finite abelian groups.
Corollary 2.5.6. For any integer n ≥ 0, and any finite abelian group A, there exist subtori
∗ r ∼ ∗ n T1 and T2 in some complex algebraic torus (C ) such that T1 ∩ T2 = (C ) × A.
2k Proof. Write A = Zd1 ⊕ · · · ⊕ Zdk . Let χ1 and χ2 be the lattices in Z spanned by the
Ik Ik columns of the matrices 0 and D , where D = diag(d1, . . . , dk). Clearly, both χ1 and
χ2 are primitive, and χ1 ∩ χ2 = 0, yet the lattice χ = χ1 + χ2 is not primitive if the group A = χ/χ is non-trivial.
∗ 2k Now consider the subtori Pi = exp(χi ⊗ C) in (C ) , for i = 1, 2. By Corollary 2.5.5, we ∼ ∗ n ∗ n+2k have that P1 ∩ P2 = A. Finally, consider the subtori Ti = (C ) × Pi in (C ) . Clearly, ∼ ∗ n T1 ∩ T2 = (C ) × A, and we are done.
In particular, if A is a finite cyclic group, there exist 1-dimensional subtori, T1 and T2,
∗ 2 ∼ in (C ) such that T1 ∩ T2 = A.
2.6 Intersections of torsion-translated subtori
In this section, we revisit Theorem 2.4.3 from the exponential point of view. In the case when the translation factors have finite order, the criterion from that theorem can be refined, to take into account certain arithmetic information about the translated tori in question.
2.6.1 Virtual belonging
We start with a definition.
Definition 2.6.1. Let χ be a primitive lattice in Zr. Given a vector λ ∈ Qr, we say λ
r r virtually belongs to χ if d · λ ∈ Z , where d = |det [χ | χ0]| and χ0 = Qλ ∩ Z .
45 Here, [χ | χ0] is the matrix obtained by concatenating basis vectors for the sublattices
r χ and χ0 of Z . Note that χ0 is a (cyclic) primitive lattice, generated by an element of the
r form λ0 = mλ ∈ Z , for some m ∈ N. In the next lemma, we record several properties of the notion introduced above.
Lemma 2.6.2. With notation as above,
1. λ virtually belongs to χ if and only if dλ ∈ χ0.
2. d = 0 if and only if λ ∈ χ ⊗ Q, which in turn implies that λ virtually belongs to χ.
3. If d > 0, then d equals the order of the determinant group χ + χ0/χ + χ0.
4. If d = 1, then λ virtually belongs to χ if and only if λ ∈ Zr, which happens if and only
if λ ∈ χ0.
Next, we give a procedure for deciding when a torsion element in a complex algebraic
torus belongs to a subtorus.
Lemma 2.6.3. Let P ⊂ (C∗)r be a subtorus, and let η ∈ (C∗)r be an element of finite order.
Write P = exp(χ ⊗ C), where χ is a primitive lattice in Zr. Then η ∈ P if and only if
η = exp(2π i λ), for some λ ∈ Qr which virtually belongs to χ.
r Proof. Set n = rank χ, and write η = exp(2π i λ), for some λ = (λ1, . . . , λr) ∈ Q . Let λ0 be
r a generator of χ0 = Qλ ∩ Z , and write λ = qλ0. Finally, set d = |det [χ | χ0]|.
First suppose d = 0. Then, by Lemma 2.6.2(2), λ virtually belongs to χ. Since λ ∈ χ⊗Q, we also have η ∈ P , and the claim is established in this case.
Now suppose d 6= 0. As in the proof of Proposition 2.2.5, we can choose a basis for Zr so
that the inclusion of χ + χ0 into χ + χ0 has matrix of the form (2.2.8), with D = d. In this
basis, the lattice χ is the span of the first n coordinates, and χ0 lies in the span of the first
2π i q n + 1 coordinates. Also in these coordinates, λn+1 = q, and so ηn+1 = e ; furthermore,
∗ r ηn+2 = ··· = ηr = 1. On the other hand, P = {z ∈ (C ) | zn+1 = ··· = zr = 1}. Therefore,
η ∈ P if and only if dq is an integer, which is equivalent to dλ ∈ χ0.
46 2.6.2 Torsion-translated tori
We are now ready to state and prove the main results of this section (Theorem ?? from the
Introduction).
r Theorem 2.6.4. Let ξ1 and ξ2 be two sublattices in Z . Set ε = ξ1 ∩ ξ2, and write ε/εc =
s ∗ r {exp(2π i µk)}k=1. Also let η1 and η2 be two torsion elements in (C ) , and write ηj =
exp(2π i λj). The following are equivalent:
1. The variety Q = η1V (ξ1) ∩ η2V (ξ2) is non-empty.
r ∨ 2. One of the vectors λ1 − λ2 − µk virtually belongs to the lattice (Z /ε) .
If either condition is satisfied, then Q = ρV (ξ1 + ξ2), for some ρ ∈ Q.
Proof. Follows from Corollary 2.4.4 and Lemma 2.6.3.
This theorem provides an efficient algorithm for checking whether two torsion-translated
tori intersect. We conclude with an example illustrating this algorithm.
3 Example 2.6.5. Fix the standard basis e1, e2, e3 for Z , and consider the primitive sub-
lattices ξ1 = span(e1, e2) and ξ2 = span(e1, e3). Then ε = span(e1) is also primitive, and
∗ 3 so s = 1 and µ1 = 0. For selected values of η1, η2 ∈ (C ) , let us decide whether the set
Q = η1V (ξ1) ∩ η2V (ξ2) is empty or not, using the above theorem.
2π i /3 1 First take η1 = (1, 1, 1) and η2 = (1, e , 1), and pick λ1 = 0 and λ2 = (1, 3 , 0). One
0 0 3 3 easily sees that d = det 0 1 1 = −3, and d(λ1 − λ2 − µ1) = (3, 1, 0) ∈ . Thus, Q 6= ∅. 1 0 0 Z 3π i /2 3 Next, take the same η1 and λ1, but take η2 = (e , 1, 1) and λ2 = ( 4 , 0, 1). In this case,
0 0 3 9 3 d = det 0 1 0 = −3 and d(λ1 − λ2 − µ1) = ( , 0, 3) ∈/ . Thus, Q = ∅. 1 0 4 4 Z
47 Chapter 3
Homological finiteness of abelian
covers
3.1 A parameter set for regular abelian covers
We start by setting up a parameter set for regular covers of a CW-complex, with special
emphasis on the case when the deck-transformation group is abelian.
3.1.1 Regular covers
Let X be a connected CW-complex with finite 1-skeleton. Without loss of generality, we
may assume X has a single 0-cell, which we will take as our basepoint, call it x0. Let
G = π1(X, x0) be the fundamental group. Since the space X has only finitely many 1-cells, the group G is finitely generated.
Consider an epimorphism ν : G A from G to a (necessarily finitely generated) group A. Such an epimorphism gives rise to a regular cover of X, which we denote by Xν. Note that Xν is also a connected CW-complex, the projection map p: Xν → X is cellular, and the group A is the group of deck transformations of Xν.
Conversely, every (connected) regular cover p:(Y, y0) → (X, x0) with group of deck
48 transformations A defines a normal subgroup p](π1(Y, y0)) / π1(X, x0), with quotient group
A. Moreover, if ν : G A is the projection map onto the quotient, then the cover Y is equivalent to Xν, that is, there is an A-equivariant homeomorphism Y ∼= Xν.
Let Epi(G, A) be the set of all epimorphisms from G to A, and let Aut(A) be the auto- morphism group of A. The following lemma is standard.
Lemma 3.1.1. Let G = π1(X, x0), and let A be a group. The set of equivalence classes of connected, regular A-covers of X is in one-to-one correspondence with
Γ(G, A) := Epi(G, A)/ Aut(A), the set of equivalence classes [ν] of epimorphisms ν : G A, modulo the right-action of Aut(A).
When the group A is finite, the parameter set Γ(G, A) is of course also finite. Efficient counting methods for determining the size of this set were pioneered by P. Hall in the 1930s.
New techniques (involving, among other things, characteristic varieties over finite fields) were introduced in [59]. In the particular case when A is a finite abelian group, a closed formula for the cardinality of Γ(G, A) was given in [59], see Theorem 3.1.3 below.
3.1.2 Functoriality properties
The above construction enjoys some (partial) functoriality properties in both arguments.
First, suppose that ϕ: G1 G2 is an epimorphism between two groups. Composition with
ϕ gives a map Epi(G2,A) → Epi(G1,A), which in turn induces a well-defined map,
ϕ∗ Γ(G2,A) / Γ(G1,A) , [ν] 7→ [ν ◦ ϕ]. (3.1.1)
Under the correspondence from Lemma 3.1.1, this map can be interpreted as follows. Let f :(X1, x1) → (X2, x2) be a basepoint-preserving map between connected CW-complexes, and suppose f induces an epimorphism ϕ = f] : G1 G2 on fundamental groups. Then the
49 ∗ ν map ϕ sends the equivalence class of the cover pν : X2 → X2 to that of the pull-back cover,
∗ ν◦ϕ pν◦ϕ = f (pν): X1 → X1. Next, recall that a subgroup K < A is characteristic if α(K) = K, for all α ∈ Aut(A).
Lemma 3.1.2. Suppose we have an exact sequence of groups, 1 → K → A −→π B → 1, with
K a characteristic subgroup of A. There is then a well-defined map between the parameter sets for regular A-covers and B-covers of X,
π˜ Γ(G, A) / Γ(G, B) , (3.1.2) which sends [ν] to [π ◦ ν].
Proof. Suppose ν1, ν2 : G A are two epimorphisms so that α ◦ ν1 = ν2, for some α ∈ Aut(A). Since the subgroup K = ker(π) is characteristic, the automorphism α induces and
automorphismα ¯ ∈ Aut(B) such thatα ¯ ◦ π = π ◦ α. Hence,α ¯ ◦ (π ◦ ν1) = π ◦ ν2, showing that q is well-defined.
The correspondence of Lemmas 3.1.1 and 3.1.2 is summarized in the following diagram:
ν pπ G / A ←→ Xν / Xπ◦ν (3.1.3)
π pπ◦ν π◦ν pν B # X
Under this correspondence, the mapπ ˜ sends the equivalence class of the cover pν to that of the cover pπ◦ν.
3.1.3 Regular abelian covers
Let H = Gab be the abelianization of our group G. Recall we are assuming G is finitely generated; thus, H is a finitely generated abelian group.
Now suppose A is any other (finitely generated) abelian group. In this case, every homo- morphism G → A factors through the abelianization map, ab: G H. Composition with
50 this map gives a bijection between Epi(H,A) and Epi(G, A), which induces a bijection
=∼ Γ(H,A) / Γ(G, A) , [ν] 7→ [ν ◦ ab]. (3.1.4)
In view of Lemma 3.1.1, we obtain a bijection between the set of equivalence classes of
connected, regular A-covers of a CW-complex X and the set Γ(H,A), where H = H1(X, Z)
is the abelianization of G = π1(X, x0). Note that this parameter set is empty, unless A is a quotient of H.
Let Tors(A) be the torsion subgroup, consisting of finite-order elements in A; clearly,
this is a characteristic subgroup of A. Let A = A/ Tors(A) be the quotient group, and let
π : A → A be the canonical projection.
Under the correspondence from (3.1.4), an epimorphism ν : H A determines a regular
ν◦ab ν cover, pν◦ab : X → X, which, for economy of notation, we will write as pν : X → X.
ν¯ There is also a free abelian cover, pν¯ : X → X, corresponding to the epimorphismν ¯ = π ◦ ν : H → A.
The projection π : A A defines a map Epi(H,A) → Epi(H, A), ν 7→ ν¯, which in turn induces maps between the parameter spaces for A and A covers,
qH Γ(H,A) / Γ(H, A) , (3.1.5)
sending the cover pν to the cover pν¯. Notice that this map is compatible with the morphism
π˜ from (3.1.2), induced by an epimorphism π : A B with characteristic kernel.
3.1.4 Splitting the torsion-free part
For a topological group G, let G → EG → BG be the universal principal G-bundle; the total
space EG a contractible CW-complex endowed with a free G-action, while the base space
BG the quotient space under this action. We will only consider here the situation when G is discrete, in which case BG = K(G, 1) and EG = K^(G, 1).
51 As before, let H be a finitely generated abelian group. Let Tors(H) be its torsion sub-
group, and let H = H/ Tors(H). Fix a splitting H → H; then H ∼= H ⊕ Tors(H).
Now consider an epimorphism ν : H A. After fixing a splitting A,→ A, we obtain a decomposition A ∼= A⊕Tors(A). We may view A as a subgroup of H by choosing a splitting
A,→ H of the projection H A. With these identifications, ν induces an epimorphism
ν˜: H/A A/A. This observation leads us to consider the set
Γ = Γ(H/A, A/A). (3.1.6)
Clearly, the set Γ is finite. Theorem 3.1 from [59] yields an explicit formula for the size
of this set. Given a finite abelian group K, and a prime p, write the p-torsion part of K as
Kp = Zpλ1 ⊕· · ·⊕Zpλs , for some positive integers λ1 ≥ · · · ≥ λs, where s = 0 if p - |K|. Thus,
Kp determines a partition π(Kp) = (λ1, . . . , λs). For each such partition λ, write l(λ) = s, Ps Ps |λ| = i=1 λi, and hλi = i=1(i − 1)λi.
Theorem 3.1.3 ([59]). Set n = rank H and r = rank A. For each prime p dividing the
order of A/A, let λ = π((A/A)p) and τ = π((Tors(H/A))p) be the corresponding partitions. Then, l(λ) − Y − p(|λ|−l(λ))(n−r)+θ(λ ,τ) pn−r+θi(λ,τ)−θi(λ ,τ) − pi−1 Y i=1 Γ(H/A, A/A) = Y , p|λ|+2hλi ϕ (p−1) p | |A/A| mk(λ) k≥1 Qm i − − where mk(λ) = #{i | λi = k}, ϕm(t) = i=1(1 − t ), λ is the partition with λi = λi − 1, Pl(τ) Pl(λ) θi(λ, τ) = j=1 min(λi, τj), and θ(λ, τ) = i=1 θi(λ, τ).
n psn−p(s−1)n n s s−1 pn−pi s Q For instance, |Γ(Z , Zp )| = ps−ps−1 , whereas Γ(Z , Zp) = i=0 ps−pi . Each homomorphism H/A → A/A defines an action of H/A on A/A. This action yields
a fiber bundle,
A/A → E(H/A) ×H/A A/A → B(H/A) (3.1.7) associated to the principal bundle H/A → E(H/A) → B(H/A). The set Γ = Γ(H/A, A/A)
parameterizes all such associated bundles.
52 3.1.5 A pullback diagram
We return now to diagram (3.1.5), which relates the parameter sets for regular A and A
covers of a connected CW-complex X. This diagram can be further analyzed by using pullbacks from the universal principal bundles over the classifying spaces for the discrete groups A and A.
Let A → EA → BA and A → EA → BA be the respective classifying bundles, and let X → BA and X → BA be classifying maps for the covers Xν → X and Xν¯ → X,
respectively. Upon identifying Tors(A) with A/A, we obtain the following diagram:
Tors(A) (3.1.8)
A A =∼ Tors(A) / A/A A A ν EA o X / E(H/A) ×H/A A/A pπ β
α EA o Xν¯ / B(H/A) pν O BA o X pν¯
BAXo / BH
Here, the map X → BH realizes the abelianization morphism, ab: π1(X, x0) → H, while α denotes the composite Xν¯ → X → BH → B(H/A).
Proposition 3.1.4. The marked square in diagram (3.1.8) is a pullback square. That is,
ν ν¯ ν¯ the cover pπ : X → X is the pullback along the map α: X → B(H/A) of the cover
β : E(H/A) ×H/A A/A → B(H/A) corresponding to the epimorphism ν˜: H/A A/A.
ν Proof. Clearly, (α ◦ pπ)](π1(X )) = im(ker ν ,→ kerν ¯), while β](π1(E(H/A) ×H/A A/A)) = ker(˜ν : H/A A/A). After picking a splitting A,→ H, and identifying the group H with kerν ¯ ⊕ A, we see that
ν (α ◦ pπ)](π1(X )) ⊆ β](π1(E(H/A) ×H/A A/A)).
53 The existence of the dashed arrow in the diagram follows then from the lifting criterion for covers. It is readily seen that this arrow is equivariant with respect to the actions on source and target by Tors(A) and A/A; thus, a morphism of covers. This completes the proof.
Using this proposition, and the discussion from §3.1.4, we obtain the following corollary.
Corollary 3.1.5. With notation as above,
qH Γ(H/A, A/A) / Γ(H,A) / Γ(H, A)
is a set fibration; that is, all fibers of qH are in bijection with the set Γ(H/A, A/A).
Let us identify topologically the fiber of qH . A regular, A-cover of our space X corre-
ν¯ sponds to an epimorphismν ¯: H A. A regular, Tors(A)-cover of X corresponds to an epimorphism kerν ¯ Tors(A). Given these data, and the chosen splitting A,→ A, we can
find an epimorphism ν : H A, such that the following diagram commutes:
ν¯ kerν ¯ / H / / A (3.1.9)
ν π Tors(A) / A / / A
Thus, any regular, Tors(A)-cover Xν → Xν¯ defines a regular A-cover Xν → X, whose corresponding free abelian cover is Xν¯. Consequently, the fiber of [¯ν] under the map qH : Γ(H,A) → Γ(H, A) coincides with the set
[ν] ∈ Γ(H,A) | Xν is a regular Tors(A)-cover of Xν¯ . (3.1.10)
3.2 Reinterpreting the parameter set for A-covers
In this section, we give a geometric description of the parameter set for regular abelian covers of a space.
54 3.2.1 Splittings
As before, let H and A be finitely generated abelian groups, and assume there is an epimor-
phism H A.
Lemma 3.2.1. The action Aut(A) on the set of all splittings A/ Tors(A) ,→ A induced by the natural action of Aut(A) on A is transitive.
Proof. Set A = A/ Tors(A), and fix a splitting s: A,→ A. Using this splitting, we may
decompose the group A as A⊕Tors(A), and view s: A,→ A⊕Tors(A) as the map a 7→ (a, 0).
An arbitrary splitting σ : A,→ A ⊕ Tors(A) is given by a 7→ (a, σ2(a)), for some homo-
morphism σ2 : A → Tors(A). Consider the automorphism of α ∈ Aut(A ⊕ Tors(A)) given by
id 0 the matrix σ2 id . Clearly, α ◦ s = σ, and we are done.
Denote by n the rank of H, and by r the rank of A. Fixing splittings H,→ H and
A,→ A, we have H = H ⊕ Tors(H), with H = Zn, and A = A ⊕ Tors(A), with A = Zr.
Now identify the automorphism group Aut(H) with the general linear group GLn(Z).
∗1 ∗2 Let P be the parabolic subgroup of GLn(Z), consisting all matrices of the form ( 0 ∗3 ),
where ∗1 is of size (n − r) × (n − r). Then GLn(Z)/ P is isomorphic to the Grassmannian
n−r Grn−r(Z). It is readily checked that the left action of P on Z , given by multiplication of ∼ n−r {∗1} = GLn−r(Z) on Z , induces an action of P on the set Γ = Γ(H/A, A/A). Also note that, if A is torsion-free, then the set Γ is a singleton.
3.2.2 A fibered product
We are now ready to state and prove the main result of this section.
Theorem 3.2.2. There is a bijection
Γ(H,A) ←→ GLn(Z) ×P Γ
55 between the parameter set Γ(H,A) = Epi(H,A)/ Aut(A) and the twisted product of GLn(Z) with the set Γ = Γ(H/A, A/A) over the parabolic subgroup P. Under this bijection, the map
q : Γ(H,A) → Γ(H, A) induced by the projection π : A → A corresponds to the canonical
projection
n GLn(Z) ×P Γ → GLn(Z)/ P = Grn−r(Z ).
Proof. Define a map θ : GLn(Z) × Γ → Γ(H,A) as follows. Given an element (M, [γ]) of
n−r GLn(Z)×Γ, let (γ1, γ2) be a representative of [γ], with γ1 : Z Tors(A) and γ2 : Tors(H)
Tors(A). Let α1, . . . , αn be the column vectors of the matrix M, which forms a basis of
∼ n n−r r n−r H = Z , we can write H = Z ⊕ Z ⊕ Tors(H), where Z is the subspace of H generated by the first n − r column vectors of M. Now define θ(M, [γ]) = [ν], where
ν : Zn−r ⊕ Zr ⊕ Tors(H) → Zr ⊕ Tors(A) is the homomorphism given by the matrix N =
0 id 0 γ1 0 γ2 . It is straightforward to check that the map θ is well-defined, i.e., θ is independent of the splitting and representative we chose.
Now let’s check that the map θ factors through GLn(Z) ×P Γ. Suppose we have two
0 0 0 elements (M, [γ1, γ2]) and (M , [γ1, γ2]) of GLn(Z) × Γ which are equivalent, that is, there is a matrix Q = Q1 Q2 ∈ P such that M = M 0Q and (γ Q−1, γ ) = (γ0 , γ0 ). By definition, 0 Q3 1 1 2 1 2
the map θ takes the pair (M, [γ1, γ2]) to the homomorphism ν given by the matrix N above. Changing the basis of H to the basis given by the column vectors of M 0, the map ν is given by the matrix
−1 −1 0 Q3 0 Q3 0 0 id 0 NQ−1 = = . −1 0 0 γ1Q1 γ1Q4 γ2 γ1Q4 id γ1 0 γ2
Q−1 0 Clearly, the matrix 3 defines an automorphism of A. Thus, the map θ factors through γ1Q4 id a well-defined map, θ : GLn(Z) ×P Γ → Γ(H,A), which is readily seen to be a bijection. It is now a straightforward matter to verify the last assertion.
56 3.2.3 Further discussion
A particular case of the above theorem is worth singling out. Recall H = Zn and A = Zr.
s t Corollary 3.2.3. Suppose Tors(H) = Zp and Tors(A) = Zp, for some prime p. Then, the
n−r+s parameter set Γ(H,A) is in bijective correspondence with GLn(Z) ×P Grt(Zp ).
n−r s t t Proof. In this case, the set Γ = Γ(H/A, A/A) is in bijection with Epi(Zp ⊕Zp, Zp)/ Aut(Zp). This bijection is established using the diagram
n−r s t Z ⊕ Zp / / Zp (3.2.1) : :
n−r s Zp ⊕ Zp
n−r+s Therefore, Γ = Grt(Zp ), and we are done.
Remark 3.2.4. Consider the projection map q : GLn(Z) ×P Γ → GLn(Z)/ P = Grn−r(Z)
from Theorem 3.2.2. It is readily seen that, for each subspace Q ∈ Grn−r(n, Z), the cardinal- ity of the fiber q−1(Q) is the same as the the cardinality of the set Γ. In particular, q−1(Q)
is finite, for all Q ∈ Grn−r(Z).
3.3 Dwyer–Fried sets and their generalizations
i In this section, we define a sequence of subsets ΩA(X) of the parameter set for regular A-
i covers of X. These sets, which generalize the Dwyer–Fried sets Ωr(X), keep track of the homological finiteness properties of those covers.
3.3.1 Generalized Dwyer–Fried sets
Throughout this section, X will be a connected CW-complex with finite 1-skeleton, and
G = π1(X, x0) will denote its fundamental group.
57 Definition 3.3.1. For each group A and integer i ≥ 0, the corresponding Dwyer–Fried set
of X is defined as
i ν ΩA(X) = [ν] ∈ Γ(G, A) | bj(X ) < ∞, for all 0 ≤ j ≤ i .
i In other words, the sets ΩA(X) parameterize those regular A-covers of X having finite Betti numbers up to degree i. In the particular case when A is a free abelian group of rank
i i r, we recover the standard Dwyer–Fried sets, Ω (X) = Ω r (X), viewed as subsets of the r Z n Grassmannian Grr(Q ), where n = b1(X).
By our assumption on the 1-skeleton on X, the group G = π1(X, x0) is finitely generated. Thus, we may assume A is also finitely generated, for otherwise Epi(G, A) = ∅, and so
i ΩA(X) = ∅, too. The Ω-sets are invariant under homotopy equivalence. More precisely, we have the fol-
lowing lemma, which generalizes the analogous lemma for free abelian covers, proved in
[81].
Lemma 3.3.2. Let f : X → Y be a (cellular) homotopy equivalence. For any group A,
∗ the homomorphism f] : π1(X, x0) → π1(Y, y0) induces a bijection f] : Γ(π1(Y, y0),A) →
i i Γ(π1(X, x0),A), sending each subset ΩA(Y ) bijectively onto ΩA(X).
Proof. Since f is a homotopy equivalence, the induced homomorphism on fundamental
∗ groups, f], is a bijection. Thus, the corresponding map between parameter sets, f] , is a
∗ i i bijection. To finish the proof, it remains to verify that f] (ΩA(Y )) = ΩA(X).
Let ν : π1(Y, y0) A be an epimorphism. Composing with f], we get an epimorphism ¯ ν ◦ f] : π1(X, x0) A. By the lifting criterion, f lifts to a map f between the respective A-covers. This map fits into the following pullback diagram:
f¯ Xν◦f] / Y ν (3.3.1)
f X / Y.
58 ¯ ν◦f] ν ν Clearly, f : X → Y is also a homotopy equivalence. Thus, bj(Y ) < ∞ if and only if
ν◦f] ∗ i i bj(X ) < ∞, which means that f] (ΩA(Y )) = ΩA(X).
Based on this lemma, we may define the Ω-sets of a (discrete, finitely generated) group
i i G as ΩA(G) := ΩA(BG), where BG = K(G, 1) is a classifying space for G.
3.3.2 Naturality properties
The Dwyer–Fried sets (or their complements) enjoy certain naturality properties in both variables, which we now describe.
Proposition 3.3.3. Let ϕ: G Q be an epimorphism of groups. Then, for each group A,
1 c 1 c there is an inclusion ΩA(Q) ,→ ΩA(G) .
Proof. Let ν : Q A be an epimorphism. Composing with ϕ, we get an epimorphism
ν ◦ ϕ: G A. So there is an epimorphism ϕ: ker(ν ◦ ϕ) ker(ν). Taking abelianizations, we get an epimorphism ϕab : ker(ν ◦ ϕ)ab ker(ν)ab. Thus, if ker(ν)ab has infinite rank,
then ker(ν ◦ ϕ)ab also has infinite rank. The desired conclusion follows.
Before proceeding, let us recall a well-known result regarding the homology of finite
covers, which can be proved via a standard transfer argument (see, for instance, [37]).
Lemma 3.3.4. Let p: Y → X be a regular cover defined by a properly discontinuous action
of a finite group A on Y , and let k be a coefficient field of characteristic 0, or a prime not
dividing the order of A. Then, the induced homomorphism in cohomology, p∗ : H∗(X; k) →
H∗(Y ; k), is injective, with image the subgroup H∗(Y ; k)A consisting of those classes α for which γ∗(α) = α, for all γ ∈ A.
Corollary 3.3.5. Let p: Y → X be a finite, regular cover. Then bi(X) ≤ bi(Y ), for all i ≥ 0.
59 Now fix a CW-complex X as above, with fundamental group G = π1(X, x0). Suppose 1 → K → A −→π B → 1 is a short exact sequence of groups, with K a characteristic subgroup
of A. As noted in Lemma 3.1.2, the homomorphism π induces a mapπ ˜ : Γ(G, A) → Γ(G, B),
[ν] 7→ [π ◦ ν], between the parameter sets for regular A-covers and B-covers of X.
Proposition 3.3.6. Suppose K = ker(π : A B) is a finite, characteristic subgroup of A.
i i Then the map π˜ : Γ(G, A) → Γ(G, B) restricts to a map π˜ :ΩA(X) → ΩB(X) between the respective Dwyer–Fried sets.
i ν Proof. Let ν : G A be an epimorphism, and suppose [ν] ∈ ΩA(X), that is, bj(X ) < ∞, for all j ≤ i. Then Xν → Xπ◦ν is a regular K-cover. By Corollary 3.3.5, we have that
π◦ν i bj(X ) < ∞, for all j ≤ i; in other words, [π ◦ ν] ∈ ΩB(X).
Proposition 3.3.6 may be summarized in the following commuting diagram:
i ΩA(X) / Γ(G, A) (3.3.2)
π˜|Ωi (X) π˜ A i ΩB(X) / Γ(G, B)
This diagram is a pullback diagram precisely when
−1 i i π˜ (ΩB(X)) = ΩA(X). (3.3.3)
As we shall see later on, this condition is not always satisfied. For now, let us just single
out a simple situation when (3.3.2) is tautologically a pullback diagram.
i i Corollary 3.3.7. With notation as above, if ΩB(X) = ∅, then ΩA(X) = ∅.
3.3.3 Abelian versus free abelian covers
Let us now consider in more detail the case when A is an abelian group. As usual, we are
only interested in the case when A is a quotient of the (finitely generated) group H = Gab, and thus we may assume A is also finitely generated.
60 Consider the exact sequence 0 → Tors(A) → A → A → 0. Clearly, Tors(A) is a finite,
characteristic subgroup of A. Thus, Proposition 3.3.6 applies, giving a map
i i q :ΩA(X) → ΩA(X). (3.3.4)
In particular, if Ωi (X) = ∅, then Ωi (X) = ∅. A A
Example 3.3.8. Let Σg be a Riemann surface of genus g ≥ 2. It is readily seen that
i Ωr(Σg) = ∅, for all r ≥ 1 and i ≥ 1, cf. [81]. Thus, if A is any finitely generated abelian
i group with rank A ≥ 1, then ΩA(Σg) = ∅, for all i ≥ 1.
Suppose now we have a short exact sequence 1 → K → A −→π B → 1, with K character-
istic. Letπ ¯ : A B be the induced epimorphism between maximal torsion-free quotients. Since K = ker(π) is finite,π ¯ is an isomorphism. Using Proposition 3.3.6 again, and the
identification from (3.1.4), we obtain the following commutative diagram:
i ΩB(X) / Γ(H,B) (3.3.5) 7 π˜ 8 q i B ΩA(X) / Γ(H,A)
qA Ωi (X) Γ(H, B) B / π¯˜ Ωi (X) Γ(H, A) A / Proposition 3.3.9. Assume the function π˜ : Γ(H,A) → Γ(H,B) is surjective. Then, if the
front square in diagram (3.3.5) is a pullback square, so is the back square; that is,
−1 i i −1 i i qA ΩA = ΩA =⇒ qB ΩB = ΩB.
Proof. Suppose the back square is not pullback square. Then there exist elements [¯ν] ∈ Ωi B i and [ν] ∈ Γ(H,B) \ ΩB such that qB([ν]) =ν ¯. By assumption, the mapπ ˜ is surjective; thus,
−1 −1 i π˜ ([ν]) is nonempty. Pick an element [σ] ∈ π˜ ([ν]). Then [σ] ∈ Γ(H,A)\ΩA, for otherwise
i [ν] =π ˜([σ]) ∈ ΩB. On the other hand,
i i qA([σ]) = qB(˜π([σ])) = qB([ν]) = [¯ν] ∈ ΩB = ΩA .
Thus, the front square is not a pullback diagram, either.
61 3.3.4 The comparison diagram
Now fix a splitting A,→ A, which gives rise to an isomorphism A ∼= A ⊕ Tors(A). Similarly, ∼ after fixing a splitting H,→ H, the abelianization H = Gab also decomposes as H = H ⊕ Tors(H). Theorem 3.2.2 yields an identification
∼ Γ(H,A) = GLn(Z) ×P Γ, (3.3.6)
where n = rank H, the group P is a parabolic subgroup of GLn(Z) so that GLn(Z)/ P =
n Grn−r(Z ), and Γ = Γ(H/A, A/A). Putting things together, we obtain a commutative diagram, which we shall refer to as
the comparison diagram,
i ∼ ΩA(X) / Γ(H,A) = GLn(Z) ×P Γ (3.3.7)
q|Ωi (X) q A Ωi (X) Γ(H, A) ∼ Gr ( n) A / = n−r Z
The next result reinterprets the condition that this diagram is a pull-back in terms of
Betti numbers of abelian covers.
Proposition 3.3.10. The following conditions are equivalent:
(i) Diagram (3.3.7) is a pull-back diagram.
(ii) q−1 Ωi (X) = Ωi (X). A A
(iii) If Xν¯ is a regular A-cover with finite Betti numbers up to degree i, then any regular
Tors(A)-cover of Xν¯ has the same finiteness property.
Proof. The equivalence (i) ⇔ (ii) is immediate. To prove (ii) ⇔ (iii), consider an epimorphis-
−1 m ν : G A, and letν ¯ = π ◦ ν : G A. We know from (3.1.10) that q ([¯ν]) coincides with the set of equivalence classes of regular Tors(A)-covers Xν → Xν¯. The desired conclusion
follows.
62 In other words, (3.3.7) is a pull-back diagram if and only if the homological finiteness of
an arbitrary abelian cover of X can be tested through the corresponding free abelian cover.
3.4 Pontryagin duality
Following the approach from [38, 82], we now discuss a functorial correspondence between
finitely generated abelian groups and abelian, complex algebraic reductive groups.
3.4.1 A functorial correspondence
Let C∗ be the multiplicative group of units in the field of complex numbers. Given a group
G, let Gb = Hom(G, C∗) be the group of complex-valued characters of G, with pointwise
multiplication inherited from C∗, and identity the character taking constant value 1 ∈ C∗ for all g ∈ G. If the group G is finitely generated, then Gb is an abelian, complex reductive algebraic group. Given a homomorphism ϕ: G1 → G2, letϕ ˆ: Gb2 → Gb1, ρ 7→ ρ ◦ ϕ be the
induced morphism between character groups. Since the group C∗ is divisible, the functor
G ; Gb = Hom(H, C∗) is exact.
Now let H = Gab be the maximal abelian quotient of G. The abelianization map,
' ab: G → H, induces an isomorphism ab:c Hb / Gb. If H is torsion-free, then Hb can be identified with the complex algebraic torus (C∗)n, where n = rank(H). If H is a finite abelian group, then Hb is, in fact, isomorphic to H. More generally, let H be the maximal torsion-free quotient of H. Fixing a splitting H →
H yields a decomposition H ∼= H ⊕ Tors(H), and thus an isomorphism Hb ∼= Hb × Tors(H).
For simplicity, write T = Hb, and T0 for the identity component of this abelian, reductive,
complex algebraic group; clearly, T0 = Hb is an algebraic torus.
ˇ ∗ Conversely, we can associate to T its weight group, T = Homalg(T, C ), where the hom set is taken in the category of algebraic groups. The (discrete) group Tˇ is a finitely generated
63 abelian group. Let C[Tˇ] be its group algebra. We then have natural identifications,
ˇ ˇ ˇ ∗ maxSpec (C[T ]) = Homalg(C[T ], C) = Homgroup(T, C ) = T. (3.4.1)
The correspondence H ! T extends to a duality V , Subgroups of H Algebraic subgroups of T (3.4.2) l where V sends a subgroup ξ ≤ H to Hom(H/ξ, C∗) ⊆ T , while sends an algebraic subgroup ˇ ˇ C ⊆ T to ker(T C) ≤ H. Both sides of (3.4.2) are partially ordered sets, with naturally defined meets and joins.
As showed in [82], the above correspondence is an order-reversing equivalence of lattices.
3.4.2 Primitive lattices and connected subgroups
Given a subgroup ξ ≤ H, set
ξ := x ∈ H | mx ∈ ξ for some m ∈ N . (3.4.3)
Clearly, ξ is again a subgroup of H, and H/ξ is torsion-free. By definition, ξ is a finite-
index subgroup of ξ; in particular, rank(ξ) = rank(ξ). The quotient group, ξ/ξ, called the
determinant group of ξ, fits into the exact sequence
0 / H/ξ / H/ξ / ξ/ξ / 0 . (3.4.4)
The inclusion ξ ,→ H induces a splitting ξ/ξ ,→ H/ξ, showing that ξ/ξ ∼= Tors(H/ξ). Since the (abelian) group ξ/ξ is finite, it is isomorphic to its character group, ξ/ξc , which in turn can be viewed as a (finite) subgroup of Hb = T .
The subgroup ξ is called primitive if ξ = ξ. Under the correspondence H ! T , primitive subgroups of H correspond to connected algebraic subgroups of T . For an arbitrary subgroup
ξ ≤ H, we have an isomorphism of algebraic groups,
V (ξ) ∼= ξ/ξc · V (ξ). (3.4.5)
64 In particular, the irreducible components of V (ξ) are indexed by the determinant group, ξ/ξ,
while the identity component is V (ξ).
3.4.3 Pulling back algebraic subgroups
Now let ν : H A be an epimorphism, and letν ¯: H A be the induced epimorphism between maximal torsion-free quotients. Applying the Hom(−, C∗) functor to the left square of (3.4.6) yields the commuting right square in the display below:
ν νˆ H / / A H o ? _ A (3.4.6) bO bO ; ν¯ ν¯ˆ H / / A Hb o ? _ Ab
The morphismν ˆ: Ab → Hb sends the identity component Ab0 to the identity component Hb0,
thereby defining a morphismν ˆ0 : Ab0 → Hb0. Fixing a splitting A → A yields an isomorphism
Ab ∼= Ab × Tors(A). The following lemma is now clear.
Lemma 3.4.1. Let ν : H A be an epimorphism. Upon identifying Ab = Ab0 and Hb = Hb0, we have:
(i) ν¯ˆ =ν ˆ0.
(ii) im(ˆν) = V (ker(ν)).
Consequently, im(ν¯ˆ) = V (ker(ν))0.
3.4.4 Intersections of translated subgroups
Before proceeding, we need to recall some results from [82], which build on work of Hironaka
[38]. In what follows, T will be an abelian, reductive complex algebraic group.
Proposition 3.4.2 ([82]). Let ξ1 and ξ2 be two subgroups of H, and let η1 and η2 be two
−1 elements in T = Hb. Then η1V (ξ1) ∩ η2V (ξ2) 6= ∅ if and only if η1η2 ∈ V (ξ1 ∩ ξ2), in which
65 case
dim η1V (ξ1) ∩ η2V (ξ2) = rank H − rank(ξ1 + ξ2).
Proposition 3.4.3 ([82]). Let C and V be two algebraic subgroups of T .
(i) Suppose α1, α2, and η are torsion elements in T such that αiC ∩ ηV 6= ∅, for i = 1, 2. Then
dim (α1C ∩ ηV ) = dim (α2C ∩ ηV ).
(ii) Suppose α and η are torsion elements in T , of coprime order. Then
C ∩ ηV = ∅ =⇒ αC ∩ ηV = ∅.
Here is a corollary, which will be useful later on.
Corollary 3.4.4. Let C and V be two algebraic subgroups of T . Suppose α and ρ are torsion elements in T , such that ρ∈ / CV , α−1ρ ∈ CV , and dim(C ∩ V ) > 0. Then C ∩ ρV = ∅ and dim(αC ∩ ρV ) > 0.
Proof. By Proposition 3.4.2,
ρ∈ / C · V ⇔ C ∩ ρV = ∅ and α−1ρ ∈ C · V ⇔ αC ∩ ρV 6= ∅.
By Proposition 3.4.3, dim(αC ∩ ρV ) = dim(C ∩ V ) > 0. The conclusion follows.
3.5 An algebraic analogue of the exponential tangent
cone
We now associate to each subvariety W ⊂ T and integer d ≥ 1 a finite collection, Ξd(W ), of subgroups of the weight group H = Tˇ, which allows us to generalize the exponential tangent cone construction from [16].
66 3.5.1 A collection of subgroups
Let T be an abelian, reductive, complex algebraic group, and consider a Zariski closed subset
W ⊂ T . The translated subtori contained in W define an interesting collection of subgroups
of the discrete group H = Tˇ.
Definition 3.5.1. Given a subvariety W ⊂ T , and a positive integer d, let Ξd(W ) be the collection of all subgroups ξ ≤ H for which the following two conditions are satisfied:
(i) The determinant group ξ/ξ is cyclic of order dividing d.
(ii) There is a generator η ∈ ξ/ξc such that η · V (ξ) is a maximal, positive-dimensional,
torsion-translated subtorus in W .
Clearly, if d | m, then Ξd(W ) ⊆ Ξm(W ). Although this is not a priori clear from the definition, we shall see in Proposition 3.5.8 that Ξd(W ) is finite, for each d ≥ 1.
To gain more insight into this concept, let us work out what the sets Ξd(W ) look like in the case when W is a coset of an algebraic subgroup of T .
Lemma 3.5.2. Suppose W = ηV (χ), where χ is a subgroup of H and η ∈ Hb is a torsion element. Write V (χ) = S ρV (χ). Then ρ∈χ/χd
n [ m o Ξd(W ) = ξ ≤ H ∃ ρ ∈ χ/χd such that ord(ηρ) | d and H/ξd = (ηρ) V (χ) . m≥1
Corollary 3.5.3. If χ is a primitive subgroup of H and η is an element of order d in Hb,
then Ξd(ηV (χ)) consists of the single subgroup ξ ≤ χ for which ξ = χ and χ/ξd = hηi.
Now note that Ξd commutes with unions: if W1 and W2 are two subvarieties of T , then
Ξd(W1 ∪ W2) = Ξd(W1) ∪ Ξd(W2). (3.5.1)
Lemma 3.5.2, then, provides an algorithm for computing the sets Ξd(W ), whenever W is a (finite) union of torsion-translated algebraic subgroups of T .
67 Example 3.5.4. Let H = Z2, and consider the subvariety W = {(t, 1) | t ∈ C∗} ∪ {(−1, t) |
∗ ∗ 2 t ∈ C } inside T = (C ) . Note that W = V (ξ1) ∪ ηV (ξ2), where ξ1 = 0 ⊕ Z, ξ2 = Z ⊕ 0, and η = (−1, 1). Hence, Ξd(W ) = {ξ1} if d is odd, and Ξd(W ) = {ξ1, 2ξ2} if d is even.
3.5.2 The exponential map
Consider now the lattice
∨ H = H := Hom(H, Z). (3.5.2)
Evidently, H ∼= H/ Tors(H). Moreover, each subgroup ξ ≤ H gives rise to a sublattice
(H/ξ)∨ ≤ H∨.
Let Lie(T ) be the Lie algebra of the complex algebraic group T . The exponential map
∨ ∗ exp: Lie(T ) → T is an analytic map, whose image is T0. Let us identify T0 = Hom(H , C ) and Lie(T ) = Hom(H∨, C). Under these identifications, the corestriction to the image of the exponential map can be written as
2π i z ∨ ∨ ∗ exp = Hom(−, e ): Hom(H , C) → Hom(H , C ), (3.5.3) where z 7→ ez is the usual exponential map from C to C∗. Finally, upon identifying
∨ Hom(H , C) with H ⊗ C, we see that T0 = exp(H ⊗ C).
The correspondence T ; H = (Tˇ)∨ sends an algebraic subgroup W inside T to the sublattice χ = (Wˇ )∨ inside H. Clearly, χ = Lie(W ) ∩ H is a primitive lattice; furthermore, exp(χ ⊗ C) = W0. As shown in [82], we have
∨ V ((H/χ) ) = exp(χ ⊗ C), (3.5.4)
where both sides are connected algebraic subgroups inside T0 = exp(H ⊗ C).
68 3.5.3 Exponential interpretation
The construction from §3.5.1 allows us to associate to each subvariety W ⊂ T and each
∨ integer d ≥ 1 a subset τd(W ) ⊆ H , given by
[ ∨ τd(W ) = (H/ξ) . (3.5.5)
ξ∈Ξd(W )
The next lemma reinterprets the set τ1(W ) in terms of the “exponential tangent cone” construction introduced in [16] and studied in detail in [81].
Lemma 3.5.5. For every subvariety W ⊂ T ,
∨ τ1(W ) = {x ∈ H | exp(λx) ∈ W, for all λ ∈ C}. (3.5.6)
Proof. Denote by τ the right-hand side of (3.5.6). Given a non-zero homomorphism x: H →
Z such that x ∈ τ, the subgroup ker(x) ≤ H is primitive and V (ker(x)) = exp(Cx) ⊆ W .
Hence, we can find a subgroup ξ ≤ H such that ξ is primitive, V (ker(x)) j V (ξ), and
V (ξ) ⊆ W is a maximal subtorus. By (3.5.4), we have that V (ξ) = exp((H/ξ)∨ ⊗ C), which implies x ∈ τ1(W ).
Conversely, for any non-zero element x ∈ τ1(W ), there is a subgroup ξ ≤ H such that
∨ ∨ ∨ ξ ∈ Ξ1(W ) and x ∈ (H/ξ) . Thus, Cx ⊆ (H/ξ) ⊗ C, and so exp(Cx) ⊆ exp((H/ξ) ⊗ C) =
V (ξ) ⊆ W . Since x 6= 0, the map x: H → Z is surjective; thus, V (ker(x)) = V ((H/χ)∨),
where χ is the rank 1 sublattice of H generated by x. Hence, V (ker(x)) = exp(χ ⊗ C) ⊆ W , and so x ∈ τ.
Using now the characterization of exponential tangent cones given in [16, 81], we obtain
the following immediate corollary.
∨ Corollary 3.5.6. τ1(W ) is a finite union of subgroups of H .
Thus, the set τ Q(W ) = S (H/ξ)∨ ⊗ is a finite union of linear subspaces in the 1 ξ∈Ξ1(W ) Q vector space Qn, where n = rank H.
69 Example 3.5.7. Let T = (C∗)n, and suppose W = Z(f), for some Laurent polynomial f
P a1 an n in n variables. Write f(t1, . . . , tn) = a∈S cat1 ··· tn , where S is a finite subset of Z , and
∗ ca ∈ C for each a = (a1, . . . , an) ∈ S. We say a partition p = (p1 | · · · | pq) of the support S is admissible if P c = 0, for each 1 ≤ j ≤ q. To such a partition, we associate the a∈pj a subgroup
n L(p) = {x ∈ Z | (a − b) · x = 0, ∀a, b ∈ pj, ∀ 1 ≤ j ≤ q}. (3.5.7)
Then τ1(W ) is the union of all subgroups L(p), where p runs through the set of admissible partitions of S. In particular, if f(1) 6= 0, then τ1(W ) = ∅.
Proposition 3.5.8. For each d ≥ 1, the set Ξd(W ) is finite.
Proof. Fix an integer d ≥ 1. For any torsion point η ∈ T whose order divides d, consider the set Ξd(W, η) of subgroups ξ ≤ H for which ξ/ξc = hηi and η · V (ξ) is a maximal, positive- dimensional, torsion-translated subtorus in W . Then [ Ξd(W ) = Ξd(W, η), (3.5.8) η where the union runs over the (finite) set of torsion points η ∈ T whose order divides d.
−1 For each such point η, we have a map Ξd(W, η) → Ξ1(η W ), ξ 7→ ξ. Clearly, this map
is an injection. Now, Corollary 3.5.6 insures that the set Ξ1(W ) is finite. Thus, the set
−1 Ξ1(η W ) is also finite, and we are done.
3.6 The incidence correspondence for subgroups of H
We now single out certain subsets σA(ξ) and UA(ξ) of the parameter set Γ(H,A), which may be viewed as analogues of the special Schubert varieties in Grassmann geometry.
3.6.1 The sets σA(ξ)
We start by recalling a classical geometric construction. Let V be a variety in Qn defined by homogeneous polynomials. Set m = dim V , and assume m > 0. Consider the locus of
70 r-planes in Qn intersecting V non-trivially,
n σr(V ) = P ∈ Grr(Q ) dim(P ∩ V ) > 0 . (3.6.1)
n This set is a Zariski closed subset of Grr(Q ), called the variety of incident r-planes to V . For all 0 < r < n − m, this an irreducible subvariety, of dimension (r − 1)(n − r) + m − 1.
Particularly simple is the case when V is a linear subspace L ⊂ Qn. The corresponding
incidence variety, σr(L), is known as the special Schubert variety defined by L. Clearly,
n−1 n σ1(L) = P(L), viewed as a projective subspace in QP := P(Q ). Now let H be a finitely generated abelian group, let A be a factor group of H, and let
Γ(H,A) = Epi(H,A)/ Aut(A).
Definition 3.6.1. Given a subgroup ξ ≤ H, let σA(ξ) be the set of all [ν] ∈ Γ(H,A) for which rank(ker(ν) + ξ) < rank H.
When A is torsion-free, we recover the classical definition of special Schubert varieties.
More precisely, set n = rank H and r = rank A. We then have the following lemma.
∼ n Lemma 3.6.2. Under the natural isomorphism Γ(H, A) = Grr(Q ), the set σA(ξ) corre-
∨ sponds to the special Schubert variety σr((H/ξ) ⊗ Q).
∨ Proof. Let Grr(H ⊗Q) be the Grassmannian of r-dimensional subspaces in the vector space ∨ ∼ n H ⊗ Q = Q . Given an epimorphism ν : H A and a subgroup ξ ≤ H, we have
∨ ∨ rank(ker(ν) + ξ) < rank H ⇐⇒ dim((H/ ker(ν)) ⊗ Q ∩ (H/ξ) ⊗ Q) > 0.
Thus, the isomorphism
' ∨ ∨ Γ(H, A) / Grr(H ⊗ Q), [ν] 7→ (H/ ker(ν)) ⊗ Q
∨ establishes a one-to-one correspondence between σA(ξ) and σr((H/ξ) ⊗ Q).
For instance, if A is infinite cyclic, then the parameter set Γ(H, Z) may be identified
∨ with the projective space P(H ), while the set σZ(ξ) coincides with the projective subspace
P((H/ξ)∨).
71 Example 3.6.3. Let ξ ≤ Z2 be the sublattice spanned by the vector (a, b) ∈ Z2. Then
2 1 σZ(ξ) ⊂ Γ(Z , Z) corresponds to the point (−b, a) ∈ QP .
The sets σA(ξ) can be reconstructed from the classical Schubert varieties σA(ξ) associated to the lattice A = A/ Tors(A) by means of the set fibration described in Theorem 3.2.2. More
precisely, we have the following proposition.
Proposition 3.6.4. Let q : Γ(H,A) → Γ(H, A) be the natural projection map. Then
(i) q(σA(ξ)) = σA(ξ).
−1 (ii) q (σA(ξ)) = σA(ξ).
Therefore, σA(ξ) fibers over the Schubert variety σA(ξ), with each fiber isomorphic to the set Γ(H/A, A/A).
3.6.2 The sets UA(ξ)
Although simple to describe, the sets σA(ξ) do not behave too well with respect to the correspondence between subgroups of H and algebraic subgroups of T = Hb. This is mainly due to the fact that the σA-sets do not distinguish between a subgroup ξ ≤ H and its primitive closure, ξ. To remedy this situation, we identify certain subsets UA(ξ) ⊆ σA(ξ) which turn out to be better suited for our purposes.
Definition 3.6.5. Given a subgroup ξ ≤ H, let UA(ξ) be the set of all [ν] ∈ σA(ξ) for which ker(ν) ∩ ξ ⊆ ξ.
In particular, if ξ = ξ, then UA(ξ) = σA(ξ). In general, though, UA(ξ) $ σA(ξ). In order to reinterpret this definition in more geometric terms, we need a lemma.
Lemma 3.6.6. Let ξ ≤ H be a subgroup such that ξ/ξ is cyclic, and let χ ≤ H be another
a subgroup. Then the following conditions are equivalent.
72 (i) χ ∩ ξ ⊆ ξ.
(ii) V (χ) ∩ ηV (ξ¯) 6= ∅, for some generator η of ξ/ξc .
(iii) V (χ) ∩ ηV (ξ¯) 6= ∅, for any generator η of ξ/ξc .
(iv) η ∈ V (ker(ν) ∩ ξ), for some generator η of ξ/ξc .
(v) η ∈ V (ker(ν) ∩ ξ), for any generator η of ξ/ξc .
Proof. Let η be a generator of the finite cyclic group ξ/ξc . We then have
(hηi) ∩ ξ = ξ. (3.6.2)
By Proposition 3.4.2, the intersection V (χ) ∩ ηV (ξ¯) is non-empty if and only if η ∈
V (χ ∩ ξ¯), that is, hηi ⊆ V (χ ∩ ξ¯), which in turn is equivalent to
(hηi) ⊇ χ ∩ ξ.¯ (3.6.3)
In view of equality (3.6.2), inclusion (3.6.3) is equivalent to χ ∩ ξ ⊆ ξ. This shows (5.1.5) ⇔
(5.2.1).
The other equivalences are proved similarly.
Corollary 3.6.7. Let ξ ≤ H be a subgroup, and assume ξ/ξ is cyclic. Let ν : H A be an epimorphism. Then
[ν] ∈ UA(ξ) ⇐⇒ dim (V (ker(ν)) ∩ ηV (ξ)) > 0
for any (or, equivalently, for some) generator η ∈ ξ/ξc .
Despite their geometric appeal, the sets UA(ξ) do not enjoy a naturality property anal- ogous to the one from Proposition 3.6.4. In particular, the projection map q : Γ(H,A) →
Γ(H, A) may not restrict to a map UA(ξ) → UA(ξ). Here is a simple example.
2 Example 3.6.8. Let ν : H A be the canonical projection from H = Z to A = Z ⊕ Z2, and let ξ = ker(ν). Then [ν] ∈ UA(ξ), but [¯ν] ∈/ UA(ξ).
73 3.7 The incidence correspondence for subvarieties of Hb
In this section, we introduce and study certain subsets ΥA(W ) ⊆ Γ(H,A), which can be
n viewed as the toric analogues of the classical incidence varieties σr(V ) ⊆ Grr(Q ).
3.7.1 The sets UA(W )
Let us start by recalling some constructions we discussed previously. In §3.5.1, we associated to each subvariety W of the algebraic group T = Hb and each integer d ≥ 1 a certain collection
Ξd(W ) of subgroups of H. In §3.6.2, we associated to each subgroup ξ ≤ H and each abelian group A a certain subset UA(ξ) of the parameter set Γ(H,A) = Epi(H,A)/ Aut(A). Putting together these two constructions, we associate now to W a family of subsets of Γ(H,A), as follows.
Definition 3.7.1. Given a subvariety W ⊂ T , an abelian group A, let
[ UA(W ) = UA,d(W ), (3.7.1) d≥1 where [ UA,d(W ) = UA(ξ). (3.7.2)
ξ∈Ξd(W ) By Proposition 3.5.8, the union in (3.7.2) is a finite union.
Lemma 3.7.2. The set UA,d(W ) consists of all [ν] ∈ Γ(H,A) for which there is a subgroup
ξ ≤ H and an element η ∈ Hb of order dividing d such that ηV (ξ) is a maximal, positive- dimensional translated subtorus in W , and dim (V (ker ν) ∩ ηV (ξ)) > 0.
Proof. Let ν : H A be an epimorphism such that [ν] ∈ UA(ξ), for some ξ ∈ Ξd(W ). According to Definition 3.5.1, this means that the group ξ/ξ is cyclic of order dividing d, and there is a generator η ∈ ξ/ξc such that ηV (ξ) is a maximal, positive-dimensional translated subtorus in W . In view of Corollary 3.6.7, the fact that [ν] ∈ UA(ξ) insures that V (ker(ν)) ∩ ηV (ξ) has positive dimension.
74 The case d = 1 is worth singling out.
Corollary 3.7.3. Let W ⊂ T be a subvariety. Set n = rank H and r = rank A. Then:
(i) U (W ) = S σ (ξ). A,1 ξ∈Ξ1(W ) A
∼ n (ii) Under the isomorphism Γ(H, A) = Grr(Q ), the set UA,1(W ) corresponds to the inci-
Q dence variety σr(τ1 (W )).
In general, the set UA(W ) is larger than UA.1(W ). Here is a simple example; a more general situation will be studied in §3.11.3.
Example 3.7.4. Let H = Z2 and let W ⊂ (C∗)2 be the subvariety from Example 3.5.4.
1 Pick A = Z⊕Z2, and identify Γ(H,A) with QP . Then UA,d(W ) = {(1, 0)} or {(1, 0), (0, 1)}, according to whether d is odd or even.
3.7.2 The sets ΥA(W )
As before, let H be a finitely-generated abelian group, and let T = Hb be its Pontryagin dual. The next definition will prove to be key to the geometric interpretation of the (generalized)
Dwyer–Fried invariants.
Definition 3.7.5. Given a subvariety W ⊂ T , and an abelian group A, define a subset
ΥA(W ) of the parameter set Γ(H,A) by setting
ΥA(W ) = [ν] ∈ Γ(H,A) | dim(V (ker ν) ∩ W ) > 0 . (3.7.3)
Roughy speaking, the set ΥA(W ) ⊂ Γ(H,A) associated to a variety W ⊂ T is the toric
n analogue of the incidence variety σr(V ) ⊂ Grr(Q ) associated to a homogeneous variety
V ⊂ Qn.
It is readily seen that ΥA commutes with unions: if W1 and W2 are two subvarieties of T , then
ΥA(W1 ∪ W2) = ΥA(W1) ∪ ΥA(W2). (3.7.4)
75 Moreover, ΥA(W ) depends only on the positive-dimensional components of W . Indeed, if Z
is a finite algebraic set, then ΥA(W ∪ Z) = ΥA(W ). The next result gives a convenient lower bound for the Υ-sets.
Proposition 3.7.6. Let A be a quotient of H. Then
UA(W ) j ΥA(W ). (3.7.5)
Proof. Let ν : H A be an epimorphism such that [ν] ∈ UA,d(W ), for some d ≥ 1. By
Lemma 3.7.2, we have that dim(V (ker ν) ∩ W ) > 0. Thus, [ν] ∈ ΥA(W ).
As we shall see in Example 3.9.8, inclusion (3.7.5) may well be strict.
3.7.3 Translated subgroups
If the variety W is a torsion-translated algebraic subgroup of T , we can be more precise.
Theorem 3.7.7. Let W = ηV (ξ), where ξ ≤ H is a subgroup, and η ∈ Hb has finite order.
Then ΥA(W ) = UA,c(W ), where c = ord(η) · c(ξ/ξ).
Proof. Inclusion ⊇ follows from Proposition 3.7.6, so we only need to prove the opposite
inclusion. Write [ V (ξ) = ρV (ξ).
ρ∈ξ/ξd
Let ν : H A be an epimorphism such that [ν] ∈ ΥA(ηV (ξ)). Hence, there is a character
ρ: ξ/ξ → C∗ such that dim (V (ker(ν)) ∩ ηρV (ξ)) > 0. Consider the subgroup
[ χ = (ηρ)mV (ξ) . m
Lemma 3.5.2 implies that χ ∈ Ξd(ηV (ξ)), where d = ord(ηρ). Using Corollary 3.6.7, we
conclude that [ν] ∈ UA(χ).
Finally, set c = ord(η) · c(ξ/ξ). Then clearly [ν] ∈ UA,c(W ), and we are done.
76 Here is an alternate description of the set ΥA(W ), in the case when W is an algebraic subgroup of T , translated by an element η ∈ T , not necessarily of finite order.
Theorem 3.7.8. Let ξ ≤ H be a subgroup, and let η ∈ Hb. Then
ΥA(ηV (ξ)) = σA(ξ) ∩ [ν] ∈ Γ(H,A) | η ∈ V (ker(ν) ∩ ξ) .
In particular, ΥA(V (ξ)) = σA(ξ).
Proof. By Proposition 3.4.2, we have
{[ν] | dim(V (ker(ν)) ∩ ηV (ξ)) > 0}
= {[ν] | V (ker(ν)) ∩ ηV (ξ) 6= ∅} ∩ {[ν] | rank(ker(ν) + ξ) < rank H}.
Moreover, V (ker(ν)) ∩ ηV (ξ) 6= ∅ ⇐⇒ η ∈ V (ker(ν) ∩ ξ), and we are done.
Remark 3.7.9. In the case when A is a free abelian group of rank r and ξ is a primitive
n subgroup of H = Z , the set ΥA(ηV (ξ)) coincides with the set σr(ξ, η) defined in [81].
When the translation factor η from Theorem 3.7.8 has finite order, a bit more can be said.
Corollary 3.7.10. Let W = ηV (ξ) be a torsion-translated subgroup of T . Then
ΥA(W ) = σA(ξ) ∩ [ν] ∈ Γ(H,A) | (hηi) ⊇ ker(ν) ∩ ξ .
3.7.4 Deleted subgroups
We now analyze in more detail the case when the variety W is obtained from an algebraic subgroup of T by deleting its identity component. First, we need to introduce one more bit of notation.
Definition 3.7.11. Given a subgroup ξ ≤ H, and a quotient A of H, consider the subset
θA(ξ) ⊆ Γ(H,A) given by
[ 0 θA(ξ) = [ν] ∈ Γ(H,A) | ν(x) 6= 0 for all x ∈ ξ \ ξ . (3.7.6) 0 ξ≤ξ0 ξ : ξ/ξ is cyclic
77 Note that the indexing set for this union is a finite set, which is empty if ξ is primitive.
On the other hand, the condition that ν(x) 6= 0 depends on the actual element x in the
(typically) infinite set ξ \ ξ0, not just on the class of x in the finite group ξ/ξ0. Thus, even
r n when A = Z , the set θA(ξ) need not be open in the Grassmannian Γ(H,A) = Grr(Q ), where n = rank(H), although each of the sets {[ν] | ν(x) 6= 0} is open.
Proposition 3.7.12. Suppose W = V (ξ) \ V (ξ), for some non-primitive subgroup ξ ≤ H.
Then
ΥA(W ) = σA(ξ) ∩ θA(ξ). (3.7.7)
Proof. Write W = S ηV (ξ). By Theorem 3.7.8 and Lemma 3.6.6, we have η∈ξ/ξd\{1} ! [ ΥA(W ) = ΥA ηV (ξ)
η∈ξ/ξd\{1} [ = σA(ξ) ∩ [ν] ∈ Γ(H,A) | η ∈ V (ker(ν) ∩ ξ)
η∈ξ/ξd\{1} [ = σA(ξ) ∩ [ν] ∈ Γ(H,A) | V (ker(ν)) ∩ ηV (ξ) 6= ∅
η∈ξ/ξd\{1}
[ 0 = σA(ξ) ∩ [ν] ∈ Γ(H,A) | ker(ν) ∩ ξ ≤ ξ . 0 ξ≤ξ0 ξ : ξ/ξ is cyclic The desired conclusion follows at once.
3.7.5 Comparing the sets ΥA(W ) and ΥA(W )
Fix a decomposition A = A ⊕ Tors(A). Clearly, the projection map q = qA : Γ(H,A) →
c c Γ(H, A) sends ΥA(W ) to ΥA(W ) . On the other hand, as we shall see in Example 3.9.8,
the map q does not always send ΥA(W ) to ΥA(W ). Nevertheless, in some special cases it does. Here is one such situation.
Proposition 3.7.13. Suppose W = ρT , where T ⊂ Hb is an algebraic subgroup, and ρ¯ ∈
H/Tb has finite order, coprime to the order of Tors(A). Then
−1 q(ΥA(W )) = ΥA(W ) and q (ΥA(W )) = ΥA(W ).
78 Therefore, ΥA(W ) fibers over ΥA(W ), with each fiber isomorphic to Γ(H/A, A/A).
Here,ρ ¯ is the image of ρ under the quotient map Hb → H/Tb .
Proof. Let ν : H A be an epimorphism such that [¯ν] = q([ν]) does not belong to ΥA(W ),
that is, the subtorus im(ν¯ˆ) = im(ˆν)1 intersects W in only finitely many points. We want to
show that im(ˆν)α ∩ W is also finite, for all α ∈ Tors(A). T T First assume im(ˆν)1 W is non-empty. If im(ˆν)α W = ∅, we are done. Otherwise, using Proposition 3.4.3(i) with C = im(ˆν)1, α1 = 1, α2 = α and ηV = ρT , we infer that dim (im(ˆν)α ∩ ρT ) = dim (im(ˆν)1 ∩ ρT ), and the desired conclusion follows. T Now assume im(ˆν)1 W is empty. Using Proposition 3.4.3(ii) with C = im(ˆν)1 and T T ηV = ρT , our assumption that im(ˆν)1 ρT = ∅ implies that im(ˆν)α ρT = ∅. Thus, the desired conclusion follows in this case, too, and we are done.
Corollary 3.7.14. For every subgroup ξ ≤ H, the set q(ΥA(V (ξ))) is contained in ΥA(V (ξ)).
In general, though, the projection map q : Γ(H,A) → Γ(H, A) does not restrict to a map
ΥA(W ) → ΥA(W ). Proposition 3.7.16 below describes a situation when this happens. First, we need a lemma, whose proof is similar to the proof of Proposition 3.3.9.
Lemma 3.7.15. Let π : A B be an epimorphism, and let π˜ : Γ(H,A) Γ(H,B) be the induced homomorphism. Then
qA(ΥA(W )) ⊆ ΥA(W ) =⇒ qB(ΥB(W )) ⊆ ΥB(W ).
Proposition 3.7.16. Let H be a finitely generated, free abelian group, and let A be a quotient
of H such that rank A < rank H. Let W be a subvariety of Hb of the form ρT ∪ Z, where Z is a finite set, T is an algebraic subgroup, and ρ is a torsion element whose order divides c(A). Then q ΥA(W ) j6 ΥA(W ).
Proof. We need to construct an epimorphism ν : H A such that [ν] ∈ ΥA(W ), yet [¯ν] ∈/
ΥA(W ).
79 Step 1. First, we assume Tors(A) is a cyclic group. In this case, we claim there exists a
subtorus C of Hb, and a torsion element α ∈ Hb, such that ord α = ord ρ and dim(C ∩ρT ) ≤ 0, yet dim(αC ∩ ρT ) > 0.
To prove the claim, set (hρi) = L and ξ = (T ). Since ρ∈ / T , we have that ξ * L.
Thus, there exist a sublattice χ of rank 1, such that χ ⊆ ξ and χ * L. Set T 0 = V (χ).
0 0 0 0 Then T is a codimension 1 subgroup with T ⊆ T ⊂ Hb, and ρ∈ / T . Thus, T0 ⊆ T0 ⊂ Hb.
0 Let C be any dimension r subtorus of T0 intersecting T with positive dimension. Then
ρ∈ / CT ⊆ T 0 and dim(C ∩ T ) > 0. Choose an element α ∈ Hb such that α−1ρ = 1 ∈ CT . Clearly, ord(α) = ord(ρ) | c(A). Using Corollary 3.4.4, we conclude that C ∩ ρT = ∅ and
dim(αC ∩ ρT ) > 0, thus finishing the proof of the claim.
S k Now, the algebraic subgroup k α C corresponds to an epimorphism H A⊕Zd, where d = ord(α). Since H is torsion-free, Tors(A) is cyclic, and d divides c(A), this epimorphism
can be lifted to an epimorphism ν : H A.
Step 2. For the general case, let B be the cyclic subgroup of Tors(A) for which |B| = c(A)
and ord(ρ) | |B|. Notice that B is a direct summand of Tors(A). We then have the following commuting diagram:
qA⊕B Γ(H/A, B) / Γ(H, A ⊕ B) / Γ(H, A) O O π1 π2
qA Γ(H/A, Tors(A)) / Γ(H,A) / Γ(H, A)
Since H/A is torsion-free, the group Aut(H/A) acts transitively on Γ(H/A, B). Using the
assumption that Γ(H/A, Tors(A)) 6= ∅, we deduce that the map π1 is surjective. Thus, the map π2 is surjective. From Step 1, we know that q ΥA⊕B(W ) is not contained in ΥA(W ). Using Lemma 3.7.15, we conclude that q ΥA(W ) is not contained in ΥA(W ), either.
80 3.8 Support varieties for homology modules
We now switch gears, and revisit the Dwyer–Fried theory in a slightly more general context.
In particular, we show that the support varieties of the homology modules of two related
chain complexes coincide.
3.8.1 Support varieties
Let H be a finitely generated abelian group, and let F be a finitely generated C-algebra. Then the group ring R = F [H] is a Noetherian ring. Let maxSpec(R) be the set of maximal
ideals in R, endowed with the Zariski topology.
Given a module M over F [H], denote by supp M its support, consisting of those maximal
ideals m ∈ maxSpec(F [H]) for which the localization Mm is non-zero.
Now let A be another finitely generated abelian group, and let ν : H A be an epi- morphism. Denote by S = F [A] the group ring of A. The extension of ν to group rings,
∗ ν : R S, is a ring epimorphism. Let ν : maxSpec(S) ,→ maxSpec(R) be the induced morphism between the corresponding affine varieties.
In the case when F = C, the group ring R = C[H] is the coordinate ring of the algebraic
group Hb = Hom(G, C∗), and maxSpec(R) = Hb. Furthermore, if M is an R-module, then
supp(M) = Z(ann M), (3.8.1) where ann M ⊂ R is the annihilator of M, and Z(ann M) ⊂ Hb is its zero-locus.
Lemma 3.8.1. If ν : H A is an epimorphism, then
(ν∗)−1(supp(M)) ∼= im(ˆν) ∩ Z(ann M). (3.8.2)
Proof. From the definitions, we see that the diagram
ν∗ maxSpec(S) / maxSpec(R) (3.8.3)
=∼ =∼ νˆ Ab / Hb
81 commutes. The conclusion readily follows.
3.8.2 Homology modules
We are now ready to state and prove the main result of this section. An abbreviated proof
was given by Dwyer and Fried in [18], in the special case when A is free abelian. For the convenience of the reader, we give here a complete proof, modeled on the one from [18].
Theorem 3.8.2. Let F be a finitely generated C−algebra, let C• be a chain complex of
finitely generated free modules over F [H], and let ν : H A be an epimorphism. Viewing F [A] as a module over F [H] by extension of scalars via ν, we have
∗ −1 supp H∗(C• ⊗F [H] F [A]) = (ν ) supp H∗(C•) . (3.8.4)
Proof. Set n = rank H and r = rank A. There are three cases to consider.
Case 1: H is torsion-free. We use induction on n − r to reduce to the case r = n − 1, in which case Tors(A) = Zq, for some q ≥ 1 (if A is torsion-free, q = 1). We then have a short exact sequence of chain complexes,
φ=xq−1 0 / C• / C• / C• ⊗F [H] F [A] / 0 ,
which yields a long exact sequence of homology groups. Consider the map φ∗ : M → M, where M = H∗(C•), viewed as a module over F [H]. Localizing at a maximal ideal m, we obtain an endomorphism φm of the finitely-generated module Mm over the Noetherian ring
F [H]m.
As a standard fact, if φm is surjective, then φm is injective. Using the exact sequence
0 → ker φm → Mm → Mm → coker φm → 0,
82 we see that coker φm = 0 ⇒ ker φm = 0. Therefore, supp ker φ∗ ⊆ supp coker φ∗, and so
supp H∗(C• ⊗F [H] F [A]) = supp coker φ∗ ∪ supp ker φ∗
= supp coker φ∗
= supp M/(xq − 1)M
= (supp M) ∩ Z(xq − 1)
= (supp M) ∩ im(ˆν).
When n−r > 1, one can change the basis of H and A so that, the epimorphism ν : H → A
is the composite
0 ν1 0 ν2 0 H ⊕ Z / / H ⊕ Zq / / A ⊕ Zq ,
id 0 ν| 0 0 where ν1 is of the form , and ν2 is of the form H . By the induction hypothesis, 0 ν|Z 0 id
equality (3.8.4) holds for ν1 and ν2. Thus, the theorem holds for the map ν = ν2 ◦ ν1.
Case 2: H is finite. In this situation, ν : H A is an epimorphism between two finite
abelian groups. As above, ν induces a ring epimorphism ν : F [H] F [A]. The correspond- ing map, i = ν∗ : maxSpec F [A] ,→ maxSpec F [H], is a closed immersion. Consider the
commuting diagram
i maxSpec F [A] / maxSpec F [H] (3.8.5)
j maxSpec C[A] / maxSpec C[H], where the morphism j is induced by ν : H A. Clearly, j is an open immersion. By commutativity of (3.8.5), we have that maxSpec F [A] is an open subset of maxSpec F [H].
It suffices to show that
∗ ˜ −1 ˜ supp(Hk(i C•)) = i (supp Hk(C•)) (3.8.6)
˜ for any k ∈ Z, where C• is the sheaf of modules over maxSpec(F [H]) corresponding to the ∗ ˜ module C• over F [H], and i C• is the sheaf of modules over maxSpec(F [A]) obtained by ˜ pulling back the sheaf C•.
83 ˜ ˜ For any m ∈ maxSpec(F [A]), we have (Hk(C•))m = Hk((C•)m), since localization is an
˜ ∗ ˜ exact functor, and also (C•)m = (i C•)m, since i is an open immersion. Thus,
∗ ˜ ∗ ˜ supp Hk(i C•) = {m ∈ maxSpec(F [A]) | (Hk(i C•))m 6= 0}
∗ ˜ = {m ∈ maxSpec(F [A]) | (i C•)m is not exact at k} ˜ = {m ∈ maxSpec(F [A]) | (Hk(C•))m 6= 0} ˜ = supp Hk(C•) ∩ maxSpec(F [A])
−1 ˜ = i (supp Hk(C•)).
Case 3: H is arbitrary. As in the proof of Theorem 3.2.2, we can choose splittings
H = H ⊕ Tors(H) and A = A ⊕ Tors(A) such that H = A ⊕ H0, and the epimorphism
ν : H A is the composite
ν1 ν2 A ⊕ H0 ⊕ Tors(H) / / A ⊕ ν(H0) ⊕ Tors(H) / / A ⊕ Tors(A) ,
id 0 0 0 id 0 0 0 ν| 0 0 where ν(H ) is finite, ν1 is of the form H , and ν2 is of the form 0 i ν|Tors(H) , with 0 0 id i: ν(H0) ,→ Tors(A) the inclusion map.
Set F = C[A ⊕ Tors(H)]; by Case 1, equality (3.8.4) holds for ν1. Now set F = C[A]; by
Case 2, equality (3.8.4) holds for ν2. Thus, the theorem holds for the map ν = ν2 ◦ ν1.
3.8.3 Finite supports
We conclude this section with a result which is presumably folklore. For completeness, we
include a proof.
Proposition 3.8.3. Let A be a finitely generated abelian group, and let M be a finitely
generated module over the group ring S = C[A]. Then M as a C-vector space is finite- dimensional if and only if supp(M) is finite.
Proof. Since M is a finitely generated S-module, we can argue by induction on the number
of generators of M. Using the short exact sequence 0 → hmi → M → M/hmi → 0, where
84 m is a generator of M, and the fact that supp(M) = supp(hmi) ∪ supp(M/hmi), we see that
it suffices to consider the case when M is a cyclic module. In this case, M = S/ ann(M)
and supp(M) = Z(ann(M)) = maxSpec(S/ ann(M)). From the assumption that supp M is
finite, and using the Noether Normalization lemma, we infer that S/ ann(M) is an integral extension of C. Thus, dimC(S/ ann(M)) < ∞. Conversely, suppose supp(M) is infinite. Then maxSpec(S/ ann(M)) is infinite, which implies maxSpec(S/ ann(M)) has positive dimension. Choose a prime ideal p contain- ing ann(M), such that the Krull dimension of S/p is positive. From the condition that dimC S/ ann(M) < ∞, we deduce that dimC S/p < ∞. By the Noether Normalization lem- ma, S/p is an integral extension of C[x1, . . . , xn], with n > 0. Thus, dimC S/p = ∞. This is a contradiction, and so we are done.
3.9 Characteristic varieties and generalized Dwyer–Fried
sets
In this section, we finally tie together several strands, and show how to determine the sets
i ΩA(X) in terms of the jump loci for homology in rank 1 local systems on X.
3.9.1 The equivariant chain complex
Let X be a connected CW-complex. As usual, we will assume that X has finite k-skeleton, for some k ≥ 1. Without loss of generality, we may assume that X has a single 0-cell e0, which we will take as our basepoint x0. Moreover, we may assume that all attaching maps
i i (S , ∗) → (X , x0) are basepoint-preserving. Let G = π1(X, x0) be the fundamental group of
X, and denote by (Ci(X, C), ∂i)i≥0 the cellular chain complex of X, with coefficients in C. Let p: Xab → X be the universal abelian cover. The cell structure on X lifts in a natural
ab −1 fashion to a cell structure on X . Fixing a liftx ˜0 ∈ p (x0) identifies the group H = Gab
85 with the group of deck transformations of Xab, which permute the cells. Therefore, we may
ab view the cellular chain complex C• = C•(X , C) as a chain complex of left-modules over
the group algebra R = C[H]. This chain complex has the form
∂˜i ∂˜2 ∂˜1 ··· / Ci / Ci−1 / ··· / C2 / C1 / C0 , (3.9.1)
1 1 The first two boundary maps can be written down explicitly. Let e1, . . . , em be the 1-cells
1 of X. Since we have a single 0-cell, each ei is a loop, representing an element xi ∈ G. Let 0 1 1 ˜ 1 0 2 e˜ =x ˜0, and lete ˜i be the lift of ei atx ˜0; then ∂1(˜ei ) = (xi − 1)˜e . Next, let e be a 2-cell,
2 and lete ˜ be its lift atx ˜0; then
m ˜ 2 X 1 ∂2(˜e ) = φ ∂r/∂xi · e˜i , (3.9.2) i=1
where r is the word in the free group Fm = hx1, . . . , xmi determined by the attaching map
of the 2-cell, ∂r/∂xi ∈ C[Fm] are the Fox derivatives of r, and φ: C[Fm] → C[H] is the ab extension to group rings of the projection map Fm → G −→ H, see [27].
3.9.2 Characteristic varieties
Since X has finite 1-skeleton, the group H = H1(X, Z) is finitely generated, and its dual,
Hb = Hom(H, C∗), is a complex algebraic group. As is well-known, the character group Hb
∗ parametrizes rank 1 local systems on X: given a character ρ: H → C , denote by Cρ the
1-dimensional C-vector space, viewed as a right R-module via a · g = ρ(g)a, for g ∈ H and
a ∈ C. The homology groups of X with coefficients in Cρ are then defined as
ab Hi(X, Cρ) := Hi C•(X , C) ⊗R Cρ . (3.9.3)
Definition 3.9.1. The characteristic varieties of X (over C) are the sets
i ∗ V (X) = { ρ ∈ Hom(H, C ) | dimC Hj(X, Cρ) 6= 0 for some 1 ≤ j ≤ i } ,
The identity component of the character group T = Hb is a complex algebraic torus,
which we will denote by T0. Let H = H/ Tors(H) be the maximal torsion-free quotient of
86 ' i H. The projection map π : H H induces an identificationπ ˆ : Hb / Hb0. Denote by W (X)
i i i the intersection of V (X) with T0 = Hb0. If H is torsion-free, then W (X) = V (X); in general, though, the two varieties differ.
For each 1 ≤ i ≤ k, the set V i(X) is a Zariski closed subset of the complex algebraic
i group T , and W (X) is a Zariski closed subset of the complex algebraic torus T0. Up to isomorphism, these varieties depend only on the homotopy type of X. Consequently, we
may define the characteristic varieties of a group G admitting a classifying space K(G, 1)
with finite k-skeleton as V i(G) = V i(K(G, 1)), for i ≤ k. It is readily seen that V 1(X) =
1 V (π1(X)). For more details on all this, we refer to [81]. The characteristic varieties of a space can be reinterpreted as the support varieties of its
Alexander invariants, as follows.
Theorem 3.9.2 ([70]). For each 1 ≤ i ≤ k, the characteristic variety V i(X) coincides with
Li ab i the support of the C[H]-module j=1 Hj(X , C), while W (X) coincides with the support Li fab of the C[H]-module j=1 Hj(X , C).
3.9.3 The first characteristic variety of a group
Let G be a finitely presented group. The chain complex (3.9.1) corresponding to a presen- ˜ tation G = hx1, . . . , xq | r1, . . . , rmi has second boundary map, ∂2, an m by q matrix, with rows given by (3.9.2). Making use of Theorem 3.9.2, we see that V 1(G) is defined by the ˜ vanishing of the codimension 1 minors of the Alexander matrix ∂2, at least away from the trivial character 1. This interpretation allows us to construct groups with fairly complicated
characteristic varieties.
Lemma 3.9.3. Let f = f(t1, . . . , tn) be a Laurent polynomial. There is then a finitely
n 1 ∗ n presented group G with Gab = Z and V (G) = {z ∈ (C ) | f(z) = 0} ∪ {1}.
Proof. Let Fn = hx1, . . . , xni be the free group of rank n, with abelianization map ab: Fn →
n Z , xk 7→ tk. Recall the following result of R. Lyndon (as recorded in [27]): if v1, . . . , vn
87 n ±1 ±1 Pn are elements in the ring Z[Z ] = Z[t1 , . . . , tn ], satisfying the equation k=1(tk − 1)vk = 0,
0 then there exists an element r ∈ Fn such that vk = ab(∂r/∂xk), for 1 ≤ k ≤ n.
0 Making use of this result, we may find elements ri,j ∈ Fn, 1 ≤ i < j ≤ n such that f · (ti − 1), if k = i ab(∂ri,j/∂xk) = f · (1 − tj), if k = j 0, otherwise.
It is now readily checked that the group G with generators x1, . . . , xn and relations rij has the prescribed first characteristic variety.
In certain situations, one may realize a Laurent polynomial as the defining equation for
the characteristic variety by a more geometric construction.
Example 3.9.4. Let L be an n-component link in S3, with complement X. Choosing
n orientations on the link components yields a meridian basis for H1(X, Z) = Z . Then
1 ∗ n V (X) = {z ∈ (C ) | ∆L(z) = 0} ∪ {1}, (3.9.4)
where ∆L = ∆L(t1, . . . , tn) is the (multi-variable) Alexander polynomial of the link.
3.9.4 A formula for the generalized Dwyer–Fried sets
Recall that, in Definition 3.7.5 we associated to each subvariety W ⊂ Hb, and each abelian group A a subset
ΥA(W ) = [ν] ∈ Γ(H,A) | dim(im(ˆν) ∩ W ) > 0 . (3.9.5)
i The next theorem expresses the Dwyer–Fried sets ΩA(X) in terms of the Υ-sets associated to the i-th characteristic variety of X.
Theorem 3.9.5. Let X be a connected CW-complex, with finite k-skeleton. Set G =
π1(X, x0) and H = Gab. For any abelian group A, and for any i ≤ k,
i i ΩA(X) = Γ(H,A) \ ΥA(V (X)).
88 ν Proof. Fix an epimorphism ν : H A, and let X → X be the corresponding cover. Recall
ab the cellular chain complex C• = C•(X , C) is a chain complex of left modules over the ring
ν R = C[H]. If we set S = C[A], the cellular chain complex C•(X , C) can be written as
C• ⊗R S, where S is viewed as a right R-module via extension of scalars by ν. Consider the S-module
i i M ν M M = Hj(X , C) = Hj(C• ⊗R S). j=1 j=1
i ν ν By definition, [ν] belongs to ΩA(X) if and only if the Betti numbers b1(X ), . . . , bi(X ) are all finite, i.e., dimC M < ∞. By Proposition 3.8.3, this condition is equivalent to supp M being finite.
Now let ν∗ : maxSpec(S) ,→ maxSpec(R) be the induced morphism between the corre- sponding affine schemes. We then have
i ∗ −1 M supp M = (ν ) supp Hj(C•) by Theorem 3.8.2 j=1 i ∼ M ab = im(ˆν) ∩ Z ann Hj(X , C) by Lemma 3.8.1 j=1
= im(ˆν) ∩ V i(X) by Theorem 3.9.2.
This ends the proof.
i i i Remark 3.9.6. If H is torsion-free, then W (X) = V (X); thus, the set ΩA(X) depends only the variety W i(X) and the abelian group A. On the other hand, if Tors(H) 6= 0, the
i i i variety W (X) may be strictly included in V (X), in which case the set ΩA(X) may depend on information not carried by W i(X). We shall see examples of this phenomenon in §3.10.3.
3.9.5 An upper bound for the Ω-sets
i We now give a computable “upper bound” for the generalized Dwyer-Fried sets ΩA(X), in terms of the sets introduced in §3.7.1.
89 Theorem 3.9.7. Let X be a connected CW-complex with finite k-skeleton. Set H =
H1(X, Z), and fix a degree i ≤ k. Let A be a quotient of H. Then
i i ΩA(X) ⊆ Γ(H,A) \ UA(V (X)). (3.9.6)
Proof. Follows at once from Proposition 3.7.6 and Theorem 3.9.5.
In general, inclusion (3.9.6) is not an equality. Indeed, the fact that V (ker ν) ∩ V i(X)
is infinite cannot guarantee there is an algebraic subgroup in the intersection. Here is a
concrete instance of this phenomenon.
Example 3.9.8. Using Lemma 3.9.3, we may find a 3-generator, 3-relator group G with abelianization H = Z3 and characteristic variety
1 ∗ 3 V (G) = (t1, t2, t3) ∈ (C ) | (t2 − 1) = (t1 + 1)(t3 − 1) .
This variety has a single irreducible component, which is a complex torus passing through the origin; nevertheless, this component does not embed as an algebraic subgroup in (C∗)3. There are 4 maximal, positive-dimensional torsion-translated subtori contained in V 1(G),
3 namely η1V (ξ1), . . . , η4V (ξ4), where ξ1, . . . , ξ4 are the subgroups of Z given by
2 0 0 0 2 −1 2 −2 ξ1 = im 0 1 , ξ2 = im 1 0 , ξ3 = im −1 0 , ξ4 = im −1 0 , 0 0 0 1 0 1 0 2
and η1 = (−1, 1, 1), η2 = η3 = (1, 1, 1), η4 = (−1, 1, −1).
2 We claim that, for A = Z ⊕ Z2, inclusion (3.9.6) from Theorem 3.9.7 is strict, i.e.,
1 1 ΩA(G) $ Γ(H,A) \ UA(V (G)).
3 2 To prove this claim, consider the epimorphism ν : Z Z ⊕ Z2 given by the matrix
1 0 0 0 ∗ 3 0 0 1 . Note that ker(ν) = im 2 , and so im(ˆν) = {t ∈ ( ) | t2 = ±1}. The intersection 0 1 0 0 C 1 of im(ˆν) with V (G) consists of all points of the form (t1, ±1, t3) with (t1 + 1)(t3 − 1) equal
1 to 0 or −2. Clearly, this is an infinite set; therefore, [ν] ∈/ ΩA(G).
90 On the other hand, im(ˆν) ∩ η1V (ξ1) = im(ˆν) ∩ η2V (ξ2) = ∅, and rank(ker(ν) + ξ3) = rank(ker(ν) + ξ4) = rank(H). Hence, im(ˆν) ∩ ηjV (ξj) is finite, for all j. In view of Lemma
1 3.7.2, we conclude that [ν] ∈/ UA(V (G)).
3.10 Comparison with the classical Dwyer–Fried in-
variants
i 3.10.1 The Dwyer–Fried invariants Ωr(X)
In the case of free abelian covers and the usual Ω-sets, Theorem 3.9.5 allows us to recover the following result from [18], [70], [81].
Corollary 3.10.1. Set n = b1(X). Then, for all r ≥ 1,
i n i Ωr(X) = [ν] ∈ Grr(Z ) | im(ˆν) ∩ W (X) is finite .
Using now the identification from Example 3.6.3, and taking into account Lemma 3.5.5,
Theorem 3.9.7 yields the following corollary.
Corollary 3.10.2 ([70, 81]). Let X be a connected CW-complex with finite k-skeleton, and
i n−1 i set n = b1(X). Then Ωr(X) ⊆ QP \ P(τ1(W (X))), for all i ≤ k and all r ≤ n.
Remark 3.10.3. As noted in [81], if the variety W i(X) is a union of algebraic subtori, then
i n the Dwyer–Fried sets Ωr(X) are open subsets of Grr(Z ), for all r ≥ 1. In general, though,
i examples from [18], [81] show that the sets Ωr(X) with r > 1 need not be open. On the other
i n n−1 hand, as noted in [18, 70, 81], the sets Ω1(X) are always open subsets of Gr1(Z ) = QP . We will come back to this phenomenon in §3.11, in a more general context.
3.10.2 The comparison diagram, revisited
As may be expected, the generalized Dwyer–Fried invariants carry more information about the homotopy type of a space and the homological finiteness properties of its regular abelian
91 covers than the classical ones. To make this more precise, let A be a quotient of the group
H = H (X, ). Fix a decomposition A = A ⊕ Tors(A), and identify Ωi (X) = Ωi (X), where 1 Z r A r = rank(A). As we saw in §3.3.4, for each i ≤ k we have a comparison diagram
i ΩA(X) / Γ(H,A) (3.10.1)
q|Ωi (X) q A i Ωr(X) / Γ(H, A)
between the respective Dwyer–Fried invariants, viewed as subsets of the parameter sets for
regular A-covers and A-covers, respectively.
i We are interested in describing conditions under which the set ΩA(X) contains more
i information than Ωr(X). This typically happens when the comparison diagram (3.10.1) is
i −1 not a pull-back diagram, i.e., there is a point [¯ν] ∈ Ωr(X) for which the fiber q ([¯ν]) is not
i i included in ΩA(X). In fact, the number of points in the fiber which lie in ΩA(X) may vary
i as we move about Ωr(X). In view of Theorem 3.9.5, we have the following criterion.
i Proposition 3.10.4. Diagram (3.10.1) fails to be a pull-back diagram if and only if q(ΥA(V (X))
i is not included in Υr(V (X), i.e., there is an epimorphism ν : H A such that
dim(imν ˆ ∩ V i(X)) > 0, yet dim(im ν¯ˆ ∩ V i(X)) = 0.
From Proposition 3.3.10, we know diagram (3.10.1) is a pull-back diagram precisely when the homological finiteness of an arbitrary A-cover of X can be tested through the corresponding A-cover. In order to quantify the discrepancy between these two types of
homological finiteness properties, let us define the “singular set”
i i −1 i −1 ΣA(X) = [¯ν] ∈ ΩA(X) | #(q ([¯ν]) ∩ ΩA(X)) < #(q ([¯ν])) . (3.10.2)
We then have:
i i c i ΣA(X) = q ΩA(X) ∩ ΩA(X). (3.10.3)
92 3.10.3 Maximal abelian versus free abelian covers
We now investigate the relationship between the finiteness of the Betti numbers of the
maximal abelian cover Xab and the finiteness of the Betti numbers of the corresponding free abelian cover Xfab of our space X.
∗ n As before, write H = H1(X, Z) and identify the character group Hb with (C ) ×Tors(H),
where n = b1(X).
Proposition 3.10.5. Suppose Tors(H) 6= 0. Furthermore, assume that W i(X) is finite,
i i whereas V (X) = W (X) ∪ (Hb \ Hb0). Then
i n (i) Ωr(X) = Grr(Z ), for all r ≥ 1.
i (ii) If rank(A) = rank(H) and Tors(A) 6= 0, then ΩA(X) = ∅.
n i Proof. By Theorem 3.9.5, an element [ν] ∈ Grr(Z ) belongs to Ωr(X) if and only if im(ˆν) ∩ W i(X) is finite. By assumption, W i(X) is finite; thus, the intersection im(ˆν) ∩ W i(X) is
also finite. This proves (i).
i Again by Theorem 3.9.5, an element [ν] ∈ Γ(H,A) belongs to ΩA(X) if and only if
i i i im(ˆν) ∩ V (X) is finite. By assumption, V (X) = W (X) ∪ (Hb \ Hb0); moreover, rank(A) = rank(H) and Tors(A) 6= 0. Thus, the intersection im(ˆν) ∩ V i(X) contains at least one
component of Hb \ Hb0, which is infinite. This proves (ii).
A similar argument yields the following result.
1 1 Proposition 3.10.6. Suppose H1(X, Z) has non-trivial torsion, W (X) is finite, and V (X)
fab ab is infinite. Then b1(X ) < ∞, yet b1(X ) = ∞.
Proposition 3.10.7. Suppose X = X1 ∨ X2, where H1(X1, Z) is free abelian and non-
1 fab trivial, and H1(X2, Z) is finite and non-trivial. If V (X1) is finite, then b1(X ) < ∞, yet
ab b1(X ) = ∞.
93 Proof. Let H = H1(X, Z); then H = H1(X1, Z) 6= 0 and Tors(H) = H1(X2, Z) 6= 0. Thus, the character group Hb decomposes as Hb0 × Tors(H), with both factors non-trivial. Now, W 1(X) = V 1(X ) × {1} is finite, and thus Ω1 (X) is a singleton. On the other 1 H 1 1 1 hand, V (X) = W (X) ∪ Hb0 × (Tors(H) \{1}) is infinite, and thus ΩH (X) = ∅.
2 2 Example 3.10.8. Consider the CW-complexes X = S1 ∨ RP and Y = S1 × RP . Then ∼ ∼ fab fab H1(X, Z) = H1(Y, Z) = Z × Z2. Clearly, the free abelian covers X and Y are rationally
i i acyclic; thus both Ω1(X) and Ω1(Y ) consist of a single point, for all i ≥ 0. On the other hand, the free abelian cover Xab has the homotopy type of a countable wedge of S1’s and
2’s, whereas Y ab ' S2. Therefore, Ω1 (X) = ∅, while Ω1 (Y ) = {point}. RP Z⊕Z2 Z⊕Z2
i This example shows that the generalized Dwyer–Fried invariants ΩA(X) may contain more information than the classical ones.
3.11 The rank 1 case
i In this section, we discuss in more detail the invariants ΩA(X) in the case when A has rank 1, and push the analysis even further in some particularly simple situations.
i 3.11.1 A simplified formula for ΩA(X)
Recall that, for a finitely generated abelian group A, the integer c(A) denotes the largest order of an element in A. Recall also that, for every subvariety W ⊂ Hb and each index d ≥ 1, we have a subset UA,d(W ) ⊂ Γ(H,A), described geometrically in Lemma 3.7.2.
Theorem 3.11.1. Let X be a connected CW-complex with finite k-skeleton. Set H =
H1(X, Z), and fix a degree i ≤ k. If rank(A) = 1, then
i i ΩA(X) = Γ(H,A) \ UA,c(A)(V (X)). (3.11.1)
94 Proof. The inclusion ⊆ follows from Theorem 3.9.7, so we only need to prove the opposite inclusion.
i Let ν : H A be an epimorphism such that [ν] ∈/ ΩA(X). By Theorem 3.9.5, the variety imν ˆ ∩ V i(X) has positive dimension. Since rank(A) = 1, there exists a component of the
1-dimensional algebraic subgroup imν ˆ contained in V i(X); that is, there exists a character
ρ ∈ Tors(\A) such that
νˆ(ρ) · im ν¯ˆ ⊆ V i(X).
Now, we may find a primitive subgroup χ ≤ H and a torsion character η ∈ Hb such that ηV (χ) =ν ˆ(ρ)V (χ) is a maximal translated subtorus in V i(X) which containsν ˆ(ρ) · im ν¯ˆ.
Set [ ξ = ηmV (χ) . m≥1 i Clearly, the subgroup ξ ≤ H belongs to Ξd(V (X)), where d := ord(ρ) divides c(A). Since
νˆ(ρ)V (χ) ⊇ νˆ(ρ) · V (ker(¯ν)), we must also have V (χ) ⊇ V (ker(¯ν)), and so [ν] ∈ UA(ξ).
i Therefore, [ν] ∈ UA,c(A)(V (X)), and we are done.
Corollary 3.11.2 ([70, 81]). Let X be a connected CW-complex with finite k-skeleton, and
i n−1 i set n = b1(X). Then Ω1(X) = QP \ P(τ1(W (X))), for all i ≤ k.
i n−1 In particular, Ω1(X) is an open subset of the projective space QP .
3.11.2 The singular set
Recall from §3.10.2 that we measure the discrepancy between the generalized Dwyer–Fried
invariant Ωi (X) and its classical counterpart, Ωi (X), by means of the “singular set,” A A Σi (X) = q Ωi (X)c ∩ Ωi (X). When the group A has rank 1, this set can be expressed A A A more concretely, as follows.
i Proposition 3.11.3. Suppose H is torsion-free, and rank(A) = 1. Then ΣA(X) consists of all [σ] ∈ Γ(H, A) satisfying the following two conditions:
95 i (i) ker(σ) ⊇ ξ, for some ξ ∈ Ξc(A)(V (X)), and
(ii) V (ker(σ)) * V i(X).
Proof. Without loss of generality, we may assume that Γ(H,A) 6= ∅. From Theorem 3.11.1,
i i i we know that ΩA(X) = Γ(H,A) \ U, where U = UA,c(A)(V (X)). It follows that ΣA(X) =
i q(U) ∩ Ω1(X). Now let S be the set of all [σ] ∈ Γ(H, A) satisfying conditions (i) and (ii). It suffices to
i show that q(U) ∩ Ω1(X) = S.
i The inclusion S ⊇ q(U) ∩ Ω1(X) is straightforward. To establish the reverse inclusion,
i let σ : H Z represent an element in S. Condition (ii) implies that [σ] ∈ Ω1(X). Recall that π : A → A is the natural projection. To prove that [σ] ∈ q(U), it is enough to find an epimorphism ν : H A such that π ◦ ν = σ and V (ker(ν)) ∩ ηV (ξ) 6= ∅, where η is a generator of ξ/ξc .
Set d := ord(η). Writing A = Zd1 ⊕ · · · ⊕ Zdk , with d1 | d2 | · · · | dk, we have that d | dk. Denote by ι the embedding of the cyclic group hηi into Hb. Under the correspondence from
(3.4.2), there is a map ˇι: H Zd; clearly, this map factors as the composite
f1 n−1 id ⊕κ1 n−1 (0 κ2) H / Z ⊕ Z / / Z ⊕ Zdk / / Zd ,
n for some isomorphism f1 : H → Z , where κ1 and κ2 are the canonical projections.
Recall we are assuming Γ(H,A) 6= ∅; thus, there is an epimorphism γ : H A. We then have a commuting diagram
γ H / / A
f2 n−1 (0 κ1) Z ⊕ Z / / Zdk n −1 for some isomorphism f2 : H → Z . The composite ν = γ ◦ f2 ◦ f1 : H A, then, is the required epimorphism.
i As we shall see in Examples 3.11.5 and 3.11.6, the singular set ΣA(X) may be non-empty; in fact, as we shall see in Example 3.11.7, this set may even be infinite.
96 3.11.3 A particular case
Perhaps the simplest situation when such a phenomenon may occur is the one when H = Z2
and A = Z ⊕ Z2. In this case, the set
Γ(H/A, A/A) = Epi(Z, Z2)/ Aut(Z2)
is a singleton, and so the map qH : Γ(H,A) → Γ(H, A) is a bijection. In other words, if
2 H1(X, Z) = Z , there is a one-to-one correspondence between regular Z ⊕ Z2-covers of X
1 2 and regular Z-covers of X, both parametrized by the projective line QP = Gr1(Z ). The comparison diagram, then, takes the form
i 1 Ω ⊕ (X) / QP (3.11.2) Z Z2 _ q|Ωi (X) = Z⊕Z2 i 1 Ω1(X) / QP
Let V i(X) ⊂ (C∗)2 be the i-th characteristic variety of X. For each pair (a, b) ∈ Z2,
± ∗ 2 a b consider the (translated) subtori Ta,b = {(t1, t2) ∈ (C ) | t1t2 = ±1}.
2 Proposition 3.11.4. Suppose H = Z and A = Z ⊕ Z2. Then,
i 1 + i − i c ΩA(X) = {(a, b) ∈ QP | T−b,a ⊂ V (X) or T−b,a ⊂ V (X)}
i 1 + i c Ω1(X) = {(a, b) ∈ QP | T−b,a ⊂ V (X)} .
Proof. Let (a, b) ∈ H and let ξ ≤ H be the subgroup generated by (−b, a). Then ξ ∈
i i −b a Ξ1(V (X)) if and only if ξ = ξ and V (X) contains a component of the form t1 t2 = 1,
i i whereas ξ ∈ Ξ2(V (X)) if and only if ξ has index at most 2 in ξ and V (X) contains a
−b a component of the form t1 t2 = ±1. The conclusions follow from Theorem 3.11.1.
In particular, Σi (X) = Ωi (X) \ Ωi (X), and diagram (3.11.2) is not a pull-back Z⊕Z2 1 Z⊕Z2 i a b diagram if and only if V (X) has a component of the form t1t2 + 1 = 0.
A nice class of examples is provided by 2-components links. Let L = (L1,L2) be such a
1 link, with complement XL. As we saw in Example 3.9.4, the characteristic variety V (XL)
97 ∗ 2 consists of the zero-locus in (C ) of the Alexander polynomial ∆L(t1, t2), together with the identity.
2 Example 3.11.5. Let L be the 2-component link denoted 41 in Rolfsen’s tables, and let XL
1 ∗ 2 −1 be its complement. Then ∆L = t1 +t2, and so V (XL) = {1}∪{(t1, t2) ∈ (C ) | t1t2 = −1}. Hence, Ω1(X ) = 1, but Ω1 (X ) = 1\{(1, 1)}. 1 L QP Z⊕Z2 L QP
3.11.4 Another particular case
3 The next simplest situation is the one when H = Z and A = Z ⊕ Z2. In this case, the set
2 ∗ Γ(H/A, A/A) = Epi(Z , Z2)/ Aut(Z2) = (Z2 ⊕ Z2)
3 consists of 3 elements. In other words, if H1(X, Z) = Z , there is a three-to-one correspon-
dence between the regular Z ⊕ Z2-covers of X and the regular Z-covers of X.
Example 3.11.6. Let G be the group from Example 3.9.8. With notation as before, we
1 1 1 have that Ξ1(V (G)) = {ξ2, ξ3} and Ξ2(V (G)) = {ξ1, ξ2, ξ3, ξ4}. Consequently, τ1(V (G)) =
1 1 {(1, 0, 0), (1, 2, 1)}, and so Ω1(G) = QP \{(1, 0, 0), (1, 2, 1)}.
Let A = Z ⊕ Z2. Given an element [ν] ∈ Γ(H,A) such that ker(ν) ⊇ ξ1, we have
1 −1 that [¯ν] = (0, 0, 1) ∈ Ω1(G). Furthermore, q ([¯ν]) consists of 3 representative classes:
0 0 1 0 0 1 0 0 1 ν1 = ( 1 0 0 ), ν2 = ( 1 1 0 ), and ν3 = ( 0 1 0 ). By calculation ν1 ∈ U, but ν2, ν3 ∈/ U. Thus,
1 −1 −1 ΩA(G) = Γ(H,A) \{q (1, 0, 0), q (1, 2, 1), ν1}.
Example 3.11.7. Consider the group from [81, Example 9.7], with presentation
2 −1 G = hx1, x2, x3 | [x1, x2], [x1, x3], x1[x2, x3]x1 [x2, x3]i.
The characteristic variety V 1(G) ⊂ (C∗)3 consists of the origin, together with the translated
∗ 3 1 2 torus {(t1, t2, t3) ∈ (C ) | t1 = −1}; hence, Ω1(G) = QP .
1 2 Let A = Z⊕Z2. The singular set Σ = ΣA(G) consists of those points [(0, b, c)] ∈ QP with ∼ 1 1 −1 b and c coprime; thus, Σ = QP is infinite. Moreover, the restriction q :ΩA(G) \ q (Σ) →
98 1 −1 Ω1(G) \ Σ is three-to-one. On the other hand, if [ν] ∈ q (Σ), then either ν is of the form
0 b c 1 0 b c 1 1 2 3 , in which case [ν] ∈/ ΩA(G), or ν is of the form 0 2 3 , in which case [ν] ∈ ΩA(G).
1 −1 Thus, the restriction q :ΩA(G) ∩ q (Σ) → Σ is one-to-one.
3.12 Translated tori in the characteristic varieties
Throughout this section, we assume all irreducible components of the characteristic varieties under consideration are (possibly translated) algebraic subgroups of the character group, a condition satisfied by large families of spaces.
3.12.1 A refined formula for the Ω-sets
i In Theorem 3.9.5 we gave a general description of the Dwyer–Freed invariants ΩA(X) in terms of the characteristic variety W = V i(X), while in Theorem 3.9.7 we gave a upper bound for those invariants, in terms of certain sets Ξd(W ) and UA(ξ), introduced in Definitions 3.5.1 and 3.6.1, respectively.
We now refine those results in the special case when all components of the characteristic variety are torsion-translated subgroups of the character group. The next theorem shows that inclusion (3.9.6) from Theorem 3.9.7 holds as equality in this case, with the union of all sets UA,d(W ) with d ≥ 1 replaced by a single constituent UA,c(W ), for some integer c depending only on W (and not on A).
Theorem 3.12.1. Let X be a connected CW-complex with finite k-skeleton. Set H =
i H1(X, Z), and fix a degree i ≤ k. Suppose V (X) is a union of torsion-translated subgroups of Hb. There is then an integer c > 0 such that, for every abelian group A,
i i ΩA(X) = Γ(H,A) \ UA,c(V (X)). (3.12.1)
i Ss Proof. By assumption, V (X) = j=1 ηjV (ξj), for some subgroups ξj ≤ H and torsion
99 elements ηj ∈ Hb. In the special case when s = 1, the required equality is proved in Theorem
3.7.7, with c = ord(η1) · c(ξ1/ξ1). The general case follows from a similar argument, with c replaced by the lowest common
multiple of ord(η1) · c(ξ1/ξ1),..., ord(ηs) · c(ξs/ξs).
3.12.2 Another formula for the Ω-sets
i We now present an alternate formula for computing the sets ΩA(X) in the case when the i-th characteristic variety of X is a union of torsion-translated subgroups of the character group.
Although somewhat similar in spirit to Theorem 3.12.1, the next theorem uses different
ingredients to express the answer.
Theorem 3.12.2. Let X be a connected CW-complex with finite k-skeleton. Suppose there
i Ss is a degree i ≤ k such that V (X) = j=1 ηjV (ξj), where ξ1, . . . , ξs are subgroups of H =
H1(X, Z), and η1, . . . , ηs are torsion elements in Hb. Then, for each abelian group A, s ! i \ c ΩA(X) = σA(ξj) ∪ [ν] ∈ Γ(H,A) | (hηji) + ker(ν) ∩ ξj . (3.12.2) j=1 Proof. The result follows from Theorems 3.9.5 and 3.7.8, as well as formula (3.7.4).
The simplest situation in which the above theorem applies is that in which there are no
translation factors in the subgroups comprising the characteristic variety.
i Corollary 3.12.3. Suppose V (X) = V (ξ1) ∪ · · · ∪ V (ξs) is a union of algebraic subgroups of Hb. Then s i [ −1 ΩA(X) = Γ(H,A) \ q (σA(ξj)). (3.12.3) j=1 In particular if Xν is a free abelian cover with finite Betti numbers up to degree i, then any
finite regular abelian cover of Xν has the same finiteness property.
Proof. By Theorem 3.12.2,
s i [ ΩA(X) = Γ(H,A) \ σA(ξj). (3.12.4) j=1
100 Indeed, for each j we have ηj = 1, and thus {[ν] | (hηji) + ker(ν) ∩ ξj} is the empty set. Applying now Proposition 3.6.4 ends the proof.
3.12.3 Toric complexes
We illustrate the above corollary with a class of spaces arising in toric topology. Let L be
a simplicial complex with n vertices, and let T n = S1 × · · · × S1 be the n-torus, with the
standard product cell decomposition. The toric complex associated to L, denoted TL, is the union of all subcomplexes of the form
σ n T = {(x1, . . . , xn) ∈ T | xi = ∗ if i∈ / σ}, (3.12.5)
1 where σ runs through the simplices of L, and ∗ is the (unique) 0-cell of S . Clearly, TL is a connected CW-complex, with unique 0-cell corresponding to the empty simplex ∅. The
fundamental group of TL is the right-angled Artin group
GL = hv ∈ V | vw = wv if {v, w} ∈ Ei, (3.12.6)
where V and E denote the 0-cells and 1-cells of L. Furthermore, a classifying space for π1(TL) is the toric complex T∆(L), where ∆(L) is the flag complex associated to L.
n Evidently, H1(TL, Z) = Z ; thus, we may identify the character group of π1(TL) with the algebraic torus (C∗)V := (C∗)n. For any subset W ⊆ V, let (C∗)W ⊆ (C∗)V be the
∗ ∅ ∗ W corresponding subtorus; in particular, (C ) = {1}. Note that (C ) = V (ξW), where ξW
n is the sublattice of Z spanned by the basis vectors {ei | i∈ / W}. From [69], we have the following description of the characteristic varieties of our toric complex:
i [ V (TL) = V (ξW), (3.12.7) W where the union is taken over all subsets W ⊆ V for which there is a simplex σ ∈ LV\W ˜ and an index j ≤ i such that Hj−1−|σ|(lkLW (σ), C) 6= 0. Here, LW denotes the subcomplex induced by L on W, and lkK (σ) denotes the link of a simplex σ ∈ L in a subcomplex K ⊆ L.
101 From the above, we see that the assumptions from Corollary 3.12.3 are true for the
classifying space of a right-angled Artin group, and, in fact, for any toric complex. Hence,
we obtain the following corollaries.
Corollary 3.12.4. Let TL be a toric complex. Then,
i [ −1 ΩA(TL) = Γ(H,A) \ q (σA(ξW)) W
n where each σA(ξW) ⊆ Grn−r(Q ) is the special Schubert variety corresponding to the coor-
dinate plane ξW ⊗ Q, and the union is taken over all subsets W ⊆ V for which there is a ˜ simplex σ ∈ LV\W and an index j ≤ i such that Hj−1−|σ|(lkLW (σ), C) 6= 0.
ν¯ Corollary 3.12.5. Let TL be a toric complex. If TL is a free abelian cover with finite Betti
ν¯ numbers up to some degree i, then all regular, finite abelian covers of TL also have finite Betti numbers up to degree i.
3.12.4 When the translation order is coprime to |Tors(A)|
We now return to the general situation, where the characteristic variety is a union of torsion-
translated subgroups. In the next proposition, we identify a condition on the order of
translation of these subgroups, insuring that diagram (3.3.7) is a pull-back diagram.
As usual, let A be a quotient of H = H1(X, Z), and let A = A/ Tors(A) be its maximal torsion-free quotient. Recall that the canonical projection, q : Γ(H,A) → Γ(H, A), restricts
i i to a map q| i :Ω (X) → Ω (X), and that resulting commuting square is a pull-back ΩA(X) A A i diagram if and only if ΩA(X) is the full pre-image of q. Using Proposition 3.7.13 and Theorem 3.9.5, we obtain the following consequence.
i S Proposition 3.12.6. Suppose the characteristic variety V (X) is of the form j ρjTj, where
each Tj ⊂ Hb is an algebraic subgroup, and each ρ¯j ∈ H/Tb j has finite order, coprime to the order of Tors(A). Then Ωi (X) = q−1 Ωi (X) . A A
102 Here is an application. As usual, let X be a connected CW-complex with finite k-skeleton.
∗ n Assume H = H1(X, Z) has no torsion, and identify the character torus Hb with (C ) , where
n = b1(X).
Corollary 3.12.7. Suppose that, for some i ≤ k, there is an (n − 1)-dimensional subspace
1 i S L ⊆ H (X; Q) such that V (X) = ( α ραT ) ∪ Z, where Z is a finite set, T = exp(L ⊗ C)
∗ n and ρα is a torsion element in (C ) of order coprime to that of Tors(A), for each α. Let r = rank A. Then, Γ(H,A), if r = 1; i −1 ΩA(X) = q (Grr(L)), if 1 < r < n; ∅, if r ≥ n.
Proof. From [81, Proposition 9.6], we have that n−1, if r = 1; QP i Ωr(X) = Grr(L), if 1 < r < n; ∅, if r ≥ n.
i −1 i On the other hand, Proposition 3.12.6 shows that ΩA(X) = q Ωr(X) , and so the desired conclusion follows.
3.12.5 When the translation order divides |Tors(A)|
To conclude this section, we give some sufficient conditions on the groups H and A, and on
the order of translation of the subgroups comprising V i(X), insuring that diagram (3.3.7) is
not a pull-back diagram.
Proposition 3.7.16 and Theorem 3.9.5 yield the following immediate application.
Corollary 3.12.8. Let X be a connected CW-complex with finite k-skeleton. Assume that
(i) The group H = H1(X, Z) is torsion-free, and A is a quotient of H.
103 (ii) There is a degree i ≤ k such that the positive-dimensional components of V i(X) form
a torsion-translated subgroup ρT inside Hb.
(iii) The rank of A is less than the rank of H, and ord(ρ) divides c(A). Then Ωi (X) q−1 Ωi (X) . A $ A
Note that the hypothesis of Corollary 3.12.8 are satisfied in Examples 3.11.5 and 3.11.7,
thus explaining why, in both cases, diagram (3.3.7) is not a pull-back diagram. Here is one
more situation when that happens.
i Corollary 3.12.9. Suppose V (X) = ρ1T1 ∪ · · · ∪ ρsTs, with each Tj an algebraic subgroup of Hb and each ρj a torsion element in Hb \ Tj. Furthermore, suppose that
0 (i) The identity component of T1 is not contained in T = T2 ··· Ts (internal product in
Hb).
(ii) The order of ρ1 divides c(A).
(iii) rank A < rank H − dim T 0. Then Ωi (X) q−1 Ωi (X) . A $ A
0 00 0 0 00 0 00 Proof. Split H as a direct sum, H ⊕ H , so that Hb = T and Hb = H/Tb . Let p: H H be the canonical projection, and letp ˆ: Hb 00 → Hb be the induced morphism. By assumption
−1 00 (1), we have thatp ˆ (T1) is a positive-dimensional algebraic subgroup of Hb . Thus, the positive-dimensional components of W = V i(X) ∩ Hb 00 form a torsion-translated subgroup of
00 −1 00 Hb , namely,p ˆ (ρ1T1). Moreover, assumptions (2) and (iii) imply that rank A < rank H
−1 and ord(ˆp (ρ1)) divides c(A).
Applying now Proposition 3.7.16 to the algebraic group Hb 00 and to the subvariety W yields
00 −1 −1 an epimorphism µ: H A such that im(µ¯ˆ) ∩ pˆ (ρ1T1) is finite, and im(ˆµ) ∩ pˆ (ρ1T1) is infinite. Setting ν = µ ◦ p, we see that im(ˆν) = im(ˆµ). Thus, im(ν¯ˆ) ∩ V i(X) is finite, while
im(ˆν) ∩ V i(X) is infinite.
104 3.13 Quasi-projective varieties
We conclude with a discussion of the generalized Dwyer–Fried sets of smooth, quasi-projec-
tive varieties.
3.13.1 Characteristic varieties
A space X is said to be a (smooth) quasi-projective variety if there is a smooth, complex
projective variety X and a normal-crossings divisor D such that X = X \ D. For instance,
d X could be the complement of an algebraic hypersurface in CP . The structure of the characteristic varieties of such spaces was determined through the
work of Beauville, Green and Lazarsfeld, Simpson, Campana, and Arapura in the 1990s, and
further refined in recent years. We summarize these results, essentially in the form proved
by Arapura.
Theorem 3.13.1 ([3]). Let X = X \ D, where X is a smooth, projective variety and D is
a normal-crossings divisor.
i (i) If either D = ∅ or b1(X) = 0, then each characteristic variety V (X) is a finite union
of unitary translates of algebraic subtori of T = H1(X, C∗).
(ii) In degree i = 1, the condition that b1(X) = 0 if D 6= ∅ may be lifted. Furthermore, each positive-dimensional component of V 1(X) is of the form ρ · S, where S is an algebraic
subtorus of T , and ρ is a torsion character.
For instance, if C is a connected, smooth complex curve of negative Euler characteristic, then V 1(C) is the full character group H1(C, C∗). For an arbitrary smooth, quasi-projective variety X, each positive-dimensional component of V 1(X) arises by pullback along a suitable
orbifold fibration (or, pencil). More precisely, if ρ · S is such a component, then S =
f ∗(H1(C, C∗)), for some curve C, and some holomorphic, surjective map f : X → C with connected generic fiber.
105 Using this interpretation, together with recent work of Dimca, Artal-Bartolo, Cogolludo, and Matei (as recounted in [81]), we can describe the variety V 1(X), as follows.
Theorem 3.13.2. Let X be a smooth, quasi-projective variety. Then
[ [ V 1(X) = V (ξ) ∪ V (ξ) \ V (ξ) ∪ Z, ξ∈Λ ξ∈Λ0 where Z is a finite subset of T = H1(X, C∗), and Λ and Λ0 are certain (finite) collections of subgroups of H = H1(X, Z).
3.13.2 Dwyer–Fried sets
Using now Theorem 3.13.2 (and keeping the notation therein), Proposition 3.7.12 yields the following structural result for the degree 1 Dwyer–Fried sets of a smooth, quasi-projective variety.
Theorem 3.13.3. Let A be a quotient of H. Then ! 1 [ −1 [ −1 ΩA(X) = Γ(H,A) \ q σA(ξ) ∪ q σA(ξ) ∩ θA(ξ) . ξ∈Λ ξ∈Λ0 In certain situations, more can be said. For instance, Proposition 3.12.6 yields the fol- lowing corollary.
Corollary 3.13.4. Suppose the order of ξ/ξ is coprime to c(A), for each ξ ∈ Λ0. Then
Ω1 (X) = q−1 Ω1 (X). A A
Similarly, Corollary 3.12.9 has the following consequence.
Corollary 3.13.5. Suppose H is torsion-free, and there is a subgroup χ ∈ Λ0 such that
0 T (i) V (χ) is not contained in T := V ( ξ∈Λ∪Λ0\{χ} ξ).
(ii) There is a non-zero element in χ/χ whose order divides c(A).
(iii) rank A < codim T 0.
106 Then Ω1 (X) q−1 Ω1 (X) . A $ A
For the remainder of this section, we shall give some concrete examples of quasi-projective
1 manifolds X for which the computation of the sets ΩA(X) can be carried out explicitly.
3.13.3 Brieskorn manifolds
n Let (a1, . . . , an) be an n-tuple of integers, with aj ≥ 2. Consider the variety X in C defined
a1 an by the equations cj1x1 +···+cjnxn = 0, for 1 ≤ j ≤ n−2. Assuming all maximal minors of the matrix cjk are non-zero, X is a quasi-homogeneous surface, with an isolated singularity at 0.
∗ ∗ ∗ The space X admits a good C -action. Set X = X \ 0, and let p: X C be the corresponding projection onto a smooth projective curve. Then p∗ : H1(C, C) → H1(X∗, C)
∗ is an isomorphism, the torsion subgroup of H = H1(X , Z) coincides with the kernel of
∗ p∗ : H1(X , Z) → H1(C, Z).
By definition, the Brieskorn manifold M = Σ(a1, . . . , an) is the link of the quasi-ho- mogenous singularity (X, 0). As such, M is a closed, smooth, oriented 3-manifold homotopy equivalent to X∗. Put
l = lcm(a1, . . . , an), lj = lcm(a1,..., abj, . . . , an), a = a1 ··· an.
The S1-equivalent homeomorphism type of M is determined by the following Seifert invari- ants associated to the projection p|M : M → C:
• The exceptional orbit data, (s1(α1, β1), ··· , sn(αn, βn)), with αj = l/lj, βjl ≡ aj
mod αj and sj = a/(ajlj), where sj = (αjβj) means (αjβj) repeated sj times, un-
less αj = 1, in which case sj = (αjβj) is to be removed from the list.
1 Pn • The genus of the base curve, given by g = 2 2 + (n − 2)a/l − j=1 sj .
• The (rational) Euler number of the Seifert fibration, given by e = −a/l2.
107 s1 sn The group H = H1(M, Z) has rank 2g, and torsion part of order α1 ··· αn · |e|. Identify
∗ 2g the character group Hb with a disjoint union of copies of Hb0 = (C ) , indexed by Tors(H),
s1 sn and set α = α1 ··· αn / lcm(α1, . . . , αn).
Proposition 3.13.6 ([17]). The positive-dimensional components of V 1(M) are as follows:
(i) α − 1 translated copies of Hb0, if g = 1.
(ii) Hb0, together with α − 1 translated copies of Hb0, if g > 1.
Denote the elements in Tors(H) corresponding to the α − 1 translated copies of Hb0 by h1, . . . , hα−1. We then have the following corollaries.
Corollary 3.13.7. Let M = Σ(a1, . . . , an) be a Brieskorn manifold, and let A be a quotient of H = H1(M, Z), with r = rank A.
(i) If g > 1, then Ω1 (M) = Ω1 = ∅. A A
(ii) If g = 1, then Ω1 (M) = Gr ( 2g), while A r Q
1 ΩA(M) = [ν] ∈ Γ(H,A) | ν(hi) = 0 for i = 1, . . . , α − 1 .
Corollary 3.13.8. Suppose g = 1 and α > 1. Then Ω1 (M) = {pt}, yet Ω1 (M) = ∅; that H H fab ab is, b1(M ) < ∞, yet b1(M ) = ∞.
Example 3.13.9. Consider the Brieskorn manifold M = Σ(2, 4, 8). Using the algorithm described by Milnor in [61], we see that the fundamental group of M has presentation
2 2 2 2 2 −1 −1 2 G = hx1, x2, x3 | x1x3 = x3x1, x2x3 = x3x2, x3(x3x1x2x1 x2 ) = 1i,
2 ∗ 2 while its abelianization is H = Z ⊕ Z4. Identifying Hb = (C ) × {±1, ±i}, we find that
V 1(M) = {1} ∪ (C∗)2 × {−1}. (The positive-dimensional component in V 1(M) arises from an elliptic orbifold fibration X → Σ1 with two multiple fibers, each of multiplicity 2.) By
fab ab Proposition 3.10.6, then, b1(M ) < ∞, while b1(M ) = ∞.
108 Now take A = Z ⊕ Z4. Applying Corollary 3.13.7 (with g = 1 and α = 2), we conclude that Ω1 (M) = 1, while Ω1 (M) consists of two copies of Γ( 2,A), naturally embedded in A QP A Z Γ(H,A).
3.13.4 The Catanese–Ciliberto–Mendes Lopes surface
We now give an example of a smooth, complex projective variety M for which the generalized
Dwyer–Fried sets exhibit the kind of subtle behavior predicted by Corollary 3.13.5.
Let C1 be a (smooth, complex) curve of genus 2 with an elliptic involution σ1 and C2
a curve of genus 3 with a free involution σ2. Then Σ1 = C1/σ1 is a curve of genus 1, and
Σ2 = C2/σ2 is a curve of genus 2. The group Z2 acts freely on the product C1 × C2 via the
involution σ1 ×σ2; let M be the quotient surface. This variety, whose construction goes back to Catanese, Ciliberto, and Mendes Lopes [11], is a minimal surface of general type with
2 pg(M) = q(M) = 3 and KM = 8.
The projection C1 × C2 → C1 descends to an orbifold fibration f1 : M → Σ1 with two
multiple fibers, each of multiplicity 2, while the projection C1 × C2 → C2 descends to a
holomorphic fibration f2 : M → Σ2. It is readily seen that H = H1(M, Z) is isomorphic
6 to Z ; fix a basis e1, . . . , e6 for this group. A computation detailed in [81] shows that the
characteristic variety V 1(M) ⊂ (C∗)6 has two components, corresponding to the above two pencils; more precisely,
1 V (M) = V (ξ1) ∪ V (ξ2) \ V (ξ2) , (3.13.1)
where ξ1 = span{e1, e2} and ξ2 = span{2e3, e4, e5, e6}.
Now suppose A is a quotient of H = Z6, and let q : Γ(H,A) → Γ(H, A) be the canonical projection. By Theorem 3.13.3,
1 −1 −1 ΩA(M) = Γ(H,A) \ q (σA(ξ1)) ∪ q (σA(ξ2)) ∩ θA(ξ2) . (3.13.2)
Let us describe explicitly this set in a concrete situation.
109 5 Example 3.13.10. Let A = Z ⊕ Z2, and identify A = Z and Γ(H, Z) = QP . The fiber of q
5 ∗ 0 is the set Γ = (Z2) . Given an epimorphism ν : H → Z ⊕ Z2, letν ¯: H → Z and ν : H → Z2 be the composites of ν with the projections on the respective factors. The terms on the right-side of (3.13.2) are as follows:
3 ∨ • σZ(ξ1) is the projective subspace QP = P((H/ξ1) ⊗ Q) spanned by e3, . . . , e6.
1 ∨ • σZ(ξ2) is the projective line QP = P((H/ξ2) ⊗ Q) spanned by e1 and e2.
−1 1 0 • q (QP ) ∩ θA(ξ2) consists of those [ν] satisfying ν(e3) = ··· = ν(e6) = 0, and ν (e3) =
0 0 0 1, ν (e4) = ν (e5) = ν (e6) = 0.
This completes the description of the set Ω1 (M). Clearly, Ω1(M) = 5 \ 3. Z⊕Z2 1 QP QP −1 1 1 Furthermore, note that the restriction map q (QP ) ∩ θA(ξ2) → QP is a 2-to-1 surjection. Thus, the restriction map Ω1 (M) → Ω1(M) is not a set fibration: the fiber over Ω1(M) \ Z⊕Z2 1 1 1 1 QP has cardinality 31, while the fiber over QP has cardinality 29.
110 Chapter 4
Algebraic elliptic cohomology theory
and flops
4.1 Oriented cohomology
4.1.1 Oriented cohomology theories and algebraic cobordism
In this subsection we collect preliminary notions and results we will use. The main goal is
to fix the notations and conventions.
Recall that a formal group law over a commutative ring R with unit is an element
F (u, v) ∈ R[[u, v]] satisfying the following conditions
1. F (u, 0) = u, F (0, v) = v;
2. F (F (u, v), w) = F (u, F (v, w));
3. F (u, v) = F (v, u).
Lazard pointed out the existence of a universal formal group law (Laz,FLaz). He also proved that the ring Laz, called Lazard ring, is a polynomial ring with integral coefficients on a
111 countable set of variables (see [57]). That is, for any formal group law (R,F ) over any ring
R, there exists a unique ring homomorphism r : Laz → R such that F = r(FLaz). Now let F (u, v) ∈ R[[u, v]] be an arbitrary formal group law over a commutative ring R; we will always assume that R is graded and that, giving both u and v degree 1, F (u, v) is homogeneous of degree 0. As the Lazard ring is generated as a Z-algebra by the coefficients of the universal formal group law, FLaz ∈ Laz[[u, v]] this convention gives Laz a uniquely defined grading, concentrated in non-positive degrees, and the classifying homomorphism
φF : Laz → R associated to F preserves the grading.
Let k be a field, and Smk be the category of smooth, quasi-projective schemes over k. Let Comm denote the category of commutative, graded rings with unit. The following definition can be found in [54].
Definition 4.1.1. An oriented cohomology theory on Smk is given by the following data.
∗ op D1 An assignment A sending any X ∈ Smk to an object in Comm.
D2 For any smooth morphism f : Y → X in Smk, a ring homomorphism
f ∗ : A∗(X) → A∗(Y ).
D3 For any projective morphism f : Y → X in Smk of relative codimension d, a homomor- phism of graded A∗(X)-modules
∗ ∗+d f∗ : A (Y ) → A (X).
These data satisfy the axioms: functoriality for f ∗ with respect to arbitrary morphisms in
Smk, functoriality for f∗ with respect to projective morphisms, base change for transverse morphisms, a projective bundle formula, and extended homotopy property. We should men- tion that, for L → X a line bundle over some X ∈ Smk, the first Chern class of L in the
A A ∗ theory A, denote c1 (L), is defined by the formula c1 (L) := s s∗(1X ), where s : X → L is
112 0 ∗ the zero section and 1X ∈ A (X) is the unit in A (X). We often write simply c1(L) if the theory A is understood.
For an oriented cohomology theory A∗, there is a unique power series
F (u, v) ∈ A∗(k)[[u, v]] satisfying
FA(c1(L), c1(M)) = c1(L ⊗ M)
for each pair line bundles L and M on a scheme X ∈ Smk; moreover, F defines a formal group law over the ring A∗(k)[54, Lemma 1.1.3]. As we will often consider cohomology theories with rational coefficients, we denote the ring A∗(X) ⊗ simply by A∗ (X). Q Q
∗ For any X ∈ Smk, Levine and Morel constructed a commutative graded ring Ω (X) as follows. Let M(X) be the set of isomorphism classes of projective morphisms f : Y → X
∗ with Y ∈ Smk. M(X) becomes a monoid under the disjoint union. Let M+(X) be its group completion, graded by the relative codimension of the map f : Y → X. Then Ω∗(X) is
∗ constructed as a quotient of M+(X)[54, Lemma 2.5.11]. The following is a resum´eof some of the main results from [54].
Theorem 4.1.2 (Levine and Morel). Assume the base field k has characteristic zero.
(i) The assignment
X 7→ Ω∗(X)
extends to define an oriented cohomology theory on Smk, called algebraic cobordism.
∗ (ii) Ω is the universal oriented cohomology theory on Smk.
∗ (iii) The canonical ring homomorphism Laz → Ω (k) induced by the formal group law FΩ of the algebraic cobordism Ω∗ is an isomorphism.
113 When the base field k has positive characteristic, the construction of Ω∗(X) described above is not known to give a oriented cohomology theory. However, the construction of
∗ Ω (X) in [54] leads to a notion of a “universal oriented Borel-Moore Laz-functor on Smk of geometric type” (see [54, Definition 2.2.1]; here we index by codimension rather than by dimension), and Ω∗ is the universal such theory [54, Theorem 2.4.13]. The structures enjoyed by such a theory A∗ are:
1. (Projective pushforward) For f : Y → X a projective morphism in Smk of relative
∗ ∗−d dimension d, there is a graded homomorphism f∗ : A (Y ) → A (X).
2. (Smooth pullback) For a smooth morphism f : Y → X in Smk, there is a graded homomorphism f ∗ : A∗(X) → A∗(Y ).
3. (1st Chern class operators) For each line bundle L → X, X ∈ Smk, there is a graded
∗ ∗+1 homomorphismc ˜1(L): A (X) → A (X). For line bundles L, M on X, the operators
c˜1(L) andc ˜1(M) commute.
∗ 4. (External products) For X,Y ∈ Smk, there is a graded homomorphism A (X) ⊗Z
∗ ∗ A (Y ) → A (X ×k Y ), which is commutative and associative in the obvious sense.
0 ∗ There is an element 1 ∈ A (k) which, together with the external product A (k) ⊗Z A∗(k) → A∗(k), makes A∗(k) into a commutative graded ring with unit.
5. (Fundamental class) For X ∈ Smk with structure morphism p : X → Spec k, we denote
∗ p (1) by 1X , and for L → X a line bundle, write c1(L) forc ˜1(L)(1X ).
6. (Dimension axiom) For each collection of line bundles L1,...,Lr on X with r > dimk X, Qr one has i=1 c˜1(Li)(1X ) = 0
∗ 7. (Formal group law) There is a homomorphism φA : Laz → A (k) of graded rings, such
∗ that, letting FA(u, v) ∈ A (k)[[u, v]] be the image of the universal formal group law
114 with respect to φA, for each X ∈ Smk and each pair of line bundles L, M on X, we have
c1(L ⊗ M) = FA(˜c1(L), c˜1(M))(1X ).
These all satisfy a number of compatibilities, detailed in [54, §2.1, 2.2].
An oriented cohomology theory has a canonical structure of an oriented Borel-Moore
Laz-functor on Smk of geometric type [54, Proposition 5.2.1, Proposition 5.2.6], where the
Chern class operatorc ˜1(L) is multiplication by the 1st Chern class c1(L). In particular, each oriented cohomology theory A∗ admits a canonical classifying map
∗ ∗ ΘA :Ω → A , (4.1.1)
∗ which is compatible with all push-forward maps f∗ for projective f, all pull-back maps f for smooth f, first Chern classes and first Chern class operators, external products and the
∗ formal group law. Explicitly, for a generator [f : Y → X] ∈ M+(X), ΘA([f : Y → X]) =
A A A ∗ 0 f∗ (1Y ), where f∗ is the pushforward in the theory A and 1A ∈ A (Y ) is the unit.
4.1.2 Motivic oriented cohomology theories
In positive characteristic, the universal oriented cohomology theory is not available; we use instead an approach via motivic homotopy theory.
Let SH(k) be the motivic stable homotopy category of P1-spectra [63, 71, 96]. SH(k) is a triangulated tensor category with tensor product (E, F) 7→ E ∧ F and unit the motivic sphere spectrum Sk. There is a functor
∞ Σ 1 − : Sm → SH(k) P + k called infinite 1-suspension; for example = Σ∞ Spec k . We have as well the auto- P Sk P1 +
1 equivalences ΣS and ΣGm , and the translation in the triangulated structure is given by
ΣS1 .
115 n,m op An object E of SH(k) and integers n, m define a functor E : Smk → Ab by
n,m ∞ n−m m E (X) := Hom (Σ 1 X , Σ Σ E). SH(k) P + S1 Gm
An object E together with morphisms µ : E ∧ E → E (multiplication) and 1 : Sk → E
∗,∗ (unit) define a ring cohomology theory on Smk if they make the bi-graded group E (X) :=
n,m ⊕n,m∈ZE (X) into a bi-graded ring which is graded commutative with respect to the first grading and commutative with respect to the second one.
Given a ring cohomology theory as above, an element ϑ ∈ E2,1(P∞/0) is called an
2,1 1 1 ∞ orientation if the restriction of ϑ to ϑ|P1/0 ∈ E (P /0) (via the embedding P → P ,
0,0 (x0 : x1) 7→ (x0 : x1 : 0 : 0 ...)) agrees with the image of 1 ∈ E (k) under the suspension
2,1 1 ∼ 0,0 n isomorphism E (P /0) = E (k). Here 0 := (1 : 0 : ... : 0) ∈ P , including n = ∞. In [75, Theorem 2.15] it is shown how an orientation ϑ for a ring cohomology theory E on
a,b a−2d,b−d Smk gives rise to functorial pushforward maps f∗ : E (Y ) → E (X) for each projective morphism f : Y → X in Smk, where d is the relative dimension of f (d := dimk Y − dimk X in case X and Y are irreducible).
Definition 4.1.3. A motivic oriented cohomology theory on Smk is an object E ∈ SH(k) together with morphisms µ : E ∧ E → E, 1 : Sk → E defining a ring cohomology theory, plus an orientation ϑ ∈ E2,1(P∞/0).
Remarks 4.1.4.
1. The notion of a motivic oriented cohomology theory on Smk is referred to as an “oriented ring cohomology theory” in [72]. We find this too similar to the term “oriented cohomology
theory”, hence our relabelling.
2. The algebraic cobordism P1-spectrum MGL ∈ SH(k) is the universal motivic oriented cohomology theory. See [71, 96] for the construction of MGL and [72] for the proof of
universality.
It follows from the results of [75] that, if (E, µ, 1, ϑ) is a motivic oriented cohomology
∗ 2∗,∗ theory on Smk, then the contravariant functor X 7→ E (X) := E (X) from Smk to Comm,
116 together with the maps f∗ for projective morphisms f : Y → X in Smk, defines an oriented
∗ ∗ cohomology theory on Smk, which we denote by E . We call E the oriented cohomology
theory on Smk represented by the motivic oriented cohomology theory (E, µ, 1, ϑ) (or just E
for short). In particular, for each motivic oriented cohomology theory E on Smk, there is a
∗ canonical homomorphism φE : Laz → E (k) classifying the formal group law of the oriented cohomology theory E∗.
Remark 4.1.5. In general, an oriented cohomology theory on Smk is not always represented by a motivic oriented theory (see [42]).
It follows from a theorem of Hopkins-Morel, recently established in detail by Hoyois
[41], that for k of characteristic zero, the ring homomorphism Laz → MGL(k) classifying the formal group law for MGL∗ is an isomorphism. When k has characteristic p > 0, it is shown in [41] that after inverting p, the classifying map Laz[1/p] → MGL∗(k)[1/p] is an
isomorphism. Conjecturally, for any field k, the classifying map is an isomorphism, but at present, this is not known.
4.1.3 Specialization of the formal group law
From the oriented cohomology theory MGL∗, and a given formal group law F (u, v) ∈ R[[u, v]]
such that the exponential characteristic p of k is invertible in R, we may construct an oriented
∗ ∗ cohomology theory R on Smk with R (k) = R and formal group law F as follows.
Take X ∈ Smk and let pX : X → Spec k be the structure morphism. The classifying
∗ ∗ ∗ map φMGL : Laz → MGL (k) composed with pX defines the ring homomorphism pX ◦ φMGL :
∗ Laz → MGL (X). Using the classifying homomorphism φF : Laz → R, we may form the
∗ ∗ tensor product ring R (X) := MGL (X)⊗LazR. Since p is invertible in R and φMGL[1/p] is an isomorphism, it follows that the canonical map R → R∗(k) is an isomorphism. Extending the
pull-back and push-forward maps for the theory MGL∗ define the pull-back and push-forward
∗ maps for the theory R . Since the functor − ⊗Laz R is additive and preserves isomorphisms,
117 it follows easily that the assigment X 7→ R∗(X) with pull-back and push-forward maps as
described above defines an oriented cohomology theory on Smk. Similarly, it is easy to see
R∗ MGL ∗ that for L → X a line bundle, c1 (L) is just the image of c1 (L) in R (X), and therefore the formal group law for R∗ is equal to F (u, v) ∈ R[[u, v]]. This gives us the following result:
Lemma 4.1.6. Let k be a perfect field, F ∈ R[[u, v]] a formal group law over a commu-
tative (graded) ring R. Assume that the exponential characteristic of k is invertible in R.
∗ ∗ ∗ Then there is an oriented cohomology theory R on Smk with R (X) = MGL (X) ⊗Laz R. Moreover, R∗(k) = R and R∗ has formal group law F . Finally, if the characteristic of k
∗ is zero, then R is the universal oriented cohomology theory on Smk with formal group law F ∈ R[[u, v]].
Proof. We have proved everything except for the last statement. This follows by noting
that the isomorphism of oriented cohomology theories Θ : Ω∗ → MGL∗ shows that R∗
∗ is isomorphic to the oriented cohomology theory X 7→ Ω (X) ⊗Laz R. The fact that this
latter theory is the universal oriented cohomology theory on Smk with formal group law F ∈ R[[u, v]] follows immediately from the fact that Ω∗ is the universal oriented cohomology
theory on Smk.
There is a nice formula for push-forward in the cohomology theory MGL∗, in the case of
the structure morphism of a projective space bundle.
Theorem 4.1.7 (Quillen). Let X be a smooth quasi-projective variety, V be some n-dimensional
vector bundle on X, and π : PX (V ) → X be the corresponding projective bundle. Let f(t) ∈ MGL∗(X)[[t]]. Then,
X f(−Ωλi) π∗(f(c1(O(1)))) = , (4.1.2) Q (λ − λ ) i j6=i j Ω i where λi are the Chern roots of V , and x +Ω y := FΩ(x, y), where FΩ is the formal group law of the cobordism theory MGL∗.
118 A proof of this theorem, in the context of complex cobordism, can be found in [95], page
50. The proof goes through word for word in our setting, so we will not repeat it here.
Clearly, for any formal group law (F,R) such that the exponential characteristic is invertible
in R, the same formula as above is valid for the push-forward in the theory R∗.
4.1.4 Landweber exactness
We now describe a sufficient condition for the theory X 7→ R∗(X) to arise from a motivic oriented cohomology theory; this is the well-known condition of Landweber exactness.
For any prime l > 0, we expand the l-series of the this formal group law x +F ··· +F x
P i (summing l copies of x), as i≥1 aix , and for all n ≥ 0 we write vn := aln . In particular we
have v0 = a1 = l.
Definition 4.1.8. The formal group (R,F ) is said to be Landweber exact if for all primes
l and for all integers n, the multiplication map
vn : R/(v0, ··· , vn−1) → R/(v0, ··· , vn−1)
is injective.
Theorem 4.1.9. Let k be a perfect field, F ∈ R[[u, v]] a formal group law. If the formal
group law (R,F ) is Landweber exact and the exponential characteristic of k is invertible in
∗ R, then the oriented cohomology theory X 7→ R (X) on Smk is represented by a motivic oriented cohomology theory.
The classical precursor of this result is due to Landweber [50]; in our setting this result
follows from [66, Theorem 7.3]. We denote the motivic cohomology theory associated to a
Landweber exact formal group law by MGL ⊗LazR and the canonical morphism given by the
universality of MGL by ΘF,R : MGL → MGL ⊗LazR.
119 4.1.5 Exponential and logarithm
Let (R,F ) be the formal group law. A logarithm of the formal group law F is a series
g(u) = u + higher order terms ∈ R[[u]] satisfying the equation
g(F (u, v)) = g(u) + g(v).
Novikov [68] showed that every formal group law with coefficients in a Q-algebra has a logarithm. The functional inverse λ(u) ∈ R[[u]] of the logarithm g(u) is called the exponential
of the formal group law. The expansion of λ(u) takes the form u + higher order terms. With
our grading conventions, if we give u degree one, then the power series g(u) and λ(u) are
P i+1 both homogeneous of degree one. Thus, if we write λ(u) = u + i≥1 τiu , then τi ∈ R has degree −i.
In fact, the formal group law and the exponential power series uniquely determine each
other, assuming that R → RQ is injective. As ring homomorphisms Laz → R are in bijection with formal group laws with coefficients in R, it is noted by Hirzebruch that ring homomor-
phisms Laz → RQ are in one to one correspondence with power series λ as above. For any
formal group law (R,F ) with exponential λ(u) ∈ R ⊗ Q[[u]], the corresponding ring homo- morphism φ : MU2∗ → R∗ (k) given by Quillen’s identification MU2∗ ∼ Laz is called the F Q =
Hirzebruch genus. In terms of algebraic geometry, φF sends the class of smooth projective ir-
Qn ξi reducible variety X of dimensional n to h , [X]i, where ξ1, ··· , ξn are the Chern roots i=1 λ(ξi) ∗ of the tangent bundle TX (for CH (X)) and h−, [X]i means evaluation on the fundamental
u P τ i class of X. Explicitly, if we let Tdτ (u) := λ(u) ∈ R[[u]] and write Tdτ (u) = 1 + n≥0 tdi u ,
τ −i τ with tdi ∈ R , then φF ([X]) = degk(Nn(TX )) · tdn, where Nn(TX ) is the Newton class,
P n Nn(TX ) := j ξj .
P n+1 Example 4.1.10. Let R = Q[b1, b2,...], with bn of degree −n. Let λb(u) = u+ n≥1 bnu
−1 and let λb (u) be the functional inverse. We let Fb(u, v) ∈ R[[u, v]] be the formal group law
−1 −1 λb(λb (u) + λb (v)). One can show (see e.g. [1, theorem 7.8]) that the subring R0 of R
120 generated by the coefficients of Fb is isomorphic to Laz via the homomorphism Laz → R0
classifying Fb; in particular, (Fb,R0) is the universal formal group law.
4.1.6 Twisting a cohomology theory
∗ Let k be an arbitrary perfect field and let R be an oriented cohomology theory on Smk.
Suppose that the coefficient ring R∗(k) is a Q-algebra. There is a twisting construction due to Quillen (see e.g. [54] for a detailed description), which enables one to construct the
oriented cohomology theory R∗ from the theory given by the Chow ring X 7→ CH∗(X).
In this section, we describe the twisting construction and its analog for motivic oriented
cohomology theories.
∗ Q∞ −i Let A be an oriented cohomology theory on Smk and τ = (τi) ∈ i=0 A (k), with P∞ i+1 τ0 = 1. Let λτ (u) = i=0 τiu . The Todd genus Tdτ (t) is by definition the power series
t Tdτ (t) := . λτ (t)
For a vector bundle E on some Y ∈ Smk, the Todd class of a vector bundle E on Y is giving by r Y Tdτ (E) = Tdτ (ξi) i=1 ∗ where ξ1, . . . , ξr are the Chern roots of E in A (Y ). The assignment E 7→ Tdτ (E) is multi- plicative in exact sequences, hence descends to a well-defined homomorphism Tdτ : K0(Y ) → (1 + A∗≥1(Y ))×.
∗ ∗ ∗ Define the twisted oriented cohomology theory Aτ on Smk with Aτ (X) = A (X), for
∗ ∗ X ∈ Smk, and with pull-backs unchanged (fτ = f ). For a projective morphism f : Y → X, we set
τ f∗ := f∗ ◦ Tdτ (Tf ), where
∗ Tf := [TY ] − [f TX ] ∈ K0(Y )
121 is the relative tangent bundle. It is not difficult to show that this does indeed define an
oriented cohomology theory on Smk.
P∞ i+1 Let λτ (u) = i=0 τiu . Then for a line bundle L, the first Chern class in the new
∗ τ cohomology theory Aτ is given by c1(L) := λτ (c1(L)), from which one easily sees that the
∗ formal group law for Aτ is given by
τ −1 −1 FA(u, v) = λτ (FA(λτ (u), λτ (v))).
In particular, if FA is the additive group law, FA(u, v) = u + v, then λτ (u) is the exponential
τ ∗ map for the twisted group law (FA,A (k)). The twisting construction is also available for motivic oriented cohomology theories.
−2i,−i Indeed, let (E, µ, 1, ϑ) be a motivic oriented cohomology theory, and (τi ∈ E (k))i a
sequence of elements, with τ0 = 1.
2,1 ∞ Form the orientation ϑτ ∈ E (P /0),
X i+1 ϑτ := λτ (ϑ) = τiϑ . i≥0
We note that the projective bundle formula implies that ϑm goes to zero in E2,1(PN /0)
2,1 ∞ for m > N, from which it follows that λτ (ϑ) is a well-defined element in E (P /0) =
2,1 N ∗ ←−limE (P /0) and that ϑτ is indeed an orientation. Let τ E be the oriented cohomology
theory corresponding to (E, µ, 1, ϑτ ).
It follows immediately from the definitions that, for L → X a line bundle, X ∈ Smk,
∗ τ E ∗ ∗ the first Chern class c1 (L) in the theory τ E is the same as in the τ-twist of E , namely
∗ ∗ τ E E τ c1 (L) = λτ (c1 (L)) = c1(L). From this it follows easily that the identity map on the graded
∗ ∗ ∗ groups Eτ (X) = E (X) defines an isomorphism of τ E with the twisted oriented cohomology
∗ ∗ theory Eτ ; we will henceforth drop the notation τ E .
Let R be a graded Q-algebra. Define
n −n HR = ⊕ Σ 1 HR n∈Z P
122 where HR−n is the P1 spectrum representing motivic cohomology with coefficients in the
Q vector space R−n. We make HR a motivic oriented cohomology theory by using the orientation induced by that of HZ. Now let (R,F ) be a formal group law, where R is given the grading following our con-
−i −2i,−i ventions and is as above a Q-algebra. Taking (τi ∈ R = HR (k))i to be the sequence such that λτ (u) is the exponential function for (R,F ), we form the twisted motivic oriented cohomology theory HRτ ; by construction HRτ has associated formal group law (R,F ). We note that (R,F ) is Landweber exact, since R is a Q-algebra.
We may also form the motivic oriented cohomology theory MGL ⊗LazR associated to the Landweber exact formal group law (R,F ). As this is the universal motivic oriented cohomol-
ogy theory with group law (R,F ), the classifying map ΘHRτ : MGL → HRτ factors through ¯ ΘF,R : MGL → MGL ⊗LazR, giving the induced classifying map ΘHRτ : MGL ⊗LazR → HRτ , unique up to a phantom map.
¯ Lemma 4.1.11. The map ΘHRτ : MGL ⊗LazR → HRτ induces an isomorphism of bi-
∗,∗ ∗,∗ graded cohomology theories MGL ⊗LazR → HRτ , in particular, we have the isomorphism of associated oriented cohomology theories on Smk
¯ ∗ ∗ ∗ ΘHRτ : R = MGL ⊗LazR → HRτ
¯ Proof. We apply the slice spectral sequence to MGL ⊗LazR and HRτ ; the map ΘHRτ induces a map of spectral sequences. For a P1 spectrum E denote the nth layer in the slice tower for
E as snE. For an abelian group A, it follows from [98] that HA for n = 0 snHA = 0 for n 6= 0.
Also s Σm E = Σn s E. As HR = ⊕ Σn HR−n, it follows that the nth layer in the slice n P1 P1 n−m n∈Z P1 tower for HR is given by s HR = Σn HR−n. By Spitzweck’s computation of the layers in n P1 the slice tower for a Landweber exact theory, we have the same s MGL ⊗ R = Σn HR−n n Laz P1
123 ¯ as well. The maps on the layers of the slice tower induced by ΘHRτ are HQ-module maps (by results of Pelaez [76]), and one knows by a result of Cisinski-Deglise [12, Theorem 16.1.4] that
n −n n −n ∼ −n −n Hom (Σ 1 HR , Σ 1 HR ) Hom (R ,R ) HQ−Mod P P = Q−Vec
In particular, the map s Θ¯ :Σn HR−n → Σn HR−n is determined by the induced map n HRτ P1 P1 after applying the functor H−2n,−n(k, −), that is, on the coefficient rings of the theories
∗,∗ ∗,∗ ¯ MGL ⊗LazR and HRτ . However, by construction, this is the map Θ: R → R induced by the classifying map Laz → R associated to the formal group Fτ (u, v) = λτ (gτ (u) + gτ (v)). As this latter formal group law is equal to F by construction, the map Θ:¯ R → R is the identity map.
Thus ΘHRτ induces an isomorphism of the (strongly convergent) slice spectral sequences, and hence an isomorphism of bi-graded cohomology theories on Smk.
4.1.7 Universality
Since MGL∗ is an oriented cohomology theory, we have the canonical comparison morphism
(4.1.1)
∗ ∗ ΘMGL :Ω (X) → MGL (X).
Relying on the Hopkins-Morel-Hoyois isomorphism Laz ∼= MGL∗(k), Levine has shown that
∗ ∗ for k a field of characteristic zero, ΘMGL :Ω → MGL is an isomorphism [53, theorem 3.1]. Thus, for a field of characteristic zero, MGL∗ is the universal oriented cohomology theory on
Smk. At present, there is no proof of the existence of a universal oriented cohomology theory on Smk if k has positive characteristic. In this subsection, we use the universality of MGL as a motivic oriented cohomology theory plus some tricks with formal group laws to show
∗ that MGLQ is the universal oriented cohomology theory for theories in Q-algebras on Smk. We also show that the two constructions of oriented cohomology theories: construction by specialisation of the formal group law from MGL∗ and construction by extending coefficients
124 ∗ for CH and then twisting, are “equivalent”, assuming the coefficient ring is a Q-algebra.
Lemma 4.1.12. Suppose k has characteristic zero. Then CH∗ is the universal oriented cohomology theory on Smk with formal group law (u+v, Z). If k has characteristic p > 0, then
∗ CHQ is the universal oriented cohomology theory on Smk with formal group law (u + v, Q).
Of course, one would expect that over an arbitrary field, CH∗ is the universal oriented cohomology theory on Smk with formal group law (u + v, Z). This does not seem to be known.
Proof. The case of characteristic zero is proven in [54, Theorem 1.2.2]. In characteristic p > 0,
∗ let A be an oriented cohomology theory with additive formal group law FA(u, v) = u+v and with A∗(k) a Q-algebra. Extend the coefficients in the theory A∗ by a Laurent polynomial ring, forming the theory A∗[t, t−1], with t of degree -1. Then take the twist with respect to the modified exponential function
−1 −tu λt(u) := t (1 − e )
i i ∗ −1 that is, τi := (−1) t /(i + 1)!. A simple computation shows that theory A [t, t ]τ has the multiplicative group law F (u, v) = u+v −tuv, and that the twisted first Chern class is given
A t −1 tc1 (L) by c1(L) = t (1 − e ). One can define the modified Chern character
A −1 ∗ −1 cht : K0[t, t ]Q → A [t, t ]τ , which sends a vector bundle E of rank r to
A t ∨ cht (E) := r − tc1(E ).
For a line bundle L, we have
A A t ∨ tc1 (L)) cht (L) = 1 − tc1(L ) = e .
125 Using the splitting principle, the fact that A∗ has the additive formal group law implies that
−1 A −1 K0[t,t ] cht is a natural transformation of functors to graded Q[t, t ]-algebras. Since c1 (L) = t−1(1 − L−1), we have
A K t cht (c1 (L)) = c1(L)
folr all line bundles L. By Panin’s Riemann-Roch theorem [73, Corollary 1.1.10], this shows
A that cht is a natural transformation of oriented cohomology theories.
We have the Adams operations ψk, k = 1, 2,..., on K0(X), which we extend to Adams
−1 −1 A ∗ −1 operations on K0(X)[t, t ]Q by Q[t, t ]-linearity. Define the operation ψk on A (X)[t, t ]
to be the Q[t, t−1]-linear map which is multiplication by kn on An(X); it is easy to see
A −1 that ψk is a natural Q[t, t ]-algebra homomorphism. As A has the additive group law,
A ⊗k A c1 (L ) = kc1 (L) and thus
A ⊗k A A A A ⊗k tc1 (L ) ktc1 (L) A tc1 (L) A A cht (ψk(L)) = cht (L ) = e = e = ψk (e ) = ψk (cht (L))
for all line bundles L. By the splitting principle, this gives the identity
A A A cht ◦ ψk = ψk ◦ cht .
∗ ∗ A If we take A = CHQ, cht is a modified version of the classical Chern character; thus by Grothendieck’s classical result, the natural transformation
CH∗ Q −1 ∗ −1 cht : K0[t, t ]Q → CHQ[t, t ]τ
is an isomorphism. This gives us the natural transformation of oriented cohomology theories
CH∗ A Q −1 ∗ −1 ∗ −1 cht ◦ (cht ) : CHQ[t, t ]τ → A [t, t ]τ
Twisting back gives us the natural transformation of oriented cohomology theories
CH t ∗ −1 ∗ −1 ϑA : CHQ[t, t ] → A [t, t ].
∗ −1 n m We give CHQ[t, t ] a bi-grading by putting CHQ ·t in bi-degree (n, m), and do the same CH∗ ∗ −1 Q A for A [t, t ]. Since both cht and cht commute with the Adams operations, we have
CH t CH A CH ϑA ◦ ψk = ψk ◦ ϑA
126 CH t and thus ϑA respects the bi-grading. Passing to the respective quotients by the ideal (t−1) gives us the natural transformation of oriented cohomology theories
CH ∗ ∗ ϑA : CHQ → A
∗ ∗ where we now use the original grading on CHQ and A . CH The uniqueness of ϑA follows from Grothendieck-Riemann-Roch. Indeed, as a natural
CH CH A transformation of oriented cohomology theories, ϑA (cn (E)) = cn (E) for all vector bundles
E on X ∈ Smk, and all n. But for irreducible X ∈ Smk, the Grothendieck-Riemann-Roch
∗ theorem implies that CH (X)Q is generated as a Q-vector space by the elements of the form
CH cn (E), E a vector bundle on X, n ≥ 1 an integer, together with the identity element
0 CH 1 ∈ CH (X). Thus ϑA is unique.
∗ We now consider the generic twist CHQ[b]b, where b = {bi}i. We have as well the motivic oriented cohomology theory HQ ∈ SH(k), representing rational motivic cohomology,
∗ 2,1 ∞ H (X, Q(∗)). The orientation vH ∈ HQ (P /0) is given by the sequence of hyperplane
CH 2,1 n 1 n classes c1 (OPn (1)) ∈ HQ (P ) = CH (P )Q. P n+1 We may form the generic twist HQ[b]b by taking the orientation λb(vH ) := n≥0 bnvH .
Proposition 4.1.13. 1. Let k be a field of characteristic zero. Then MGL∗ is the uni-
versal oriented cohomology theory on Smk.
∗ 2. Let k be a perfect field of characteristic p > 0. Then MGLQ is the universal oriented
cohomology theory in Q-algebras on Smk.
P n+1 3. Let (F,R) be a formal group law, with R a Q-algebra and let λτ (u) = n≥0 τnu be the associated exponential function. Let k be a perfect field. Then the classifying map
∗ ∗ MGLQ ⊗LazR → (CH ⊗R)τ is an isomorphism.
∗ 4. Let k be a perfect field, and consider the generic twist CHQ[b]b. For an arbitrary perfect ∗ ∗ field k, the classifying map MGLQ → CHQ[b]b is an isomorphism
127 ∗ Proof. (1) follows from the isomorphism of oriented cohomology theories on Smk,Ω → MGL∗ [53, Theorem 3.1], and the universal property of Ω∗ [54, Theorem 7.1.3]
(4) follows from (3), noting that the classifying map Laz → Z[b1, b2,...] for the formal
−1 −1 group law Fb(u, v) := λb(λb (u) + λb (v)) induces an isomorphism LazQ → Q[b1, b2,...] (see e.g. [1, Theorem 7.8]), hence MGL∗ → MGL∗ ⊗ [b] is an isomorphism. The assertion Q Q LazQ Q ∗ ∼ 2∗,∗ (3) follows immediately from Lemma 4.1.11, as the isomorphism CH = H (−, Z) gives ∗ ∼ ∗ rise to a canonical isomorphism HRτ = (CH ⊗R)τ .
∗ To prove (2), it suffices by (4) to show that CHQ[b]b is the universal oriented coho-
mology theory in Q-algebras on Smk. This follows by applying the twisting construction.
∗ ∗ Indeed, let A be an oriented cohomology theory on Smk, such that A (k) is a Q-algebra.
−1 −n Let (τn ∈ A )n be the sequence such that λτ −1 (u) is the logarithm of the formal group
∗ ∗ law (FA∗ ,A (k)). The twist Aτ −1 thus has the additive formal group law and hence by Lem-
∗ ∗ −1 ma 4.1.12 we have the (unique) classifying map θτ : CHQ → Aτ −1 . As we have already
mentioned above, the map LazQ → Q[b1, b2,...] classifying the formal group law Fb(u, v) is
∗ an isomorphism, hence there is a unique ring homomorphism φ : Q[b1, b2,...] → A (k) with
∗ ∗ ∗ −1 −1 φ(Fb(u, v)) = FA (u, v). Extend θτ to Θτ : CHQ[b] → Aτ −1 by using φ, and then twist ∗ ∗ back by b and φ(b) = τ to get the map ΘA : CHQ[b]b → A of oriented cohomology theories on Smk. The uniqueness of ΘA follows from the uniqueness of θτ −1 and that of φ. This completes the proof.
Given a formal group law (R,F ) with exponential λ(t) ∈ RQ[[t]]. Write λ(t) = t + P τ ti+1 with τ ∈ R−i; we set τ = 1. We have two methods of constructing an ori- i≥1 i i Q 0 ented cohomology theory with formal group law (RQ,F ): the specialisation construction
∗ ∗ MGL ⊗LazRQ or the twisting construction (CH ⊗RQ)τ . As these two oriented cohomology theories are canonically isomorphic, we will denote both of them by R∗ : Q
∗ ∗ ∗ MGL ⊗LazRQ =: RQ := (CH ⊗RQ)τ . (4.1.3)
128 4.2 The algebraic elliptic cohomology theory
4.2.1 The elliptic formal group law
An algebraic elliptic cohomology theory is the cohomology theory corresponding to an elliptic
formal group law. More precisely, let R be a ring and let p : E → Spec R be a smooth projective morphism with section e : Spec R → E and addition map E ×R E → E defining a commutative group scheme over R with geometrically connected fibers of dimension one.
We assume in addition that we have chosen a local uniformizer t around the identity section.
The expansion of the group law of E in terms of the coordinate t gives a formal group law
FE with coefficients in R.
There is a well-studied elliptic formal group law and the corresponding cohomology the- ory. Namely, taking R = Z[µ1, µ2, µ3, µ4, µ6] and taking E to be the Weierstrass curve
2 3 2 y + µ1xy + µ3y = x + µ2x + µ4x + µ6
over the ring R, and using t = y/x as the local uniformizer. This formal group law will
be referred as the TMF elliptic formal group law. It has been studied by Franke, Hopkins,
Landweber, Miller, Ravenel, Stong, etc. See [42] and [40] for survey of this theory. In
particular, it has been shown that, the map of rings
−1 FE : Laz → R[∆ ]
given by this formal group law is Landweber exact, where ∆ is the discriminant.
The elliptic formal group law we are using is called the Krichever’s elliptic formal group
law in literature. It is related to, but different from the TMF elliptic formal group law. We
recall the genus corresponding to this formal group law, following the convention in [84].
Let
2 Y z + (z/w) σ(z, τ) := z (1 − z/w)e w 2 w∈Z+Zτ,w6=0
129 be the Weierstrass sigma function. Note that the sigma function defined above, as a function in z, is an entire odd function, with zero set equal to the lattice Z + Zτ ⊆ C. It has the following property
η (z+ 1 ) σ(z, τ) = −e 1 2 σ(z + 1, τ), and
η (z+ 1 τ) σ(z, τ) = −e 2 2 σ(z + τ, τ),
for some functions ηj = ηj(τ), j = 1, 2. We define the algebraic elliptic genus as the ring homomorphism
k φ : Laz → ((e2πiz))[[e2iπτ , ]] E Q 2πi
t 1 associated to the power series λE(t) := Q(t) under the Hirzebruch correspondence, where
t t t σ( − z, τ) kt ζ(z) 2πi 2πi Q(t) := e e t , 2πi σ( 2πi , τ)σ(−z, τ)
d log σ(z) here ζ(z) = dz is the zeta function and the function
t t σ( − z, τ) ζ(z) 2πi 2πi Φ(t, z) = e t σ( 2πi , τ)σ(−z, τ) is the Baker-Akhiezer function. Explicitly, the elliptic genus of a n-dimensional smooth Qn variety X is defined by φ(X) := h i=1 Q(ξi), [X]i, where the ξi are the Chern roots of the tangent bundle TX of X. The coefficient ring Ell˜ of the corresponding formal group law, i.e., the elliptic formal group law, is by definition the image of φE. Moreover, the elliptic formal group law can be written as
−1 −1 x +E y = λ(λ (x) + λ (y)),
1The series Q(t) we use here is slightly different from the one used in [43] and [84], where the series is
t t t σ( − z, τ) kt η1z 2πi 2πi Q(t) := e e t , 2πi σ( 2πi , τ)σ(−z, τ) for the eta function η1. Note that they both enjoy the rigidity property.
130 t for the series λ(t) = Q(t) as above. The algebraic elliptic cohomology theory associated to ∗ ˜ ˜ ∗ this formal group law, MGL ⊗LazEll, is denoted by Ell .
∗ Remark 4.2.1. The coefficient ring Ell˜ (k) depends only on the characteristic of k. For k ∗ ∗ having characteristic zero, Ell˜ (k) = Ell˜ and for k having characteristic p > 0, Ell˜ (k) =
Ell[1˜ /p].
When k = C, the elliptic genus has the rigidity property, proved by Krichever in [49] and H¨ohnin [43]. A consequence of the rigidity property is that given a fiber bundle F → E → B of closed connected weakly complex manifolds, with structure group a compact connected
Lie group G, and if F is a SU-manifold, then the elliptic genus is multiplicative with respect to this fibration.
The coefficient ring Ell˜ has been studied in [9]. We summarize their results here. The
˜ 2πiz 2iπτ k subring Ell of Q((e ))[[e , 2πi ]] is contained in a polynomial subring Z[a1, a2, a3, a4] of
2πiz 2iπτ k Q((e ))[[e , 2πi ]]. The elements ai can be described explicitly in terms of elements in
2πiz 2iπτ k Q((e ))[[e , 2πi ]] as follows. Let gi be the weight 2i Eisenstein series where i = 2, 3, and let X be the Weierstrass p-function and Y its derivative. The power series expansions of
these functions are as follows.
y Y Y X(τ, z) = 1/12 + − 2 nqmn + nqmn(yn + y−n); (1 − y)2 m,n≥1 m,n≥1 Y qmy(1 + qmy) Y qm/y(1 + qm/y) Y (τ, z) = − ; (1 − qmy)3 (1 − qm/y)3 m≥0 m≥1 " # Y m3qm g (τ) = 1/12 1 + 240 , 2 1 − qm m≥1
2πiz 2πiτ k 1 where y = e and q = e . Then a1 = − 2πi , a2 = X, a3 = Y , a4 = 2 g2.
˜ Remark 4.2.2. We shall see below that EllQ = Q[a1, a2, a3, a4]. However, as noted by
Totaro [84, §6], Ell˜ itself is not even finitely generated over Z.
1 Let R = Z[a1, a2, a3, 2 a4]. We let ER → Spec R be the elliptic curve over R defined as
the base change from the Weierstrass curve on A = Z[µ1, µ2, µ3, µ4, µ6] via the map of rings
131 ϕ : A → R
µ1 7→ 2a1
2 µ2 7→ 3a2 − a1
µ3 7→ −a3 1 µ 7→ − a + 3a2 − a a 4 2 4 2 1 3
µ6 7→ 0.
The chosen parameter t on the Weierstrass curve gives by base-change a parameter tR along
the zero section of ER. This gives us a formal group law over R, this just being the one induced from the TMF formal group law through change of coefficient ring via the map ϕ. It
is shown in [9, Lemma 44] that this formal group law is isomorphic to the Krichever’s elliptic
formal group law (after extended the coefficient ring from Ell˜ to R). This isomorphism is explicitly given in [9].
1 −1 Let Ell = Z[a1, a2, a3, 2 a4][∆ ], where ∆ is the discriminant.
Theorem 4.2.3 (Theorem ??). The Krichever’s elliptic formal group law (Ell[1/2],FKr) is
∗ ∗ ∗ Landweber exact. Therefore, the oriented cohomology theory Ell[1/2] := MGL ⊗Laz Ell[1/2]
on Smk is represented by a motivic oriented cohomology theory E``[1/2] on Smk.
This theorem can be proved using the same idea as the proof of the Landweber exactness
of the TMF formal group law. Nevertheless, for sake of completeness and the reader’s
convenience, we include a sketch of the proof here.
Proof. According to Theorem 4.1.9, it suffices to show that for all primes l > 2 and integers
n ≥ 0, the multiplication map
vn : Ell[1/2]/(v0, . . . , vn−1) → Ell[1/2]/(v0, . . . , vn−1)
is injective.
132 Note that v0 = l and Ell[1/2] is an integral domain, hence, multiplication by v0 is always injective.
Consider the ring
Rl := Fl[a1, ··· , a4] = R/(l)
and the family of elliptic curves El := ER ⊗R Rl over Rl. The injectivity of vi for i > 0 is related to the height of the formal group law of these curves. Note that the only possible
height of these curves are 1, 2, or infinity. We need first to remove the curves with infinite
height formal group laws. This can be done by inverting the discriminant ∆. If we fix a
−1 geometric point x ∈ Spec Rl[∆ ], the fiber is a supersingular elliptic curve if and only if the
corresponding formal group law has height 1, i.e., v1 vanishes when restricted to the residue
−1 field of x. As Rl[∆ ] is an integral domain, v1 is injective if and only if it is not zero. This,
−1 in turn, is equivalent to the condition that on Rl[∆ ] there is at least one geometric fiber which is not supersingular. Note that the residue field of x has characteristic l > 2. The
family El → Spec Rl contains the Legendre family, which is dominant over the moduli space
when passing to the separable closure. Therefore, the family El → Spec Rl is non-constant, hence contains a non-supersingular member.
−1 Finally, we claim that v2 in Rl[∆ ]/(v1) is a unit. This claim implies that multiplication
−1 by v2 is injective and that Rl[∆ ]/(v1, v2) = 0, which implies the required injectivity for
n ≥ 3. To verify the claim, assume otherwise, then v2 is contained in a maximal ideal
−1 m ⊂ Rl[∆ ]/(v1). Therefore, the fiber of the family of curves over this closed point has associated formal group law with height greater than 2. This contradicts with the fact that
the height of the formal group law of elliptic curves over a field of characteristic l > 0 can
only be 1 or 2.
Remark 4.2.4. The Landweber exactness condition for the prime l = 2 fails, since v1 vanishes in Ell /(2). To see this, one can easily calculate the j-invariant of the family of elliptic curves EEll ⊗Ell Ell /(2) over Ell /(2) to see that the j-invariant is a constant. Passing
133 to the separable closure, one finds that the curve y2 + y = x3, which is supersingular, is a
member in the family, and hence every member in this family is supersingular.
Remark 4.2.5. Inverting ∆ is an overkill, as the locus ∆ = 0 contains not only the
curves with infinite height formal group law, but also certain curves whose formal group
law has height 1. This can be fixed using the modern approach [42]. Taking the j-
invariant for the family ER ⊗R R[1/2] we obtain a morphism of Spec Z[1/2][a1, a2, a3, a4] to the coarse moduli space of elliptic curves and thereby obtain a stack with coarse moduli
space Spec Z[1/2][a1, a2, a3, a4]. Let Kr be the open substack obtained by removing the lo- cus consisting of curves with infinite height formal group laws. This stack Kr is flat over
the stack of formal group laws, by the same argument as in the proof of Theorem 4.2.3.
Therefore, any flat morphism of stacks from a commutative ring R to Kr defines a motivic
oriented Krichever elliptic cohomology theory.
4.3 Flops in the cobordism ring
4.3.1 The double point formula
Following [55], we define the ring ω∗(X) of double-point cobordism to be the graded abelian
∗ group M+(X) of smooth projective schemes Y → X as in Subsection 4.1.1, modulo the relations generated by the following double point relation.
1 Let Y ∈ Smk be of pure dimension. A morphism π : Y → P is a double point degeneration
over 0 ∈ P1, if π−1(0) = A ∪ B with A and B smooth, of codimension one, intersecting transversely along A ∩ B = D.
We denote the normal bundles of D in A and B by NDA and NDB respectively. Then, the two projective bundles
P(OD ⊕ NDA) → D and P(OD ⊕ NDB) → D
134 are isomorphic and will be denoted by P(π) → D. Let g : Y → X × P1 be a projective
1 1 morphism for which π = p2 ◦ g : Y → P is a double point degeneration over 0 ∈ P . The double point relation associated to g is
[Yζ → X] − [A → X] − [B → X] + [P(π) → X],
1 where Yζ is the fiber over an arbitrary regular value ζ ∈ P . It is shown in [55] that if k has characteristic zero, then ω∗(X) is isomorphic to the cobordism ring Ω∗(X). We will use a weaker version of this theorem, which holds in a characteristic free fashion.
Proposition 4.3.1 (Levine and Pandharipande). For any field k and scheme X ∈ Smk of
∗ ∗ ∗ finite type over k, the natural projection Π: M+(X) → Ω (X) factors through ω (X).
The proof in [55] uses only the existence of smooth pull-back, projective push-forward, the first Chern class of a line bundle, and external product, which means it does not depend on any assumption on k. Thus, the double point relation also holds when the field k has positive characteristic.
We note that ω∗ has the following structures:
∗ ∗ ∗ 1. pullback maps f : ω (X) → ω (Y ) for each smooth morphism f : Y → X in Smk.
∗ ∗−d 2. push-forward maps f∗ : ω (Y ) → ω (X) for each projective morphism f : Y → X of
relative dimension d in Smk.
3. associative, commutative external products, and an identity element 1 ∈ ω0(k).
The pullback and pushforward maps are functorial, and are compatible with the external products.
∗ ∗ Composing the map ω → Ω with the natural transformation ΘMGL gives us the trans- formation
∗ ∗ θMGL : ω → MGL ,
135 natural with respect to smooth pullback, projective push-forward, external products and
unit.
Let F ⊆ X be a smooth closed subscheme of some X ∈ Smk. The double point relation yields the following blow-up formula in ω∗(X), and hence in Ω∗(X) and MGL∗(X):
1X = [BlF X → X] + [P(NF X ⊕ O) → X] − [P(ONF X (1) ⊕ O) → X] (4.3.1)
where BlF X is blow-up of X along F , and NF X is the normal bundle of F . This is proved by the usual method of deformation to the normal cone. In case X is projective over k, pushing forward to Spec k gives the relation in ω∗(k), Ω∗(k) and MGL∗(k)
[X] = [BlF X] + [P(NF X ⊕ O)] − [P(ONF X (1) ⊕ O)] (4.3.2)
Let Q3 ⊂ P4 denote the 3-dimensional quadric with an ordinary double point v, defined
by the equation x1x2 = x3x4. We say two smooth projective n-folds X1 and X2 are related by a flop if we have the following diagram of projective birational morphisms:
Xe (4.3.3)
~ X1 X2 p1 p2
Y ~
Here Y is a singular projective n-fold with singular locus Z such that along Z, the pair (Y,Z)
is ´etalelocally isomorphic to an ´etaleneighbourhood of (v, 0) in the pair (Q3×An−3, v×An−3). In particular, Z is smooth of dimension n−3. We assume in addition that there exist rank 2
1 vector bundles A and B on Z, such that the exceptional locus F1 in X1 is the P -bundle P(A)
over Z, with normal bundle NF1 X1 = B ⊗ O(−1). Similarly, the the exceptional locus F2 in
X2 is P(B), with normal bundle NF2 X2 = A⊗O(−1). We say that X1 and X2 are related by a classical flop if in addition along Z,(Y,Z) is Zariski locally isomorphic to (Q3 × Z, v × Z).
We assume now X1 and X2 are related by a flop. Let X = X1 and F = F1, the terms on
136 the right hand side of formula (4.3.2) become:
BlF X = X,e
P(NF X ⊕ O) = PP(A)(B ⊗ OP(A)(−1) ⊕ O),
P(ONF X (1) ⊕ O) = P(OP(B⊗O(−1))(1) ⊕ O),
where PP(A)(B⊗OP(A)(−1)⊕O) is a projective bundle over P(A), which in turn is a projective
bundle over Z; and P(OP(B⊗O(−1))(1)⊕O) is a projective bundle over P(B⊗OP(A)(−1)), which
is a projective bundle over P(A). Thus, we get the following immediate lemma.
Lemma 4.3.2. In the cobordism ring MGL∗(k), we have
X1 − X2 =PP(A)(B ⊗ OP(A)(−1) ⊕ O) − P(OP(B⊗O(−1))(1) ⊕ O)
− PP(B)(A ⊗ OP(B)(−1) ⊕ O) + P(OP(A⊗O(−1))(1) ⊕ O).
∗ ∗ In particular, X1 − X2 in MGL (k) comes from an element in MGL (Z).
The second claim follows from the observation that each of the projective bundles are
∗ smooth varieties over Z. We will abuse notation by denoting a lifting of X1 −X2 to MGL (Z)
by X1 − X2 itself.
4.3.2 Flops in the cobordism ring
Since each term in the formula in Lemma 4.3.2 is a iterated projective bundle over Z, we will apply Quillen’s formula iteratively to each term, to calculate the fundamental class of the iterated projective bundles in MGL∗(Z).
Proposition 4.3.3. Let a1, a2 be the Chern roots of the bundle A over Z and let b1, b2 be the Chern roots of the bundle B over Z, all Chern roots to be taken in MGL∗. Then in
137 MGL∗(Z), we have
1 1 X1 − X2 = + (a1 +Ω b1)(b2 +Ω a1)(a2 −Ω a1) (a2 +Ω b1)(b2 +Ω a2)(a1 −Ω a2) 1 1 − − . (b1 +Ω a1)(a2 +Ω b1)(b2 −Ω b1) (b2 +Ω a1)(a2 +Ω b2)(b1 −Ω b2)
The rest of this subsection is devoted to prove this proposition.
The term P(OP(B⊗O(−1))(1) ⊕ O)
We first prove the following
∗ Lemma 4.3.4. In MGL (Z), we have P(OP(B⊗O(−1))(1) ⊕ O) = P(OP(A⊗O(−1))(1) ⊕ O).
Let π1 : P(OP(B⊗O(−1))(1) ⊕ O) → P(B ⊗ O(−1)) be the natural projection. Let the first
Chern class of the bundle OP(B⊗O(−1))(1) be uB. Then the two Chern roots of OP(B⊗O(−1))(1)⊕
O are uB and 0. Applying Quillen’s formula (4.1.2) with f1(t) ≡ 1 being the fundamental class, we have
1 1 1 1 π1∗([P(OP(B⊗O(−1))(1) ⊕ O)]) = + = + . (4.3.4) 0 −Ω uB uB −Ω 0 −ΩuB uB
Next, let π2 : PP(A)(B ⊗ OP(A)(−1)) → P(A) be the projection. The two Chern roots of the bundle B ⊗ OP(A)(−1) are b1 −Ω vA and b2 −Ω vA, where vA := c1(OP(A)(1)), and b1, b2
1 1 are two Chern roots of bundle B. Applying Quillen’s formula (4.1.2) with f2(t) = + , −Ωt t we get
1 + 1 1 + 1 −Ω(−Ω(b1−ΩvA)) −Ω(b1−ΩvA) −Ω(−Ω(b2−ΩvA)) −Ω(b2−ΩvA) π2∗(f2) = + (b2 −Ω vA) −Ω (b1 −Ω vA) (b1 −Ω vA) −Ω (b2 −Ω vA) 1 + 1 1 + 1 = b1−ΩvA −Ωb1+ΩvA + b2−ΩvA −Ωb2+ΩvA . b2 −Ω b1 b1 −Ω b2
Finally, let π3 : P(A) → Z be the projection. Recall the two Chern roots of bundle A are 1 + 1 1 + 1 b1−Ωt −Ωb1+Ωt b2−Ωt −Ωb2+Ωt denoted by a1, a2. Let f3(t) := + , then Quillen’s formula (4.1.2) b2−Ωb1 b1−Ωb2
138 yields
1 + 1 1 + 1 b1+Ωa1 −Ωb1−Ωa1 b2+Ωa1 −Ωb2−Ωa1 π3(f3) = + (a2 −Ω a1)(b2 −Ω b1) (a2 −Ω a1)(b1 −Ω b2) 1 + 1 1 + 1 + b1+Ωa2 −Ωb1−Ωa2 + b2+Ωa2 −Ωb2−Ωa2 , (a1 −Ω a2)(b2 −Ω b1) (a1 −Ω a2)(b1 −Ω b2)
which is π∗(P(OP(B⊗O(−1))(1) ⊕ O)).
0 Similarly, for the projective bundle π : P(OP(A⊗O(−1))(1) ⊕ O) → Z, we have:
1 + 1 1 + 1 0 a1+Ωb1 −Ωa1−Ωb1 a2+Ωb1 −Ωa2−Ωb1 π∗([P(OP(A⊗O(−1))(1) ⊕ O)]) = + (b2 −Ω b1)(a2 −Ω a1) (b2 −Ω b1)(a1 −Ω a2) 1 + 1 1 + 1 + a1+Ωb2 −Ωa1−Ωb2 + a2+Ωb2 −Ωa2−Ωb2 . (b1 −Ω b2)(a2 −Ω a1) (b1 −Ω b2)(a1 −Ω a2)
0 Now comparing the two formulas of π∗(P(OP(B⊗O(−1))(1) ⊕ O)) and π∗([P(OP(A⊗O(−1))(1) ⊕ O)]), the lemma follows.
The term PP(A)(B ⊗ OP(A)(−1) ⊕ O)
Let π1 : PP(A)(B ⊗ OP(A)(−1) ⊕ O) → P(A) be the natural projection. The three Chern roots
of the bundle B ⊗ OP(A)(−1) ⊕ O are b1 −Ω vA, b2 −Ω vA, and 0, where vA := c1(OP(A)(1)),
and b1, b2 are two Chern roots of the bundle B. Applying Quillen’s formula (4.1.2) with
f1(t) ≡ 1, we get
1 1 π1∗(P(B ⊗ OP(A)(−1) ⊕ O)) = + (b2 −Ω b1)(vA −Ω b1) (b1 −Ω b2)(vA −Ω b2) 1 + . (b1 −Ω vA)(b2 −Ω vA)
Now let π2 : P(A) → Z be the natural projection. Recall the two Chern roots of bundle
1 1 1 A are denoted by a1, a2. Let f2(t) := + + , then (b2−Ωb1)(t−Ωb1) (b1−Ωb2)(t−Ωb2) (b1−Ωt)(b2−Ωt)
139 Quillen’s formula (4.1.2) gives 1 1 π2∗(f2) = + (b2 −Ω b1)(a2 −Ω a1)(−Ωa1 −Ω b1) (b1 −Ω b2)(a2 −Ω a1)(−Ωa1 −Ω b2) 1 1 + + (a1 +Ω b1)(b2 +Ω a1)(a2 −Ω a1) (b2 −Ω b1)(a1 −Ω a2)(−Ωa2 −Ω b1) 1 1 + + . (b1 −Ω b2)(a1 −Ω a2)(−Ωa2 −Ω b2) (a2 +Ω b1)(b2 +Ω a2)(a1 −Ω a2)
0 Similarly, for the bundle π : PP(B)(A ⊗ OP(B)(−1) ⊕ O) → Z, we have:
0 π∗([PP(B)(A⊗OP(B)(−1) ⊕ O)]) 1 1 = + (a2 −Ω a1)(b2 −Ω b1)(−Ωb1 −Ω a1) (a1 −Ω a2)(b2 −Ω b1)(−Ωb1 −Ω a2) 1 1 + + (b1 +Ω a1)(a2 +Ω b1)(b2 −Ω b1) (a2 −Ω a1)(b1 −Ω b2)(−Ωb2 −Ω a1) 1 1 + + (a1 −Ω a2)(b1 −Ω b2)(−Ωb2 −Ω a2) (b2 +Ω a1)(a2 +Ω b2)(b1 −Ω b2)
Therefore,
0 π∗(PP(A)(B ⊗ OP(A)(−1) ⊕ O)) − π∗(PP(B)(A ⊗ OP(B)(−1) ⊕ O)) 1 1 = + (a1 +Ω b1)(b2 +Ω a1)(a2 −Ω a1) (a2 +Ω b1)(b2 +Ω a2)(a1 −Ω a2) 1 1 − − . (b1 +Ω a1)(a2 +Ω b1)(b2 −Ω b1) (b2 +Ω a1)(a2 +Ω b2)(b1 −Ω b2) This finishes the proof of the proposition.
4.3.3 Flops in the elliptic cohomology ring
In this subsection, we prove the following Proposition.
Proposition 4.3.5. Suppose X1 and X2 are smooth projective varieties related by a flop.
∗ Notations as above, in the ring EllQ(Z), we have
PP(A)(B ⊗ OP(A)(−1) ⊕ O) = PP(B)(A ⊗ OP(B)(−1) ⊕ O).
140 ∗ ˜ ∗ In particular, we have X1 − X2 = 0 in EllQ(Z), and hence X1 − X2 = 0 in Ell (k).
∗ Remark 4.3.6. Once we know that X1 − X2 = 0 in EllQ(Z), it follows by pushing forward ∗ ˜ ∗ to Spec k that X1 − X2 = 0 in EllQ(k). But X1 − X2 is a well-defined element in Ell (k) ˜ ∗ ∗ ˜ ∗ and Ell (k) → EllQ(k) is injective, since Ell (k) is by construction Z-torsion free, hence ˜ ∗ X1 − X2 = 0 in Ell (k), as claimed above.
Proof of the proposition. Thanks to Proposition 4.3.3, we can reduce X1 − X2 to an explicit element in MGL∗(Z). Applying the canonical map MGL∗(Z) → Ell∗(Z) to this element, we have, in the ring Ell∗(Z),
1 1 X1 − X2 = + (a1 +E b1)(b2 +E a1)(a2 −E a1) (a2 +E b1)(b2 +E a2)(a1 −E a2) 1 1 − − , (b1 +E a1)(a2 +E b1)(b2 −E b1) (b2 +E a1)(a2 +E b2)(b1 −E b2)
here the ai’s and bi’s are the Chern roots in the elliptic cohomology. We would like to show
∗ the above expression is 0 in EllQ(Z).
−1 −1 Recall that x+E y = λ(λ (x)+λ (y)) where the exponential λ(t) is given by the power
t series Q(t) .
−1 −1 ∗ Let λ (ai) = Ai, and λ (bi) = Bi for i = 1, 2. Then in EllQ(Z), X1 − X2 becomes:
1 1 1 1 1 1 (A + B ) (B + A ) (A − A ) + (B + A ) (B + A ) (A − A ) λ 1 1 λ 2 1 λ 2 1 λ 1 2 λ 2 2 λ 1 2 1 1 1 1 1 1 − (A + B ) (A + B ) (B − B ) − (A + B ) (B + A ) (B − B ). λ 1 1 λ 2 1 λ 2 1 λ 1 2 λ 2 2 λ 1 2
Plugging-in
t 1 Q(t) 1 t σ( − z, τ) kt ζ(z) 2πi 2πi (t) = = e e t , λ t 2πi σ( 2πi , τ)σ(−z, τ)
141 and cancelling some obvious factors, X1 − X2 becomes
σ(A1 + B1 − z) σ(B2 + A1 − z) σ(A2 − A1 − z)
σ(A1 + B1) σ(B2 + A1) σ(A2 − A1) σ(B + A − z) σ(B + A − z) σ(A − A − z) + 1 2 2 2 1 2 σ(B1 + A2) σ(B2 + A2) σ(A1 − A2) σ(A + B − z) σ(A + B − z) σ(B − B − z) − 1 1 2 1 2 1 σ(A1 + B1) σ(A2 + B1) σ(B2 − B1) σ(A + B − z) σ(B + A − z) σ(B − B − z) − 1 2 2 2 1 2 . σ(A1 + B2) σ(B2 + A2) σ(B1 − B2)
Now let
x1 = −A1, x2 = B2, x3 = B1, x4 = −A2. and
y1 = B2 − z, y2 = B1 − z, y3 = −A2 + z, y4 = −A1 + z.
Using the fact that σ(z) is an odd function, we get
σ(A1 + B1 − z)σ(B2 + A1 − z)σ(A2 − A1 − z)σ(−z)
σ(A1 + B1)σ(B2 + A1)σ(A2 − A1) σ(x − y )σ(x − y )σ(x − y )σ(x − y ) = 1 1 1 2 1 3 1 4 ; σ(x1 − x2)σ(x1 − x3)σ(x1 − x4)
σ(A + B − z)σ(B + A − z)σ(B − B − z)σ(−z) − 1 2 2 2 1 2 σ(A1 + B2)σ(B2 + A2)σ(B1 − B2) σ(x − y )σ(x − y )σ(x − y )σ(x − y ) = 2 1 2 2 2 3 2 4 ; σ(x2 − x1)σ(x2 − x3)σ(x2 − x4)
σ(A + B − z)σ(A + B − z)σ(B − B − z)σ(−z) − 1 1 2 1 2 1 σ(A1 + B1)σ(A2 + B1)σ(B2 − B1) σ(x − y )σ(x − y )σ(x − y )σ(x − y ) = 3 1 3 2 3 3 3 4 ; σ(x3 − x1)σ(x3 − x2)σ(x3 − x4)
σ(B1 + A2 − z)σ(B2 + A2 − z)σ(A1 − A2 − z)σ(−z)
σ(B1 + A2)σ(B2 + A2)σ(A1 − A2) σ(x − y )σ(x − y )σ(x − y )σ(x − y ) = 4 1 4 2 4 3 4 4 . σ(x4 − x1)σ(x4 − x2)σ(x4 − x3)
142 The Proposition follows from the following classical identity of sigma-function with n = 4. Pn Pn (See § 20.53, Example 3 of [100].) Assuming r=1 xr = r=1 yr, we have
n X σ(xr − y1)σ(xr − y2) ··· σ(xr − yn) = 0 σ(x − x )σ(x − x ) · · · ∗ · · · σ(x − x ) r=1 r 1 r 2 r n with the ∗ denoting that the vanishing factor σ(xr − xr) is to be omitted.
4.4 The algebraic elliptic cohomology ring with ratio-
nal coefficients
∗ Let Ifl be the ideal in MGLQ(k) generated by differences X1 − X2, where X1 and X2 are ∗ related by a flop and Iclfl ⊂ Ifl the ideal in MGLQ(k) generated by differences X1 − X2, where X1 and X2 are related by a classical flop. Section 4.3.3 shows that the elliptic genus
∗ ∗ ∗ φ : MGL (k) → Ell (k) factors through the quotient MGL (k)/Ifl.
∗ Proposition 4.4.1. The ideal Iclfl in MGLQ(k) contains a system of polynomial generators ∗ xn of MGLQ(k) in degree n ≤ −5.
This proposition follows from same the calculation as in [84]. Nevertheless, for the con- venience of the readers and to be as explicit as possible, we include the calculation in the
Appendix.
∗ Following H¨ohn([43]), we define the four elements Wi, i = 1, 2, 3, 4 in MGL (k) via their Chern numbers as follows,
c1[W1] = 2;
2 c1[W2] = 0, c2[W2] = 24;
3 c1[W3] = 0, c1c2[W3] = 0, c3[W3] = 2;
4 2 2 c1[W4] = 0, c1c2[W4] = 0, c2[W4] = 2, c1c3[W4] = 0, c4[W4] = 6.
143 In fact, solving the systems of linear equations, one can write down Wi explicitly as rational linear combinations of products of projective spaces,
1 W1 := [P ];
2 1 1 W2 := −16[P ] + 18[P × P ]; 3 5 W := [ 3] − 4[ 2 × 1] + [( 1)3]; 3 2 P P P 2 P 4 3 1 2 2 2 1 2 1 4 W4 := −4[P ] + 12.5[P × P ] + 6[P × P ] − 26[P × (P ) ] + 11.5(P ) .
2 Write the Hirzebruch characteristic power series Q(t) = 1 + f1t + f2t + ··· . Let A, B, C, and D be such that 1 f = A; 1 2 1 f = (6A2 − B); 2 24 · 3 1 f = (2A3 − AB + 16C); 3 25 · 3 1 f = (60A4 − 60A2B + 1920AC + 7B2 − 1152D). 4 29 · 32 · 5 Then, a calculation shows (see, e.g., § 2.2 of [43]), the lower degree parts of the elliptic genus
φ are as follows 1 K = Ac ; 1 2 1 1 K = ((6A2 − B)c2 + 2Bc ); 2 24 · 3 1 2 1 K = ((2A3 − AB + 16C)c3 + (2AB − 48C)c c + 48Cc ); 3 25 · 3 1 2 1 3 1 K = ((60A4 − 60A2B + 1920AC + 7B2 − 1152D)c4 4 29 · 32 · 5 1 2 2 2 2 2 +(24B − 2304D)c2 + (120A B − 5760AC − 28B + 4608D)c1c2
2 2 (5760AC + 8B − 4608D)c3c1 + (−8B + 4608D)c4),
where Ki is the homogeneous degree i part of the elliptic genus φ. Therefore, φ(W1) =
A, φ(W2) = B, φ(W3) = C and φ(W4) = D.
∗ We then compare the elements A, B, C, D with the polynomial generators of EllQ(k). Let
gi be the weight 2i Eisenstein series where i = 2, 3, and let X be the Weierstrass p-function
144 and Y its derivative. The same calculations as in §2 of [43] show B = 24X, C = Y , and
2 1 1 D = 6X − g2/2. Recall that a2 = X, a3 = Y , a4 = 2 g2. Note that the image of W1 = P , k which is πi , involves non-trivially the variable k, but the images of Wi for i = 2, 3, 4 lies in the subring Q((e2πiz))[[e2iπτ ]]. In particular, A, B, C, and D are algebraically independent. Summarizing all the above, we have proved the following proposition.
∗ 2πiz 2iπτ k Proposition 4.4.2. The classifying map φE : MGL (k) → Q((e ))[[e , 2πi ]] descends ∗ to define an isomorphism of MGLQ(k)/Iclfl with a polynomial subalgebra Q[a1, a2, a3, a4] of 2πiz 2iπτ k Q((e ))[[e , 2πi ]], with ai in degree −i.
This implies the following corollary.
∗ ∗ Corollary 4.4.3. The natural ring homomorphism MGLQ(k)/Iclfl → EllQ(k) is injective
and Iclfl = Ifl.
Proof. The first assertion follows immediately from proposition 4.4.2; the second from the
∗ ∗ ∗ first, noting that MGLQ(k)/Iclfl → EllQ(k) factors through the quotient MGLQ(k)/Ifl by proposition 4.3.5.
4.5 Birational symplectic varieties
In this section we work over a base field k with characteristic zero.
4.5.1 Specialization in algebraic cobordism theory
We will use the specialization morphism in algebraic cobordism theory. The existence of a
specialization morphism in algebraic cobordism theory is folklore; for lack of a reference, we
sketch a construction here.
Proposition 4.5.1. Let C be a smooth curve, and p : X → C a smooth projective morphism.
Let o ∈ C be a closed point with fiber Xo and η be the generic point of C whose fiber is denoted
145 by Xη. Let i : Xo → X and j : Xη → X be the natural embeddings. Then there is a natural
∗ ∗ ∗ ∗ ∗ ∗ morphism σ :Ω (Xη) → Ω (Xo) such that σ ◦ j = i where j is the pull-back and i is the Gysin morphism.
Proof. Let R be the local ring of C at o ∈ C. Although in this case neither XR nor Xη are k-schemes of finite type, they are both projective limits of such, which allows us to define
∗ ∗ Ω (XR) and Ω (Xη) as
∗ ∗ −1 ∗ ∗ −1 Ω (XR) := lim Ω (p (U)); Ω (Xη) := lim Ω (p (U)). 0∈U⊂C ∅6=U⊂C
Here U ⊂ C is an open subscheme. We may then replace C with Spec R, X with XR.
For any integer n, there is a localization short exact sequence
∗ n−1 i∗ n j n Ω (Xo) −→ Ω (X ) −→ Ω (Xη) → 0.
∗ n n ∗ Let i :Ω (X ) → Ω (Xo) be the pull-back. In order to show it factors through j , it
∗ ∗ ∼ suffices to check that i ◦ i∗ = 0. This is true since i ◦ i∗ = c1OXo |Xo = 0 (see [54, Lemma 3.1.8]).
∗ ∗ We note that the specialization map σ :Ω (Xη) → Ω (Xo) is a ring homomorphism, and is natural with respect to pullback and push-forward in the following sense: Let q : Y → C
be a smooth projective morphism, with C as above, let f : Y → X be a morphism over C and
∗ ∗ let fo : Yo → Xo, fη : Yη → Xη be the respective restrictions of f. Let σX :Ω (Xη) → Ω (Xo),
∗ ∗ σY :Ω (Yη) → Ω (Yo) be the respective specialization maps. Then
∗ ∗ 1. fo ◦ σX = σY ◦ fη .
2. Suppose f is projective. Then fo∗ ◦ σY = σX ◦ fη∗.
Indeed, as the respective restriction maps j∗ are surjective, the fact that σ is a ring homo-
morphism and the compatibility (1) follows from the fact that pullback maps are functorial
146 ring homomorphisms. For (2), we note that the diagram
iY Yo / Y
fo f Xo / X iX
∗ is cartesian, and then the compatibility (2) follows from the base-change identity iX ◦ f∗ =
∗ fo∗ ◦ iY and the surjectivity mentioned above.
4.5.2 Cobordism ring of birational symplectic varieties
Consider two birational irreducible symplectic varieties X1 and X2 satisfying the following condition: There exist smooth projective algebraic varieties X1 and X2, flat over a smooth quasi-projective curve C with a closed point o ∈ C, such that:
(i) the fiber of Xi over o is (Xi)o = Xi;
(ii) there is an isomorphism Ψ : (X1)C\{o} → (X2)C\{o} over C.
The counterpart of the following Proposition in Chow theory is proved in [32]. Let Ω∗ be the algebraic cobordism with Z coefficient. When the base field k has characteristic zero, then Ω∗ = MGL∗.
Proposition 4.5.2. Let X1 and X2 be two birational symplectic varieties satisfying condi- tions (i) and (ii). The deformation data induce an isomorphism
∗ ∼ ∗ Ω (X1) = Ω (X2).
It was proved in [44] that the above conditions (i) and (ii) hold, when:
• either X1 and X2 are connected by a general Mukai flop,
• or X1 and X2 are isomorphic in codimension two, that is, there exist isomorphic open
subsets U1 ⊂ X1, and U2 ⊂ X2 with codimXi (Xi \ Ui) ≥ 3, for i = 1, 2.
147 The proof follows the same idea as in [32], nevertheless, for the convenience of the readers, we include the proof.
∗ ∗ Let σi :Ω (Xiη) → Ω (Xio) be the specialization map, where i = 1, 2. Let ∆ ⊂ (X1η ×X2η) be the diagonal, or the graph of the isomorphism Ψ as in condition (ii) restricted to the generic fiber. We define an element
∗ Z := (σ1 × σ2)([∆]) ∈ Ω (X1 × X2).
∗ The projection X1 × X2 → Xi for i = 1, 2 is denoted by pi. The element Z ∈ Ω (X1 × X2) defines a map
∗ ∗ [Z]:Ω (X1) → Ω (X2)
∗ op ∗ ∗ by α 7→ p2∗(p1(α) ∩ Z). By symmetry, we also have a map [Z ]:Ω (X2) → Ω (X1), given
∗ op op ∗ by β 7→ p1∗(p2(β) ∩ Z ), where Z ∈ Ω (X2 × X1) is the image of Z by the symmetry morphism X1 × X2 → X2 × X1.
We summarize the notations in the following diagram:
X1 × X2 × X1 (4.5.1) p12 p23 p13 t * X1 × X2 X1 × X1 X2 × X1 pr1 pr2 p2 p2 p1 p1 $ x * t & z X1 X2 X1
op ∗ Now we check that [Z ] ◦ [Z] = 1, i.e., for any α ∈ Ω (X1) we have
∗ ∗ op p1∗ p2 p2∗(p1α ∩ Z) ∩ Z = α.
148 The diagonal in Xi × Xi will be denoted by ∆Xi . Similarly we have ∆Xiη ⊆ Xiη × Xiη as the diagonal. We have: ∗ ∗ op ∗ ∗ op p1∗ p2 p2∗(p1α ∩ Z) ∩ Z = p1∗ p23∗ p12(p1α ∩ Z) ∩ Z ∗ ∗ ∗ op = (p1p23)∗ (p13pr1) α ∩ p12Z ∩ p23Z ∗ ∗ ∗ op = (pr2)∗ pr1α ∩ p13∗ p12Z ∩ p23Z
∗ = (pr2)∗(pr1α ∩ ∆X1 )
= α.
The only less clear equality is the second last one. It follows from the following lemma.
Lemma 4.5.3. Notations as in the diagram, we have:
∗ ∗ op p13∗(p12Z ∩ p23Z ) = ∆X1 ∈ Ω(X1 × X1).
Proof. Consider the same diagram as (4.5.1) with Xi replaced by Xiη. We make the con- vention here in the proof that for any map p in diagram (4.5.1), the corresponding map for generic fibers will be denoted by pe. Note that
p (p ∗∆ ∩ p ∗∆op) = ∆ ∈ Ω((X ) × (X )) . f13∗ f12 f23 X1η 1 η 1 η
Applying the specialization map σ on both sides, we get:
∗ ∗ op p13∗(p12Z ∩ p23Z ) = ∆X1 ∈ Ω(X1 × X1).
Note that the following proposition has no direct counterpart in the Chow theory.
∗ ∗ Let πi :Ω (Xi) → Ω (k) be the structure map of Xi → k, where i = 1, 2.
∗ ∗ Proposition 4.5.4. Let [Z]:Ω (X1) → Ω (X2) be the map constructed as above. We have,
1. [Z](1X1 ) = 1X2 ,
149 2. π1∗(1X1 ) = π2∗(1X2 ).
Proof. We have
p (p ∗(1 ) ∩ ∆) = 1 ∈ Ω((X ) ). e2∗ e1 (X1)η (X2)η 2 η
Applying the specialization map σ on both sides, we get:
∗ ∗ p2∗(p1(1X1 ) ∩ Z) = 1X2 ∈ Ω (X2).
This finishes the proof of (1). ∼ Let Ψ : X1η = X2η be the isomorphism as in condition (ii), we have Ψ∗(1X1η ) = 1X2η , and π (1 )) = π (1 )). Applying the specialization map, (2) follows. e1∗ X1η e2∗ X2η
i1,...,ir For a vector bundle E on some X ∈ Smk, let c (E) denote the product ci1 (E) ··· cir (E) in CH∗(X). For X a smooth projective variety over k, the Chern number cI (X) associated
P I to an index I = (i1, . . . , ir) with j ij = dimk X and ij > 0 is degk(c (TX )).
Corollary 4.5.5. Let X1 and X2 be two birational symplectic varieties satisfying conditions
(i) and (ii). Then X1 and X2 have the same Chern numbers.
Proof. For an integer d > 0, let Pd denote the number of partitions of d. It is well-known
Q I that the function X 7→ I c (X) on smooth projective irreducible k-schemes of dimension d over k descends to well-defined homomorphism c∗,d :Ω−d(k) → ZPd via the map sending
X to its class [X] ∈ Ω−d(k). In fact, c∗,d is injective and has image a subgroup of ZNd of finite index, but we will not use this. The result is now an immediate consequence of
Proposition 4.5.4.
4.6 Appendix: The ideal generated by differences of
flops
∗ Let Iclfl be the ideal in MGLQ(k) generated by those [X1] − [X2] with X1 and X2 related by a classical flop.
150 ∗ Proposition 4.6.1. The ideal Iclfl in MGLQ(k) contains a system of polynomial generators ∗ xn of MGLQ(k) in all degrees n ≤ −5.
This proposition was originally proved in Section 5 of [84]. Their proof is based on explicit calculation that lends itself with slight adjustment to our setting. Note that this proof is independent of the fact that the Krichever’s elliptic formal group law is defined over
Z[a1, a2, a3, a4]. The way to show this is to use the fact that an element x of MGL−n(k) is a polynomial Q ∗ n generator of the ring MGLQ(k) if and only if the Chern number s is not zero on x. Here n n n for a smooth irreducible projective variety X over k , s (X) = hξ1 + ··· + ξn , [X]i, with ξi being the Chern roots of the tangent bundle of X (in the Chow ring CH∗(X)).
The following lemma can be found in [28], Page 47.
Lemma 4.6.2. For any space X with a vector bundle V of rank r, let π : P(V ) → X be the
1 projective bundle and let u = c1(O(1)) ∈ CH (P(V )). Then, for all i ≥ 0,
i π∗(u ) = si−(r−1)(V ),
where sk is the k-th Segr´eclass.
Recall that we have shown that, in MGL∗(k),
X1 − X2 = PP(A)(B ⊗ OP(A)(−1) ⊕ O) − PP(B)(A ⊗ OP(B)(−1) ⊕ O).
For each smooth Z and rank 2 vector bundles A and B, there is a pair X1 and X2, related by a classical flop with exceptional fibers equal to P(A) and P(B) respectively. Indeed, consider
1 1 3 the P × P bundle q : P(A) ×Z P(B) → Z, which we embed in the P bundle P(A ⊕ B) → Z
∗ ∗ 0 via the line bundle p1OA(1) ⊗ p2OB(1). We then take Y to be the affine Z cone in A ⊕ B
0 associated to P(A) ×Z P(B) ⊂ P(A ⊕ B), and Y the closure of Y in P(A ⊕ B ⊕ OZ ). Y thus
2 contains the P bundles P1 := P(A ⊕ OZ ) and P2 := P(B ⊕ OZ ); we take Xi → Y to be the ˜ blow-up of Y along Pi, i = 1, 2 and X the blow-up of Y along Z.
151 We will, for each n ≥ 5, find an n − 3-fold Z and rank two vector bundles A and B over
n n Z, such that s (PP(A)(B ⊗ OP(A)(−1) ⊕ O)) 6= s (PP(B)(A ⊗ OP(B)(−1) ⊕ O)). We start with an arbitrary choice of Z and A, B. Recall we denote the Chern roots
∗ (in CH ) of A by a1, a2 and Chern roots of B by b1, b2. Set vA := c1(OP(A)(1)), vB = c1(OP(B)(1)), wA := c1(OP(A⊗O(−1)⊕O)(1)), and wB = c1(OP(B⊗O(−1)⊕1)(1)). Let the Chern roots of the tangent bundle of Z be z1 ··· , zn−3. Then, the Chern roots of the tangent bundle of PP(A)(B ⊗ OP(A)(−1) ⊕ O) are
b1 − vA + wB, b2 − vA + wB, wB, a1 + vA, a2 + vA, z1, . . . , zn−3.
For the bundle π1 : PP(A)(B ⊗ OP(A)(−1) ⊕ O) → P(A), Lemma 4.6.2 yields
i X i1 i2 π1∗(wB) = si−2(B ⊗ OP(A)(−1) ⊕ O) = (−b1 + vA) (−b2 + vA) . i1+i2=i−2
Similarly, for π2 : P(A) → Z, we have
i X i1 i2 π2∗(vA) = si−1(A) = (−a1) (−a2) . i1+i2=i−1
Thus
n π1∗s PP(A)(B ⊗ OP(A)(−1) ⊕ O) " n−3 # n n n n n X n =π1∗ (b1 − vA + wB) +(b2 − vA + wB) +wB +(a1 + vA) +(a2 + vA) + zi i=1 n n X n X n = (b − v )n−iπ (wi ) + (b − v )n−iπ (wi ) + π (wn ) i 1 A 1∗ B i 2 A 1∗ B 1∗ B i=0 i=0 n X n X = (b − v )n−i (−b + v )i1 (−b + v )i2 i 1 A 1 A 2 A i=2 i1+i2=i−2 n X n X + (b − v )n−i (−b + v )i1 (−b + v )i2 i 2 A 1 A 2 A i=2 i1+i2=i−2
X i1 i2 + (−b1 + vA) (−b2 + vA) .
i1+i2=n−2
152 In the case when a2 = b1 = b2 = 0, we have:
n n X n n−i X i−2 X n−2 π1∗s (A)(B ⊗ O (A)(−1) ⊕ O) = 2 (−vA) (vA) + (vA) PP P i i=2 i1+i2=i−2 i1+i2=n−2 n X n = 2 (−v )n−i(i − 1)(v )i−2 + (n − 1)(v )n−2 i A A A i=2
n−2 n = (vA) (2(−1) + n − 1).
n−2 n−3 According to Lemma 4.6.2, π2∗(vA) = (−a1) , since a2 = 0. Therefore,
n n−3 n π2∗π1∗s PP(A)(B ⊗ OP(A)(−1) ⊕ O) = (−a1) (2(−1) + n − 1)
n−3 n−3 = (a1) (−2 + (−1) (n − 1)).
0 0 We do the same for the projection π1 : PP(B)(A⊗OP(B)(−1)⊕1) → P(B), and π2 : P(B) →
Z, in the case of a = b = b = 0. By Lemma 4.6.2, π0 (wi ) = P (−a +v )i1 (v )i2 , 2 1 2 1∗ A i1+i2=i−2 1 B B and hence,
0 n π1∗s PP(B)(A ⊗ OP(B)(−1) ⊕ 1) n−3 0 n n n n X n = π1∗((a1 − vB + wA) + (−vB + wA) + wA + 2(vB) + (zi) ) i=1 n n X n X n = (a − v )n−iπ (wi ) + (−v )n−iπ (wi ) + π (wn ) i 1 B 1∗ A i B 1∗ A 1∗ A i=0 i=0 n X n X = (a − v )n−i (−a + v )i1 (v )i2 i 1 B 1 B B i=2 i1+i2=i−2 n X n X X + (−v )n−i (−a + v )i1 (v )i2 + (−a + v )i1 (v )i2 . i B 1 B B 1 B B i=2 i1+i2=i−2 i1+i2=n−2
0 0 0 i For π2∗, Lemma 4.6.2 tells us that π2∗(vB) = 1, and π2∗(vB) = 0 for i 6= 1. Finally, we obtain:
0 0 n n−3 2 n n−3 n−3 π π s (B)(A ⊗ O (B)(−1) ⊕ 1) = a (−(n − 1) + − (−1) + (n − 1)(−1) ) 2∗ 1∗ PP P 1 2 n = an−3(−(n − 1)2 + + (n − 2)(−1)n−3) 1 2
153 1 To show this can be non-vanishing for certain choice of Z and a1 ∈ CH (Z), we simply
n−3 take Z = P and a1 = c1(OPn−3 (1)). We then have
n sn[X − X ] = han−3(−2 + (−1)n−3(n − 1) + (n − 1)2 − − (n − 2)(−1)n−3), [ n−3]i 1 2 1 2 P n2 − 3n − 2 + 2(−1)n−1 = , 2
n so for n ≥ 5, s [X1 − X2] 6= 0, as desired.
154 Chapter 5
On deformed double current algebras
for simple Lie algebras
5.1 Deformed double current algebras
For a simple Lie algebra g over C with rank ≥ 3, let ∆ be the set of roots for g, and ∆+ be a
N set of positive roots. Let C = (cij)i,j=0 be the affine Cartan matrix of the affine Lie algebra
N bg and the numbers d0, d1, . . . , dN are such that (dicij)i,j=0 is a symmetric matrix.
Proposition 5.1.1 ([35], Lemma 4.1, 4.2). The universal central extension bg(C[u, v]) of + g(C[u, v]) := g ⊗C C[u, v] is isomorphic to the Lie algebra generated by elements Xi,r, Hi,r
+ + for 1 ≤ i ≤ N, r = 0, 1 and X0,0, X0,1 subject to the following relations:
[H ,H ] = 0, [H ,X± ] = ±d c X± , i1,r1 i2,r2 i1,0 i3,r3 i1 i1,i3 i3,r3
for1 ≤ i1, i2 ≤ N, 0 ≤ i1, i2 ≤ N, r1, r2, r3 = 0 or 1 (5.1.1)
[H ,X± ] = [H ,X± ], [X± ,X± ] = [X± ,X± ], 0 ≤ i , i ≤ N (5.1.2) i1,1 i2,0 i1,0 i2,1 i1,1 i2,0 i1,0 i2,1 1 2
[X+ ,X− ] = δ H , 0 ≤ i ≤ N, 1 ≤ i ≤ N, r + r = 0, 1 (5.1.3) i1,r1 i2,r2 i1i2 i1,r1+r2 1 2 1 2
[X± , [X± ,..., [X± ,X± ] ... ]] = 0. (5.1.4) i1,0 i1,0 i1,0 i2,0
155 In (5.1.2), (5.1.4), when i1 = 0, i2 = 0 or i3 = 0, there is a relation only in the +–case.
Assume that g is not of Dynkin type A. Then there is a unique k ∈ {1,...,N} such that
c0k 6= 0. In other words, k is the label of the unique vertex in the Dynkin diagram of bg to which the zero node is connected. Let θ be the highest root of g. Let
1 X 1 ω± := ± S [X±,X±],X∓ − S(X±,H ) ∈ U(g) (5.1.5) i 4 i α α 4 i i α∈∆+
+ − and νi := [ωi ,Xi ] ∈ U(g).
Definition 5.1.1 ([35], Definition 5.1). The deformed double current algebra D(g) is the
± + algebra generated by Xi,r, Hi,r, X0,r for 1 ≤ i ≤ N, r = 0, 1 subjected to the same relations
+ as those in Proposition 5.1.1, except that the following relations involving X0,r must be modified: d [X+ , X+ ] − [X+ , X+ ] = − 0 S(X+ , X−) + [ω+, X−] + [X+ , ω+] (5.1.6) k,1 0,0 k,0 0,1 2 k,0 θ k θ k,0 0 d [H , X+ ] − [H , X+ ] = − 0 S(H+ , X−) + d ω+ + [ν , X−] (5.1.7) k,1 0,0 k,0 0,1 2 k,0 θ 0 0 k θ
+ − − + + + + − [X0,1, Xk,0] = [Xk,0, ω0 ], [X0,1, X0,0] = 2d0X0,0Xθ (5.1.8)
+ ± − ± + ± + ± [X0,0, Xi,1] = [Xθ , ωi ], [X0,1, Xi,0] = −[ω0 , Xi,0] for i 6= k (5.1.9)
The elements X− and ω+ are defined in the following way. We write X− as X− = [X− ,X− ] θ 0 θ θ k,0 θ−αk (it may be necessary to rescale X± to achieve this) and θ−αk X− = [X− , X− ] ∈ D(g). (Here, X− ∈ g ⊂ D(g).) Set ω+ = −[ω−,X− ]. θ k,0 θ−αk θ−αk 0 k θ−αk
Let’s consider another algebra:
Definition 5.1.2. 1 The algebra D(g) is generated by elements X,K(X),Q(X),P (X),X ∈
g, such that
•{ K(X),X | X ∈ g} generate a subalgebra which is an image of g⊗C C[u] with X ⊗u 7→ K(X);
1 This definition is first found in [34] in the case of sln, for n ≥ 4.
156 •{ Q(X),X | X ∈ g} generate a subalgebra which is an image of g⊗C C[v] with X ⊗v 7→ Q(X);
• P (X) is linear in X, and for any X,X0 ∈ g,[P (X),X0] = P [X,X0],
and the following relation holds for all root vectors Xβ1 ,Xβ2 ∈ g with β1 6= −β2:
(β1, β2) 1 X [K(X ),Q(X )] = P ([X ,X ]) − S(X ,X ) + S([X ,X ], [X ,X ]), β1 β2 β1 β2 4 β1 β2 4 β1 α −α β2 α∈∆ (5.1.10)
Theorem 5.1.3. There exists an algebra isomorphism ϕ : D(g) → D(g) given by
± ± ϕ(Xi,0) = Xi , ϕ(Hi,0) = Hi
± ± ϕ(Xi,1) = Q(Xi ), ϕ(Hi,1) = Q(Hi), for 1 ≤ i ≤ N
− + − + ϕ(X0,0) = K(Xθ ), ϕ(X0,1) = P (Xθ ) − ω0 .
In particular, the presentation in Definition 5.1.2 gives a double loop presentation of the
deformed double current algebra.
5.2 The map ϕ is a homomorphism of algebras
This section is devoted to prove that the map ϕ defined in Theorem 5.1.3 is a homomorphism
of algebras.
± Lemma 5.2.1. Let ωi be the element in U(g) defined in (5.1.5), we have
1 X ω± = ∓ S [X±,X∓],X±. (5.2.1) i 4 i α α α∈∆+
Proof. The Casimir element of g is
N X + − X i Ω = S(Xα ,Xα ) + eh ehi α∈∆+ i=1
157 1 N where {eh1,..., ehN } is a basis of h and {eh ,..., eh } is the dual basis. Ω is in the center of U(g), so 1 1 X 0 = ± [X±, Ω] = ω± ± S(X±, [X±,X∓]) 4 i i 4 α i α α∈∆+ because
N N ± X j X ± j j ± [Xi , eh ehj] = ([Xi , eh ]ehj + eh [Xi , ehj]) j=1 j=1 N X j ± j ± ± = ∓ ((Hi, eh )Xi ehj + eh (ehj,Hi)Xi ) = ∓S(Xi ,Hi). j=1 The conclusion follows.
We have to verify that ϕ is a homomorphism of algebras, that is, that it respects the
defining relations of D(g).
Let’s start with (5.1.6). We have to check that
d0 1 X [Q(X+),K(X−)] − [X+,P (X−)] = − S(X+,X−) − S([X+,X−], [X+,X−]) k θ k θ 2 k θ 4 k α α θ α∈∆+ 1 − S([[X+,X−],X−],X+). 4 k k θ k
− − + − − (Observe that [Xk ,Xθ ] = 0 and that if [[Xk ,Xα ],Xθ ] is a root vector (where α is positive), then αk − α − θ ≥ −θ, so α ≤ αk: since αk is simple and both are positive, this forces α to be equal to αk.) The previous equality is equivalent to
d0 1 X [Q(X+),K(X−)] − [X+,P (X−)] = − S(X+,X−) − S([X+,X−], [X+,X−]) k θ k θ 4 k θ 4 k α α θ α∈∆+ and to
d0 1 X [K(X−),Q(X+)] = P ([X−,X+]) + S(X+,X−) + S([X+,X−], [X+,X−]) θ k θ k 4 k θ 4 k α α θ α∈∆+ This is the same as
(−θ, αk) 1 X [K(X−),Q(X+)] = P ([X−,X+]) − S(X+,X−) + S([X−,X ], [X ,X+]) θ k θ k 4 k θ 4 θ α −α k α∈∆
158 which is true in D(g). Thus, ϕ preserves relation (5.1.6).
(5.1.7) follows from (5.1.6). For the same reason, the first relation in (5.1.8) holds because
− − [Xθ ,Xk,0] = 0.
− + − For the second relation in (5.1.8), we need to compute [P (Xθ ) − ω0 ,K(Xθ )]. Notice that if [X−,X− ] 6= 0 for a positive root α, then it is a root vector in g for the root α θ−αk
−α − θ + αk, so −α − θ + αk ≥ −θ and thus α ≤ αk: this forces α = αk because αk is a simple root. Similarly, if [[[X−,X+],X−],X− ] 6= 0 then it is a root vector in g for the k α θ θ−αk
root −αk + α − θ − θ + αk, so −αk + α − θ − θ + αk ≥ −θ hence α ≥ θ: this implies that α = θ. These two observations serve to obtain the third equality below.
[ω+,K(X−)] = −[[ω−,X− ],K(X−)] = −[[ω−,K(X−)],X− ] 0 θ k θ−αk θ k θ θ−αk 1 X = − [S([[X−,X+],K(X−)],X−),X− ] 4 k α θ α θ−αk α∈∆+ 1 1 = − S([[[X−,X+],K(X−)],X− ],X−) − S([[X−,X+ ],K(X−)], [X− ,X− ]) 4 k θ θ θ−αk θ 4 k αk θ αk θ−αk 1 − − − 1 − − = − S([[X ,K(Hθ)],X ],X ) + S([Hk,K(X )],X ) 4 k θ−αk θ 4 θ θ 1 − − − 1 − − = − d0S([K(X ),X ],X ) − d0S(K(X ),X ) 4 k θ−αk θ 4 θ θ 1 1 = − d S(K(X−),X−) − d S(K(X−),X−) 4 0 θ θ 4 0 θ θ − − = −d0K(Xθ )Xθ (5.2.2)
− − − − We show that [P (Xθ ),K(Xθ )] = d0K(Xθ )Xθ as follows. Start with
− − − (αk, θ − αk) − − 1 X − − [K(X ),Q(X )] = P (X )− S(X ,X )+ S([X ,Xα], [X−α,X ]) αk θ−αk θ 4 αk θ−αk 4 αk θ−αk α∈∆ (5.2.3)
Observe that if [X− ,X ] 6= 0 and [[X ,X− ],X−] 6= 0, then both are roots vectors for αk α −α θ−αk θ the roots −αk + α and −α − θ + αk − θ, so −αk + α ≥ −θ and −α − θ + αk − θ ≥ −θ, hence
α ≥ −θ + αk, −θ + αk ≥ α and α = −θ + αk. Similarly, if [[X− ,X ],X−] 6= 0 and [X ,X− ] 6= 0, then both are root vectors for the αk α θ −α θ−αk roots −αk + α − θ and −α − θ + αk; it follows that −αk + α − θ ≥ −θ and −α − θ + αk ≥ −θ, so α ≥ αk and αk ≥ α, thus α = αk.
159 − Using these two observations, we now apply [·,K(Xθ )] to (5.2.3) to get
[K(X− ), [Q(X− ),K(X−)]] (5.2.4) αk θ−αk θ
− − 1 − − − = [P (X ),K(X )] + S([X ,X−θ+α ], [[Xθ−α ,X ],K(X )]) θ θ 4 αk k k θ−αk θ 1 − − − + S([[X ,Xα ],K(X )], [X−α ,X ]) 4 αk k θ k θ−αk 1 = [P (X−),K(X−)] + S(X , [H ,K(X−)]) θ θ 4 −θ θ−αk θ d + 0 S(K(X−),X−) 4 θ θ d d = [P (X−),K(X−)] − 0 S(X ,K(X−)) + 0 S(K(X−),X−) θ θ 4 −θ θ 4 θ θ − − = [P (Xθ ),K(Xθ )] (5.2.5)
We compute the left-hand side of the above equality (5.2.5).
− − (θ − αk, θ) − − 1 − − [Q(X ),K(X )] = S(X ,X ) − S([X ,Xα ], [X−α ,X ]) θ−αk θ 4 θ−αk θ 4 θ k k θ−αk d0 − − 1 − − = S(X ,X ) − S([X ,Xα ],X ) 4 θ−αk θ 4 θ k θ so
− − − d0 − − − 1 − − [K(X ), [Q(X ),K(X )]] = S(K([X ,X ]),X ) + S([X ,K(Hk)],X ) αk θ−αk θ 4 αk θ−αk θ 4 θ θ d = 0 S(K(X−),X ) = d K(X−)X . (5.2.6) 2 θ −θ 0 θ −θ
− − − − Substituting (5.2.6) into (5.2.5) yields [P (Xθ ),K(Xθ )] = d0K(Xθ )Xθ and it follows from
− + − − (5.2.2) that [P (Xθ ) − ω0 ,K(Xθ )] = 2d0K(Xθ )X−θ as desired.
Finally, we check the first relation in (5.1.9), so with i 6= 0, k (hence (θ, αi) = 0). 1 X [K(X−),Q(X−)] = −[ω−,X−] = S([X−,X−], [X+,X−]) θ i i θ 4 i α α θ α∈∆+ using (5.1.5) and
1 X [K(X−),Q(X+)] = −[ω+,X−] = S([X+,X−], [X+,X−]) θ i i θ 4 i α α θ α∈∆+ this time using (5.2.1). Both of these relations do hold in D(g).
− ± − ± The second relation in (5.1.9) is also satisfied since [P (Xθ ),Xi,0] = P ([Xθ ,Xi,0]) = 0 when i 6= k.
160 5.3 The map ϕ is an isomorphism of algebras.
In this section, we construct the inverse map ψ : D(g) → D(g) of ϕ.
Let us first describe the images ψ(X), ψ(K(X)), ψ(Q(X)) and ψ(P (X)) of the generators
of D(g) under the map ψ, where X is any element in g.
Lemma 5.3.1 ([]). The current Lie algebra g[u] is isomorphic to the Lie algebra generated
by elements ei, fi, hi for 1 ≤ i ≤ N and e0 subject to the following relations: for 1 ≤ i ≤ N,
ei, fi, hi satisfy the usual relations for Serre’s Theorem for g;
[hi, e0] = dici0e0, [e0, fi] = 0,;
1−ci0 1−c0i ad(ei) (e0) = 0 and ad(e0) (ei) = 0.
Note that the above relations of g[u] are the defining relations of the Kac-Moody algebra bg, except those involving f0, h0. In the defining relations of D(g), {K(X),X | X ∈ g} generate a subalgebra g[u]. By
Lemma 5.3.1, we have homomorphisms U(g[u]) → D(g), given by X 7→ X, for X ∈ g, and
− K(Xθ ) 7→ X0,0. Likewise, we have homomorphisms U(g[u]) → D(g), given by X 7→ X, for
± ± X ∈ g, and Q(Xi )) 7→ Xi,1, for 1 ≤ i ≤ N. The above homomorphisms tell us how to define ψ(K(X)) and ψ(Q(X)) in D(g), for any X ∈ g.
Lemma 5.3.2. There is an ad(g)-module morphism P : g → D(g), such that X−θ 7→
+ + P (X−θ) := X0,1 + ω0 . In particular, let P (X) to be the image of X ∈ g under the morphism P , then
1. P (X) is linear in X;
2. [X,P (X0)] = P ([X,X0]), for any X,X0 ∈ g.
Proof. 2 Let g = n− ⊕ h ⊕ n be the triangular decomposition of g, and b− := n− ⊕ h.
Let M(−θ) := U(g) ⊗U(b−) C−θ be the Verma module with lowest weight −θ, where C−θ 2This proof is given by Valerio Toledano Laredo.
161 is the 1–dimensional b−–module with trivial n−-action. Thanks to the PBW Theorem, ∼ ∼ U(g) = U(n) ⊗b− C−θ, which allows us to write M(−θ) = U(n) ⊗ C−θ. Denote the lowest weight vector of M(−θ) by v− := 1 ⊗ 1. Then, M(−θ) as a U(g) is generated by v−, such that
n− · v− = 0 and h · v− = h(−θ)v−, for all h ∈ h. (5.3.1)
P As in the following diagram M(−θ) e / D(g) , we first construct a morphism Pe : = P # g − + + M(−θ) → D(g), given by v 7→ P (X−θ) := X0,1 + ω0 , then show Pe factors through the adjoint representation of g, which gives the desired map P : g → D(g).
To show the morphism Pe is well-defined, it suffices to verify that P (X−θ) satisfies (5.3.1).
− Rewriting the defining relation (5.1.8) of D(g), we have: [P (X−θ),Xk,0] = 0, and similarly,
± relation (5.1.9) gives the following relation: for i 6= k,[P (X−θ),Xi,0] = 0. It remains to check
[Hi,0,P (X−θ)] = (αi, −θ)P (X−θ), (5.3.2)
+ + + This follows from the relation in Proposition 5.1.1 that [Hi,0,X0,1] = dici,0X0,1 = (αi, −θ)X0,1 and
[H , ω+] = −[H , [ω−,X− ]] i,0 0 i,0 k θ−αk
= −[[H , ω−],X− ] − [ω−, [H ,X− ]] i,0 k θ−αk k i,0 θ−αk
= −(α , −α )[ω−,X− ] − (α , −θ + α )[ω−,X− ] i k k θ−αk i k k θ−αk
= −(α , −θ)[ω−,X− ] i k θ−αk
+ = −(αi, θ)ω0
Thus, we have a (non-zero) g-equivariant map from the Verma module M(−θ) with lowest
− weight −θ to D(g) mapping the lowest weight vector v to P (X−θ). We show D(g) is locally finite as an ad(g)-module, that is, any vector is contained in
a finite-dimensional g-module. Note that the algebra D(g) is bigraded, with the following
grading:
162 ± • deg(Xi,0) = (0, 0), for i = 1, . . . , n.
± + • deg(Xi,1) = (1, 0), for i = 1, . . . , n. and deg(X0,0) = (0, 1).
+ • deg(λ) = (1, 1) and deg(X0,1) = (1, 1).
Now each graded piece is invariant under the adjoint action of g, since the degree of the ele- ments in g is (0, 0). The degree (0, 0) piece of D(g) coincide with the envoloping algebra U(g)
and each graded piece as a U(g)-module is finite generated. Since we know the envoloping algebra U(g) as ad(g)-module is locally finite, thus, each graded piece of D(g) is also locally
finite as an ad(g)-module.
Thus, this g-equivariant map Pe must factor through a finite dimensional quotient of the Verma module M(−θ), and there is only one such quotient, the adjoint representation of g.
The remainder of the proof consists in checking that, for any two roots β1 6= −β2,
[ψ(K(Xβ1 )),ψ(Q(Xβ2 ))] = ψ(P ([Xβ1 ,Xβ2 ])
(β1, β2) 1 X − S(ψ(X ), ψ(X )) + S([ψ(X ), ψ(X )], [ψ(X ), ψ(X )]). 4 β1 β2 4 β1 α −α β2 α∈∆
From the defining relations of D(g), this is known to be true in the cases when β1 = −θ and
β2 = ±α1,..., ±αN , where the αi are the simple roots of g. In order to see that it’s true in general, we can use the operators si which have the property that si(Xα) is a root vector for the root si(α).
Suppose that (5.1.10) holds when β1 = −θ and β2 = ±α1,..., ±αN . (These cases correspond under ϕ to defining relations of D(g) given in definition 5.1.1.) The goal is to show, using this assumption and [K(X1),X2] = K([X1,X2]), [Q(X1),X2] = Q([X1,X2]),
[P (X1),X2] = P ([X1,X2]), that (5.1.10) must hold in full generality for any two roots β1, β2 with β1 6= −β2.
163 Let m : U(g) ⊗C U(g) −→ U(g) be the multiplication map. The following observation will be useful below:
! X X S([Xβ1 ,Xα], [X−α,Xβ2 ]) = m [Xβ1 ⊗ 1,Xα ⊗ X−α], 1 ⊗ Xβ2 α∈∆ α∈∆ ! X +m [1 ⊗ Xβ1 ,X−α ⊗ Xα],Xβ2 ⊗ 1 α∈∆ N ! X i = m [Xβ1 ⊗ 1, Ω], 1 ⊗ Xβ2 − [Xβ1 ⊗ 1, eh ⊗ ehi], 1 ⊗ Xβ2 i=1 N ! X i +m [1 ⊗ Xβ1 , Ω],Xβ2 ⊗ 1 − [1 ⊗ Xβ1 , eh ⊗ ehi],Xβ2 ⊗ 1 i=1 where we view Ω as an element of g ⊗C g. Consequently, for any X ∈ g, we have [Ω,X ⊗ 1 + 1 ⊗ X] = 0 and
X X [S([Xβ1 ,Xα], [X−α,Xβ2 ]),Xγ] − S([[Xβ1 ,Xγ],Xα], [X−α,Xβ2 ]) α∈∆ α∈∆ X − [S([Xβ1 ,Xα], [X−α, [Xβ2 ,Xγ]]) (5.3.3) α∈∆
= − (γ, β2)S([Xβ1 ,Xγ],Xβ2 ) − (γ, β1)S(Xβ1 , [Xβ2 ,Xγ]), (5.3.4)
(5.3.5)
here we write X = Xγ ∈ gγ, for γ ∈ ∆ ∪ {0}. The above equality follows from the following computation:
N X h i i (6.16.1) = − m Xβ1 ⊗ 1, [eh ⊗ ehi,Xγ ⊗ 1 + 1 ⊗ Xγ] , 1 ⊗ Xβ2 i=1 N X h i i − m 1 ⊗ Xβ1 , [eh ⊗ ehi,Xγ ⊗ 1 + 1 ⊗ Xγ] ,Xβ2 ⊗ 1 i=1 N X h i i i = − m Xβ1 ⊗ 1, (eh ,Hγ)Xγ ⊗ ehi + (ehi,Hγ)eh ⊗ Xγ , 1 ⊗ Xβ2 i=1 N X h i i i − m 1 ⊗ Xβ1 , (eh ,Hγ)Xγ ⊗ ehi + (ehi,Hγ)eh ⊗ Xγ ,Xβ2 ⊗ 1 i=1
164 h i = − m Xβ1 ⊗ 1,Xγ ⊗ Hγ + Hγ ⊗ Xγ , 1 ⊗ Xβ2 h i − m 1 ⊗ Xβ1 ,Xγ ⊗ Hγ + Hγ ⊗ Xγ ,Xβ2 ⊗ 1
= − m ([Xβ1 ,Xγ] ⊗ [Hγ,Xβ2 ] + [Xβ1 ,Hγ] ⊗ [Xγ,Xβ2 ])
− m ([Xγ,Xβ2 ] ⊗ [Xβ1 ,Hγ] + [Hγ,Xβ2 ] ⊗ [Xβ1 ,Xγ])
= − m ((γ, β2)[Xβ1 ,Xγ] ⊗ Xβ2 + (γ, β1)Xβ1 ⊗ [Xβ2 ,Xγ])
− m ((γ, β1)[Xβ2 ,Xγ] ⊗ Xβ1 + (γ, β2)Xβ2 ⊗ [Xβ1 ,Xγ])
= − (γ, β2)S([Xβ1 ,Xγ],Xβ2 ) − (γ, β1)S(Xβ1 , [Xβ2 ,Xγ]) = (6.16.2).
We are supposing that (5.1.10) holds when β1 = −θ and β2 = ±αk. For convenience, let’s write down these two relations here:
(θ, αk) 1 X [K(X−),Q(X+)] = P ([X−,X+]) + S(X−,X+) + S([X−,X ], [X ,X+]) θ k θ k 4 θ k 4 θ α −α k α∈∆ (5.3.6) and
(θ, αk) 1 X [K(X−),Q(X−)] = − S(X−,X−) + S([X−,X ], [X ,X−]). θ k 4 θ k 4 θ α −α k α∈∆ + Let’s apply [Xθ , ·] to this second relation to obtain, note that:
+ − + − + − + − ([Xθ ,Xk ],X−θ+αk ) = (Xθ , [Xk ,X−θ+αk ]) = (Xθ ,Xθ ) = 1 =⇒ [Xθ ,Xk ] = Xθ−αk .
(θ, α ) (θ, α ) [K(H ),Q(X−)] + [K(X−),Q(X )] = − k S(H ,X−) − k S(X−,X ) θ k θ θ−αk 4 θ k 4 θ θ−αk 1 X 1 X + S([H ,X ], [X ,X−]) + S([X−,X ], [X ,X ]) 4 θ α −α k 4 θ α −α θ−αk α∈∆ α∈∆
(θ, αk) − d0 − + + S(Hθ,X ) + S(X ,X ) by (6.16.2) 4 k 2 θ θ−αk 1 X − 1 X − d0 − + = S([Hθ,Xα], [X−α,X ]) + S([X ,Xα], [X−α,Xθ−α ]) + S(X ,X ) 4 k 4 θ k 4 θ θ−αk α∈∆ α∈∆ (5.3.7)
q 2 + q 2 − Set ei = X and fi = X for i 6= 0. Consider the operators in the (αi,αi) i (αi,αi) i adjoint representation of g given by si = exp(ad(fi)) exp(−ad(ei)) exp(ad(fi)). It is known
165 that X ∈ gα =⇒ si(Xα) ∈ gsi(α) (where gα is the root subspace of g for the root α) and
2(αi,αj ) si(Hj) = Hj − Hi. We will use si to denote either a simple reflection in the Weyl (αi,αi) group W of g or the corresponding operator in its adjoint representation.
Let w0 be the longest element of the Weyl group. Then −w0 is a permutation of the simple roots α1, . . . , αk and it can be checked that −w0(αk) = αk in all types (except type
A, which we are not considering). Express w0 as a product of simple reflections w0 = si1 ··· si`
and denote also by w0 the corresponding operator in the adjoint representation of g for this
choice of decomposition. Applying w0 to (5.3.6) shows that the following relation also holds (after rescaling):
(θ, αk) 1 X [K(X+),Q(X−)] = P ([X+,X−]) + S(X+,X−) + S([X+,X ], [X ,X−]). θ k θ k 4 θ k 4 θ α −α k α∈∆ (5.3.8)
(For any w ∈ W , one can choose a decomposition into simple transpositions w = sj1 ··· sjk and obtain a corresponding operator in the adjoint representation with the property that w(Xα) is a scalar multiple of Xw(α); moreover, since the Killing form is invariant for the ad-
−1 joint action, (w(Xα), w(X−α)) = (Xα,X−α) = 1, so w(Xα) = aXw(α), w(X−α) = a X−w(α)
+ − + − for some non-zero scalar a and S([Xθ ,Xw(α)], [X−w(α),Xk ]) = S([Xθ , w(Xα)], [w(X−α),Xk ]).
+ A similar observation applies to w(X−θ) and w(Xk ).)
− Applying [·,Xθ ] to (5.3.8) gives
(θ, αk) 1 X [K(H ),Q(X−)] = P ([H ,X−]) + S(H ,X−) + S([H ,X ], [X ,X−]) θ k θ k 4 θ k 4 θ α −α k α∈∆ d − 0 S(H ,X−) 4 θ k 1 X = P ([H ,X−]) + S([H ,X ], [X ,X−]) because (−θ, α ) = d c = −d (5.3.9) θ k 4 θ α −α k k 0 0k 0 α∈∆
166 (6.16.2) was useful again here to obtain this relation. Indeed,
X + − − X + − − [S([Xθ ,Xα], [X−α,Xk ]),Xθ ] − S([[Xθ ,Xθ ],Xα], [X−α,Xk ]) α∈∆ α∈∆
X + − − − [S([Xθ ,Xα], [X−α, [Xk ,Xθ ]]) α∈∆
+ − − + − − = − (θ, αk)S([Xθ ,Xθ ],Xk ) − (−θ, θ)S(Xθ , [Xk ,Xθ ])
− = − d0S(Hθ,Xk )
Combining (5.3.9) with (5.3.7) yields
− − 1 X − d0 − + [K(X ),Q(Xθ−α )] = −P ([Hθ,X ]) + S([X ,Xα], [X−α,Xθ−α ]) + S(X ,X ) θ k k 4 θ k 4 θ θ−αk α∈∆ which is equivalent to
− [K(Xθ ),Q(Xθ−αk )]
(θ, αk − θ) 1 X = P ([X−,X ]) − S(X−,X ) + S([X−,X ], [X ,X ]) (5.3.10) θ θ−αk 4 θ θ−αk 4 θ α −α θ−αk α∈∆
− − − − − because [Hθ,Xk ] = −(θ, αk)Xk = d0c0kXk = −d0Xk and [Xθ ,Xθ−αk ] is a scalar multiple
− − + of Xk with scalar given by ([Xθ ,Xθ−αk ],Xk ), which in turn equals:
− + − + − + ([Xθ ,Xθ−αk ],Xk ) = −([Xθ−αk ,Xθ ],Xk ) = −(Xθ−αk , [Xθ ,Xk ])
= (θ, α )(X ,X− ) = (θ, α ) = d , k θ−αk θ−αk k 0
where the third equality follows from [X−,X+] = −(θ, α )X− , since θ k k θ−αk
− + + − + − ([Xθ ,Xk ],Xθ−αk ) = −([Xk ,Xθ ],Xθ−αk ) = −(Xk , [Xθ ,Xθ−αk ])
+ − + − = −(Xk , [Xθ , [Xθ ,Xk ]])
+ + − − + − − + = (Xk , [Xθ , [Xk ,Xθ ]]) + (Xk , [Xk , [Xθ ,Xθ ]])
+ − = −(θ, αk)(Xk ,Xk ) = −(θ, αk).
Now let β be any positive root different from θ −αk. Then there exist simple roots β1, . . . , β` such that [X− , [..., [X− ,X ]]] is a root vector in g ; let’s denote it by X . (This is true β` β1 θ−αk β eβ
167 for Xθ instead of Xθ−αk because θ is the highest root of g, but the only simple root βi such that [X−,X+] 6= 0 is β = α , so we might as well start with X .) Applying [X−, ·] to βi θ i k θ−αk βi (5.3.10) and using that [X−,X−] = 0, we obtain in the end (after perhaps rescaling since βi θ
Xβ and Xeβ may differ by a non-zero constant) that
(θ, β) 1 X [K(X−),Q(X )] = P ([X−,X ])+ S(X−,X )+ S([X−,X ], [X ,X ]) (5.3.11) θ β θ β 4 θ β 4 θ α −α β α∈∆
Let’s prove this by induction on `. We already know that this is true when ` = 0 by (5.3.10), so let’s assume it’s true for ` − 1. Set X = [X− , [..., [X− ,X ]]], so X = [X− , X ] and eβe β`−1 β1 θ−αk eβ β` eβe
− − (θ, βe) − 1 X − [K(X ),Q(Xe )] = P ([X , Xe ]) + S(X , Xe ) + S([X ,Xα], [X−α, Xe ]) θ βe θ βe 4 θ βe 4 θ βe α∈∆
We want to apply [X− , ·] to both sides and the only term which is unclear is the last one. The β` following computation helps to understand the difference D between P [X− ,S([X−,X ], [X ,X ])] α∈∆ β` θ α −α βe P − and α∈∆ S([Xθ ,Xα], [X−α, Xeβ]). Using (6.16.2), we obtain that
D = (−β , β)S([X−,X− ], X ) + (β , θ)S(X−, [X ,X− ]) ` e θ β` eβe ` θ eβe β`
− = −(β`, θ)S(Xθ , Xeβ)
To complete the induction step, note that β = βe − β`. We still have to prove (5.3.11) when β is a negative root. Write X = [X− , [..., [X− ,X−]]] β β` β1 i for some simple roots αi, β1, . . . , β`. This can be done by induction as above, this time starting with relation (5.1.10) with β = −θ, β = −α and applying successively [X− , ·], j = 1 2 i βj 1 . . . , ` to both sides. We have now proved that (5.3.11) holds for any root β of g different from θ. Let β1 be any long root for g. Then there exists w ∈ W (W being the Weyl group of g) such that β1 = w(−θ).
Let w = si1 si2 ··· si` be a decomposition of w into a product of simple reflections. Let’s denote also by w the corresponding operator in the adjoint representation of g for this choice
168 of decomposition. Relation (5.1.10) when β1 is a long root now follows by applying w to
−1 (5.3.11) with β = w (β2).
Therefore, (5.1.10) is true when β1 is a long root. Using (5.3.8), it could also be proved
that it is true when β2 is a long root. If g is simply laced, this means that it holds for any
two roots β1, β2 with β1 6= −β2.
It remains to deal with the case when both β1, β2 are short roots, with β1 6= −β2.
+ Apply [Xθ , ·] to relation (5.3.11) in the case that β is any positive root but β 6= θ. Then,
[K(Hθ),Q(Xβ)] (5.3.12) (θ, β) 1 X (θ, β) =P ([H ,X ]) + S(H ,X ) + S([H ,X ], [X ,X ]) − S(H ,X ) θ β 4 θ β 4 θ α −α β 4 θ β α∈∆ 1 X =P ([H ,X ]) + S([H ,X ], [X ,X ]) θ β 4 θ α −α β α∈∆
In the case that β is a negative root with β 6= −θ, we can apply [·,X−θ] to relation (5.1.10)
with β1 = θ, and β2 = β, the same argument as above shows that for any root β, such that β 6= ±θ, then
1 X [K(H ),Q(X )] = P ([H ,X ]) + S([H ,X ], [X ,X ]). θ β θ β 4 θ α −α β α∈∆
We are in a similar situation as before, use the action of Tits extension Wf of Weyl group on g, we obtain for any long root γ, for any root β 6= ±γ,
1 X [K(H ),Q(X )] = P ([H ,X ]) + S([H ,X ], [X ,X ]). (5.3.13) γ β γ β 4 γ α −α β α∈∆
For any root η ∈ ∆ of g, we will take η to be a short root later, there exist some long root γ ∈ ∆, such that (γ, η) 6= 0. For η 6= −β, apply [·,Xη] to relation (5.3.13) with such γ,
169 denote [Xβ,Xη] = aXβ+η, where a ∈ C. Then,
[[K(Hγ),Q(Xβ)],Xη] 1 X =(γ, η)P ([X ,X ]) + aP ([H ,X ]) + [S([H ,X ], [X ,X ]),X ] η β γ β+η 4 γ α −α β η α∈∆ 1 X =(γ, η)P ([X ,X ]) + aP ([H ,X ]) + (γ, η) S([X ,X ], [X ,X ]) η β γ β+η 4 η α −α β α∈∆ 1 X 1 + a S([H ,X ], [X ,X ]) − (γ, η)(β, η)S(X ,X ) by (6.16.2) 4 γ α −α β+η 4 η β α∈∆ (5.3.14)
On the other hand, if β + η 6= ±γ, using (5.3.13), we get:
[[K(Hγ),Q(Xβ)],Xη] = (γ, η)[K(Xη),Q(Xβ)] + a[K(Hγ),Q(Xβ+η)] 1 X =(γ, η)[K(X ),Q(X )] + aP ([H ,X ]) + a S([H ,X ], [X ,X ]) (5.3.15) η β γ β+η 4 γ α −α β+η α∈∆
Combine (5.3.14) with (5.3.15), if β + η 6= ±γ,(β and η could be short roots), we get
(β, η) 1 X [K(X ),Q(X )] = P ([X ,X ]) − S(X ,X ) + S([X ,X ], [X ,X ]) (5.3.16) η β η β 4 η β 4 η α −α β α∈∆
Note that the root γ we are choosing is not unique, we can vary the root γ as long as γ being a long root and (η, γ) 6= 0, which implies (5.3.16) holds for all roots η, and β, with η 6= −β.
Therefore, the relation (5.1.10) holds in D(g).
5.4 Some properties of the deformed double current
algebras
Corollary 5.4.1. For any roots β1, β2, the following relations hold in the deformed double current algebra D(g):
1 X [K(H ),Q(X )] = P ([H ,X ]) + S([H ,X ], [X ,X ]), (5.4.1) β1 β2 β1 β2 4 β1 α −α β2 α∈∆
170 and 1 X [K(X ),Q(H )] = P ([X ,H ]) + S([X ,X ], [X ,H ]). β1 β2 β1 β2 4 β1 α −α β2 α∈∆
Proof. In the case β 6= −β , we can assume [X ,X− ] = aX , for some a ∈ , when 1 2 β2 β1 β2−β1 C β − β is not a root, the number a = 0. Now applying [·,X− ] to relation (5.1.10), we then 2 1 β1 get
[K(Hβ1 ),Q(Xβ2 )] + [K(Xβ1 ), aQ(Xβ2−β1 )] = P ([Hβ1 ,Xβ2 ]) + P ([Xβ1 , aXβ2−β1 ])
(β1, β2) (β1, β2) 1 X − − S(Hβ ,Xβ ) − S(Xβ , aXβ −β ) + [S([Xβ ,Xα], [X−α,Xβ ]),X ] 4 1 2 4 1 2 1 4 1 2 β1 α∈∆ (β , β ) (β , β ) =P ([H ,X ]) + P ([X , aX ]) − 1 2 S(H ,X ) − a 1 2 S(X ,X ) β1 β2 β1 β2−β1 4 β1 β2 4 β1 β2−β1 1 X 1 X S([H ,X ], [X ,X ]) + S([X ,X ], [X , aX ]) 4 β1 α −α β2 4 β1 α −α β2−β1 α∈∆ α∈∆ (β , β ) (β , β ) − 1 2 S(H ,X ) + a 1 1 S(X ,X ) 4 β1 β2 4 β1 β2−β1 (β , β − β ) =P ([H ,X ]) + P ([X , aX ]) − a 1 2 1 S(X ,X ) β1 β2 β1 β2−β1 4 β1 β2−β1 1 X 1 X S([H ,X ], [X ,X ]) + a S([X ,X ], [X ,X ]) 4 β1 α −α β2 4 β1 α −α β2−β1 α∈∆ α∈∆
Relation (5.1.10) gives the following:
[K(Xβ1 ),Q(Xβ2−β1 )]
(β1, β2 − β1) 1 X = P ([X ,X ]) − S(X ,X ) + S([X ,X ], [X ,X ]) β1 β2−β1 4 β1 β2−β1 4 β1 α −α β2−β1 α∈∆
Combining the above calculations, for β1 6= −β2, we obtain:
1 X [K(H ),Q(X )] = P ([H ,X ]) + S([H ,X ], [X ,X ]), β1 β2 β1 β2 4 β1 α −α β2 α∈∆
The first relation in the corollary follows from linearity of the factor Hβ1 . Similar argument implies the second relation in the Corollary.
Corollary 5.4.2. For any roots β1, β2, such that (β1, β2) = 0, the following relations hold
171 in the deformed double current algebra D(g):
1 X [K(H ),Q(H )] = S([H ,X ], [X ,H ]) β1 β2 4 β1 α −α β2 α∈∆ 1 X = (β , α)(β , α)S(X ,X ), 2 1 2 α −α α∈∆+
where S(Xα,X−α) = XαX−α +X−αXα is called the truncated Casimir operator corresponding to root α.
Proof. Applying [·,X− ] to the relation (5.4.1), we have, β2
[K(H ),Q(H )] + (β , −β )K(X− ),Q(X )] β1 β2 1 2 β2 β2
− 1 X − =(β1, −β2)P ([X ,Xβ ]) + [ S([Hβ ,Xα], [X−α,Xβ ]),X ] β2 2 4 1 2 β2 α∈∆ 1 X = S([H ,X ], [X ,H ]) by (6.16.2). 4 β1 α −α β2 α∈∆
The second equlity follows from the fact that [Hβ,Xα] = (β, α)Xα.
∼ Proposition 5.4.1. We have an isomorphism D(g)λ=0 = U(bg[u, v]). In particular, (add some conditions of g) the universal central extension of g[u, v] is generated by elements
X,K(X),Q(X),P (X),X ∈ g, such that
•{ K(X),X | X ∈ g} generate a subalgebra which is an image of g⊗C C[u] with X ⊗u 7→ K(X);
•{ Q(X),X | X ∈ g} generate a subalgebra which is an image of g⊗C C[v] with X ⊗v 7→ Q(X);
• P (X) is linear in X, and for any X,X0 ∈ g,[P (X),X0] = P [X,X0],
and the following relation holds for all root vectors Xβ1 ,Xβ2 ∈ g with β1 6= −β2:
[K(Xβ1 ),Q(Xβ2 )] = P ([Xβ1 ,Xβ2 ]).
172 Proof. It is obvious that the algebra Dλ=0(g) is isomorphic to the universal central extension
U(bg[u, v]) of g[u, v] by the presentations of the two algebras. The assertion follows from Theorem 5.1.3.
Corollary 5.4.3 (See [34], Proposition 12.1). The following relation holds in D(g):
[K(Xβ1 ),Q(Xβ2 )] + [Q(Xβ1 ),K(Xβ2 )] = 2P ([Xβ1 ,Xβ2 ]), if β1 6= −β2.
Proof. The conclusion follows directly from relation (5.1.10), using the property that the
S(a1, a2) = S(a2, a1).
Proposition 5.4.2 ([91]). 3 There is a well defined automorphism of D(g) of order 4, which
is defined by z 7→ z, K(z) 7→ −Q(z), Q(z) 7→ K(z), and P (z) 7→ −P (z), for z ∈ g.
More generally, see [91], there is a right action of SL2(C) on D(g). For an element a11 a12 g A = ∈ SL2(C), the action of A is defined by z 7→ z, for any z ∈ and a21 a22
K(z) 7→ a11K(z) + a12Q(z),Q(z) 7→ a22Q(z) + a21K(z),
P (z) 7→ (a11a22 + a12a21)P (z) + a11a21[K(y),K(w)] + a12a22[Q(y),Q(w)], where z = [y, w].
In particular,
Proof. We first check that under the action of A preserves the defining relations of D(g).
We only check it preserves relation (5.1.10), other relations are obvious. We set
1 1 X C(X ,X ) := − (β , β )S(X ,X ) + S([X ,X ], [X ,X ]), β1 β2 4 1 2 β1 β2 4 β1 α −α β2 α∈∆
The property S(a1, a2) = S(a2, a1) implies that C(Xβ1 ,Xβ2 ) = C(Xβ2 ,Xβ1 ). The relation (5.1.10) can be rewritten as [K(x),Q(y)] = P ([x, y]) + C(x, y).
3The proof is copied from [91].
173 Now under the action of A ∈ SL2(C),
[K(x),Q(y)] 7→ [a11K(x) + a12Q(x), a22Q(y) + a21K(y)]
= a11a22[K(x),Q(y)] + a12a21[Q(x),K(y)] + a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)]
= a11a22(P ([x, y]) + C(x, y)) − a12a21(P ([y, x]) + C(y, x))
+ a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)]
= (a11a22 + a12a21)P ([x, y]) + a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)] + C(x, y). and by definition
P ([x, y]) + C(x, y) 7→ (a11a22 + a12a21)P ([x, y])
+a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)] + C(x, y).
Thus, the action of A preserves the relation (5.1.10).
We then check it defines a right action of SL2. It’s obvious that z·Id = z, for any z ∈ D(g), where Id is the 2 × 2–identity matrix.
For any A, B ∈ SL2(C), and for any z ∈ U(g), it’s a direct calculation to show (z·A)·B = z·(AB). We show the less obvious case when z = P ([x, y]) as follows. a a b b 11 12 11 12 Set A = and B = , one can check that a21 a22 b21 b22
(P ([x, y])·A)·B = P ([x, y])·(AB).
5.5 A central element of D(g)
We introduce some elements in D(g). Set
1 X C(β , β ) := [K(H ),Q(H )] − S([H ,X ], [X ,H ]), 1 2 β1 β2 4 β1 α −α β2 α∈∆
174 we denote C(β) := C(β, β), and set
(β, β) 1 X B(β) := [K(X ),Q(X )]−P (H )− S(X ,X )− S([X ,X ], [X ,X ]) ∈ D(g). β −β β 4 β −β 4 β α −α −β α∈∆
Proposition 5.5.1. (i) For any simple Lie algebra g, the following relations holds in
D(g),
C(β1, β2) = (β1, β2)B(β2).
C(β) In particular, when β1 = β2 = β, the above relation gives (β,β) = B(β).
(ii) For any two roots β1, β2 in ∆, we have
C(β ) C(β ) 1 = 2 . (β1, β1) (β2, β2)
Proof. To show (i), applying [·,X− ] to relation (5.4.1), we then get: β2
[K(H ),Q(H )] + (β , −β )[K(X− ),Q(X )] β1 β2 1 2 β2 β2
− 1 X − =P ([Hβ ,Hβ ]) + (β1, −β2)P ([X ,Xβ ]) + [ S([Hβ ,Xα], [X−α,Xβ ]),X ] 1 2 β2 2 4 1 2 β2 α∈∆ 1 X = S([H ,X ], [X ,H ]) + (β , β )P (H ) 4 β1 α −α β2 1 2 β2 α∈∆
1 X − − + (β1, −β2) S([X ,Xα], [X−α,Xβ ]) − (β2, β2)(β1, β2)S(X ,Xβ ). 4 β2 2 β2 2 α∈∆
The claim (i) follows from rewriting the above equality.
To show (ii), using (i), it suffices to show B(β1) = B(β2), for any two roots β1, β2.
Assume β1 + β2 is a root, and [Xβ2 ,X−β1−β2 ] = aX−β1 , for some scalar a ∈ C. Then
[Xβ1 ,X−β1−β2 ] = −aX−β2 , since
([Xβ1 ,X−β1−β2 ],Xβ2 ) =(Xβ1 , [X−β1−β2 ,Xβ2 ])
= − a(Xβ1 ,X−β1 ) = −a.
175 Applying [·,X−β1−β2 ] to relation (5.1.10), we obtain,
[K(Xβ1 ),Q(X−β1 )] − [K(X−β2 ),Q(Xβ2 )] (β , β ) (β , β ) =P ([X ,X ]) − P ([X ,X ]) − 1 2 S(X ,X ) + 1 2 S(X ,X ) β1 −β1 −β2 β2 4 β1 −β1 4 β2 −β2 1 X + [ S([X ,X ], [X ,X ]),X ] 4 β1 α −α β2 −β1−β2 α∈∆ (β , β ) (β , β ) =P ([X ,X ]) − P ([X ,X ]) + 1 1 S(X ,X ) − 2 2 S(X ,X ) β1 −β1 −β2 β2 4 β1 −β1 4 β2 −β2 1 X 1 X + S([X ,X ], [X ,X ]) − [ S([X ,X ], [X ,X ]),X ]. 4 β1 α −α −β1 4 −β2 α −α β2 −β1−β2 α∈∆ α∈∆
Rewriting the above equality, we conclude that if β1 + β2 is a root, then B(β1) = B(−β2).
Note by definition C(β) = C(−β), the above implies
C(β1) C(β2) = , if β1 + β2 is a root. (*) (β1, β1) (β2, β2)
It’s straightforward to check that when the root system ∆ has rank 2, the condition (*)
C(β1) C(β2) implies = , for any two roots β1, β2 ∈ ∆. In general, we use induction on the (β1,β1) (β2,β2) rank of the root system. For any root system ∆ of rank n > 2 of a simple Lie algebra. There
exists a decomposition ∆ = ∆1 ∪∆2, such that ∆1 ∩∆2 6= ∅, and ∆i is a root system of some simple Lie algebra, for i = 1, 2. Therefore, the condition (*) implies the assertion (ii).
Theorem 5.5.1. Some conditions on g, for any root β of g, the element
1 X C(β) = [K(H ),Q(H )] − S([H ,X ], [X ,H ]) β β 4 β α −α β α∈∆
is a central element of the algebra D(g).
Proof. It suffices to show C(β) commutes with the generators of D(g).
Lemma 5.5.2. For any X ∈ g, we have [C(β),X] = 0.
Proof. It suffices to take X = Xγ ∈ gγ ⊂ g, for a root γ. We then have,
176 [[K(Hβ),Q(Hβ)],Xγ] = (β, γ)[K(Xγ),Q(Hβ)] + (β, γ)[K(Hβ),Q(Xγ)] (β, γ) X =(β, γ)P ([X ,H ]) + S([X ,X ], [X ,H ]) + (β, γ)P ([H ,X ]) γ β 4 γ α −α β β γ α∈δ (β, γ) X + S([H ,X ], [X ,X ]) 4 β α −α γ α∈δ 1 X 1 X = S([[H ,X ],X ], [X ,H ]) + S([H ,X ], [X , [H ,X ]]) 4 β γ α −α β 4 β α −α β γ α∈δ α∈δ 1 X =[ S([H ,X ], [X ,H ]),X ] by (6.16.2) 4 β α −α β γ α∈δ
Thus, the conclusion follows.
Lemma 5.5.3. Assume there exist two roots β, γ in ∆, such that (β, γ) = 0, then [C(β),K(Hγ)] = 0.
The above condition holds for g = sln, for n ≥ 4, and any non-type A simple Lie algebra g.
Proof. From the defining relations of D(g), the elements {K(X),X | X ∈ g} generate the subalgebra g ⊗C C[u] of D(g), with X ⊗ u = K(X). Therefore, we have [K(Hβ),K(Hγ)] = 0 in D(g). As a result,
[K(Hγ), [K(Hβ),Q(Hβ)]]
=[K(Hβ), [K(Hγ),Q(Hβ)]] 1 X =[K(H ), (γ, α)(β, α)S(X ,X )] by Corollary 5.4.2 β 4 α −α α∈∆ 1 X 1 X = (γ, α)(β, α)S([K(H ),X ],X ) + (γ, α)(β, α)S(X , [K(H ),X ]) 4 β α −α 4 α β −α α∈∆ α∈∆ 1 X 1 X = (γ, α)(β, α)2S(K(X ),X ) − (γ, α)(β, α)2S(X ,K(X )) 4 α −α 4 α −α α∈∆ α∈∆ 1 X = (γ, α)(β, α)2S(K(X ),X ) 2 α −α α∈∆
177 On the other hand,
1 X [K(H ), S([H ,X ], [X ,H ])] γ 4 β α −α β α∈∆ 1 X =[K(H ), (β, α)2S(X ,X )] γ 4 α −α α∈∆ 1 X 1 X = (β, α)2S([K(H ),X ],X ) + (β, α)2S(X , [K(H ),X ]) 4 γ α −α 4 α γ −α α∈∆ α∈∆ 1 X 1 X = (γ, α)(β, α)2S(K(X ),X ) + (γ, −α)(β, α)2S(X ,K(X )) 4 α −α 4 α −α α∈∆ α∈∆ 1 X = (γ, α)(β, α)2S(K(X ),X ) 2 α −α α∈∆ Combining the above computations, the assertion follows.
As a corollary, under the assumption of Lemma 5.5.3, we have
[C(β),K(X)] = 0, for any X ∈ g.
Since by Lemma 5.5.2, C(β) commutes with g, and the elements K(X) satisfies the relation
[K(X1),X2] = K[X1,X2], for any elements X1,X2 ∈ g. Similar argument shows
[C(β),Q(X)] = 0, for any X ∈ g.
Note that D(g) can be generated by X,K(X),Q(X), for X ∈ g, thus, C(β) is central in
D(g).
Remark 5.5.4. Consider a quotient algebra of D(g), by modulo the central element C(β).
Then, by Proposition 5.5.1, the presentation of the quotient algebra is the same as D(g)
with the following modified relations:
(β1, β2) 1 X [K(X ),Q(X )] = P ([X ,X ]) − S(X ,X ) + S([X ,X ], [X ,X ]), β1 β2 β1 β2 4 β1 β2 4 β1 α −α β2 α∈∆ for any roots β1, β2, without the assumption that β1 6= −β2.
178 For any β1, β2, in the quotient algebra, one has
1 X [K(H ),Q(H )] = (β , α)(β , α)κ , β1 β2 2 1 2 α α∈∆+
where κα := S(Xα,X−α) is the truncated Casimir operator.
5.6 Deformed double current algebras with two param-
eters
The following definition of deformed double current algebras in type A with two parameters are introduced in [34] Definition 12.1. We first fix the notations. Let the Lie algebra g = sln,
n and {1, . . . , n} be the standard orthogonal basis of C . The set of roots of sln is denoted by
+ ∆ = {i − j | 1 ≤ i 6= j ≤ n}, with a choice of positive roots ∆ = {i − j | 1 ≤ i < j ≤ n}.
The longest positive root θ equals 1 − n. The elementary matrices will be written as
+ − Eij ∈ sln. So Xi = Ei,i+1, Xi = Ei,i−1 and Hi = Eii − Ei+1,i+1 for 2 ≤ i ≤ n − 1. We set
Eθ = E1n.
Definition 5.6.1. Let λ, β ∈ C, and n ≥ 4. We define Dλ,β(sln) to be the algebra generated by elements
z, K(z),Q(z),P (z), z ∈ sln,
which satisfy the following relations: the elements z1,K(z2), ∀z1, z2 ∈ sln, satisfy the relations for U(sln[u]) so that we have a map U(sln[u]) → Dλ,β(sln) given by z ⊗ u 7→ K(z). Similarly, z1,Q(z2) satisfy the relations of U(sln[v]), so that we have a map U(sln[v]) → Dλ,β(sln) given by z ⊗ v 7→ Q(z).
The elements z1,P (z2) satisfies the relations of the Yangians of finite type An−1. For
179 a 6= b, c 6= d, and (a, b) 6= (d, c), the following relations hold:
λ λ [K(E ),Q(E )] = P ([E ,E ]) + (β − )(δ E +δ E ) − ( − , − )S(E ,E ) ab cd ab cd 2 bc ad ad cb 4 a b c d ab cd
(5.6.1) λ X + S([E ,E ], [E ,E ]). 4 ab ij ji cd 1≤i6=j≤n
λ When β = 2 , the relation (5.6.1) coincides with the relation (5.1.10) of the deformed
double current algebra Dλ(sln) in Definition 5.1.2.
In the above definition, the elements z1,P (z2) satisfies the relations of the Yangians
Y (sln). While, in Definition 5.1.2 for a general simple Lie algebra g 6= sln, it’s not known
that the elements z1,P (z2) satisfies the relations of the Yangians Y (g).
We first list some relations of Dλ,β(sln), which are parallel to Corollary 5.4.1. For the convenience of the reader, we include the proof.
Corollary 5.6.2. For any a 6= b, and c 6= d, the following relations hold in the algebra
Dλ,β(sln):
λ X λ [K(E ),Q(H )] = P ([E ,H ])+ S([E ,E ], [E ,H ])+(β− )( + , − )E . ab cd ab cd 4 ab ij ji cd 2 a b c d ab 1≤i6=j≤n
and
λ X λ [K(H ),Q(E )] = P ([H ,E ])+ S([H ,E ], [E ,E ])+(β− )( − , + )E . ab cd ab cd 4 ab ij ji cd 2 a b c d cd 1≤i6=j≤n
λ P In particular, [K(Eab),Q(Hab)] = −2P (Eab) + 4 1≤i6=j≤n S([Eab,Eij], [Eji,Hab]).
180 Proof. Assume that (a, b) 6= (d, c), acting [·,Edc] to relation (5.6.1), we then get
[K(Eab),Q(Hcd)] + [K([Eab,Edc]),Q(Ecd)] λ X =P ([E ,H ]) + S([E ,E ], [E ,H ]) + P ([[E ,E ],E ]) ab cd 4 ab ij ji cd ab dc cd 1≤i6=j≤n λ X + S([[E ,E ],E ], [E ,E ]) 4 ab dc ij ji cd 1≤i6=j≤n λ − ( − + − , − )S([E ,E ],E ) 4 a b d c c d ab dc cd λ + (β − )(δ − δ )E , 2 bc ad ab
λ where the term involving β − 2 is obtained by:
[δbcEad + δadEcb,Edc] =δbc(Eac − δcaEdd) + δad(δbdEcc − Edb)
=δbcEac − δadEdb = (δbc − δad)Eab.
On the other hand, under the assumption of (a, b) 6= (d, c), the relation (5.6.1) gives
[K([Eab,Edc]),Q(Ecd)] λ X =P ([[E ,E ],E ]) + S([[E ,E ],E ], [E ,E ]) ab dc cd 4 ab dc ij ji cd 1≤i6=j≤n λ λ − ( − + − , − )S([E ,E ],E ) + (β − )(δ − δ )E , 4 a b d c c d ab dc cd 2 bd ac ab
λ where the term involving (β − 2 ) is obtained by the following:
Writing [Eab,Edc] = δbdEac − δacEdb, note that under the assumption (a, b) 6= (d, c), the two numbers δbd, δac can not both be 1.
δbd(Ead + δadEcc) − δac(δbcEdd + Ecb)
=δbdEad − δacEcb = (δbd − δac)Eab
λ gives the term (β − 2 )(δbd − δac)Eab.
181 Combining the above calculations, under the assumption of (a, b) 6= (d, c), we obtain
λ X [K(E ),Q(H )] = P ([E ,H ]) + S([E ,E ], [E ,H ]) ab cd ab cd 4 ab ij ji cd 1≤i6=j≤n λ + (β − )(δ − δ − δ + δ )E . 2 bc ad bd ac ab
The first relation follows from the linearity of the factor Hcd.
The proof of the second relation is similar. Applying [·,Eba] to [K(Eab),Q(Ecd)]. then we
λ get a relation of [K(Hab),Q(Ecd)] + [K(Eab),Q([Ecd,Eba])], where the coefficient of (β − 2 ) is giving by:
[δbcEad + δadEcb,Eba] =δbc(δdbEaa − Ebd) + δad(Eca − δacEbb)
= − δbcEbd + δadEca = (δad − δbc)Ecd.
λ While in [K(Eab),Q([Ecd,Eba])] = [K(Eab), δdbQ(Eca) − δacQ(Ebd)], the coefficient of (β − 2 ) is giving by:
δdb(δbcEaa + Ecb) − δac(Ead + δadEbb)
=δdbEcb − δacEad = (δdb − δac)Ecd.
λ Thus, combining the above computations, the term of (β − 2 ) in [K(Hab),Q(Ecd)] is λ λ (β − )(δ − δ − δ + δ )E = (β − )( − , + )E , 2 ad bc db ac cd 2 a b c d cd which gives the second relation in the Corollary.
In the reminder of this section, we construct a central element of Dλ,β(sln). Set
λ X Z := [K(H ),Q(H )] − S([H ,E ], [E ,H ]). ab,cd ab cd 4 ab ij ji cd 1≤i6=j≤n and denote Zab,ab by Zab for short. Set
λ X λ W := [K(E ),Q(E )] − P (H ) − S([E ,E ], [E ,E ]) − S(E ,E ) ab ab ba ab 4 ab ij ji ba 2 ab ba 1≤i6=j≤n The following Proposition is parallel to Proposition 5.5.1, we include the proof for the con-
venience of the reader.
182 Proposition 5.6.1. (i) For any 1 ≤ a 6= b ≤ n, and 1 ≤ c 6= b ≤ n, the following relations
hold in D(sln), n ≥ 4.
λ Z = −( − , − )W + (β − )( + , − )H . ab,cd c d b a ab 2 a b c d ab
In particular, we have Zab = 2Wab, and when a, b, c, d are distinct, the element Zab,cd = 0.
(ii) For 1 ≤ a 6= b ≤ n, and 1 ≤ c 6= d ≤ n, in D(sln), n ≥ 4, we have:
λ W − W = (β − )(H + H ). ab cd 2 ac bd
(iii) For 1 ≤ a 6= b ≤ n, and 1 ≤ c 6= d ≤ n, in D(sln), n ≥ 4, we have:
λ Z − Z = 2(β − )(H + H ). ab cd 2 ac bd
Proof. The relation in (i) is obtained by applying [·,Eba] to the first relation in Corollary 5.6.2.
The relation in (iii) is obtained by (ii) and the fact Zab = 2Wab in (i).
It remains to show (ii). Let’s start with (5.6.1) and apply [·,Eca] to [K(Eab),Q(Ebc)] to obtain:
[K(Eab),Q(Eba)] − [K(Ecb),Q(Ebc)] λ λ λ = P (H ) + (β − )H + S(E ,E ) − S(E ,E ) ac 2 ac 2 ab ba 2 cb bc λ X λ X − S([E ,E ], [E ,E ]) + S([E ,E ], [E ,E ]) 4 cb ij ji bc 4 ab ij ji ba 1≤i6=j≤n 1≤i6=j≤n
Starting with (5.6.1) and applying [·,Ebd] to [K(Ecb),Q(Edc)], we obtain:
[K(Ecd),Q(Edc)] − [K(Ecb),Q(Ebc)] λ λ λ = P ([H ]) − (β − )H + S(E ,E ) − S(E ,E ) bd 2 bd 2 cd dc 2 cb bc λ X λ X + S([E ,E ], [E ,E ]) − S([E ,E ], [E ,E ]). 4 cd ij ij dc 4 cb ij ji bc 1≤i6=j≤n 1≤i6=j≤n
183 Therefore,
[K(Eab),Q(Eba)] − [K(Ecd),Q(Edc)] λ λ = P (H ) − P (H ) + (β − )H + S(E ,E ) ab cd 2 ac 2 ab ba λ X λ + S([E ,E ], [E ,E ]) + (β − )H 4 ab ij ji ba 2 bd 1≤i6=j≤n λ λ X − S(E ,E ) − S([E ,E ], [E ,E ]) 2 cd dc 4 cd ij ji dc 1≤i6=j≤n
The conclusion in (ii) follows from rewriting the above equality.
Theorem 5.6.3. Set
n n ! X X λ X Z := Z = [K(H ),Q(H )] − S([H ,E ], [E ,H ]), a,a+1 a a 4 a ij ji a a=1 a=1 1≤i6=j≤n where (a, a + 1) = (n, 1), when a = n. The element Z is central in Dλ,β(sln).
Proof. We first show the element Z commutes with elements Ecd in sln. We have
λ [Z ,E ] = 2(β − )( − , − )( + , − )E , ab cd 2 a b c d c d a b cd
Set ab := a − b, and ab := a + b, we have
[[K(Hab),Q(Hab)],Ecd]
=(ab, cd)[K(Ecd),Q(Hab)] + (ab, cd)[K(Hab),Q(Ecd)] ! λ X λ =( , ) P ([E ,H ]) + S([E ,E ], [E ,H ]) + (β − )( , )E ab cd cd ab 4 cd ij ji ab 2 cd ab cd 1≤i6=j≤n ! λ X λ +( , ) P ([H ,E ]) + S([H ,E ], [E ,E ]) + (β − )( , )E ab cd ab cd 4 ab ij ji cd 2 ab cd cd 1≤i6=j≤n λ X λ = [S([H ,E ], [E ,H ]),E ] + 2(β − )( , )( , )E 4 ab ij ji ab cd 2 ab cd cd ab cd 1≤i6=j≤n Thus,
n n X λ X [Z,E ] = [Z ,E ] = 2(β − ) ( − , − )( + , − )E . i,i+1 a,a+1 i,i+1 2 a a+1 i i+1 i i+1 a a+1 i,i+1 a=1 a=1
184 It is not hard to show that, for any 1 ≤ i < n,
n X (a − a+1, i − i+1)(i + i+1, a − a+1) ≡ 0, a=1 where (a, a + 1) = (n, 1), if a = n.
Since the summand (a −b, i −i+1)(i +i+1, a −b) is non-zero, only if either a+1 = i, or a = i + 1. In the case a + 1 = i, the summand is 1, while the summand is −1, in the case a = i + 1.
Therefore, we have [Z,Ei,i+1] = 0, for any 1 ≤ i < n. Thus, the element Z commutes with any elements in sln. In the following, we show the element Z commutes with K(h), for some diagonal matrix h of sln.
0 By the defining relations of Dλ,β(sln), we have [K(h), h ] = 0, for any two diagonal matrices h, h0. By Proposition 5.6.1(iii),
λ Z − Z = 2(β − )(H + H ). ab cd 2 ac bd
Thus, [Zab,K(h)] = [Zcd,K(h)], for any diagonal matrix h.
To show [Z,K(Hcd)] = 0, it suffices to show [Zab,K(Hcd)] = 0, for distinct integers 1 ≤ a 6= b 6= c 6= d ≤ n, with (a, b) = (a, a + 1). On one hand,
[K(Hcd), [K(Hab),Q(Hab)]]
=[K(Hab), [K(Hcd),Q(Hab)]] 1 X =[K(H ), ( − , − )( − , − )S(E ,E )] by Proposition 5.6.1(i) ab 4 c d i j a b i j ij ji 1≤i,j≤n 1 X = ( − , − )( − , − )(S([K(H ),E ],E ) + S(E , [K(H ),E ])) 4 c d i j a b i j ab ij ji ij ab ji 1≤i6=j≤n 1 X = ( − , − )( − , − )2 (S(K(E ),E ) − S(E ,K(E ))) 4 c d i j a b i j ij ji ij ji 1≤i6=j≤n 1 X = ( − , − )( − , − )2S(K(E ),E ) 2 c d i j a b i j ij ji 1≤i6=j≤n
185 On the other hand, we have:
1 X [K(H ), S([H ,E ], [E ,H ])] cd 4 ab ij ji ab 1≤i6=j≤n 1 X =[K(H ), ( − , − )2S(E ,E )] cd 4 a b i j ij ji 1≤i6=j≤n 1 X = ( − , − )2 (S([K(H ),E ],E ) + S(E , [K(H ),E ])) 4 a b i j cd ij ji ij cd ji 1≤i6=j≤n 1 X = ( − , − )2( − , − )(S(K(E ),E ) − S(E ,K(E ))) 4 a b i j c d i j ij ji ij ji 1≤i6=j≤n 1 X = ( − , − )( − , − )2S(K(E ),E ) 2 c d i j a b i j ij ji 1≤i6=j≤n
Combine the above computations, we conclude that [K(Hcd),Zab] = 0. Thus, [K(Hcd),Z] = 0.
As a corollary, we have
[Z,K(X)] = 0, for any X ∈ sln.
Since Z commutes with sln, and the elements K(X) satisfies the relation [K(X1),X2] =
K[X1,X2], for any elements X1,X2 ∈ sln. Similar argument shows
[Z,Q(X)] = 0, for any X ∈ sln.
Note that Dλ,β(sln) can be generated by X,K(X),Q(X), for X ∈ sln, thus, Z is central in
Dλ,β(sln).
186 Chapter 6
Universal KZB Equations and the
elliptic Casimir connection
6.1 The generalized holonomy Lie algebra Aell
Let h be a Euclidean vector space, Φ ⊂ h∗ a reduced, crystallographic root system. Let
Q ⊂ h∗ be the lattice generated by the roots α, α ∈ Φ and P ∨ ⊂ h the dual lattice, called the coweight lattice. Let Q∨ ⊂ h be the lattice generated by the coroots α∨, with the inner product (α∨, α) = 2, α ∈ Φ and P ⊂ h∗ the dual weight lattice.
∗ For a subset Ψ ⊂ Φ and subring R ⊂ R, let hΨiR ⊂ h be the R−span of Ψ.
Definition 6.1.1. A root subsystem of Φ is a subset Ψ ⊂ Φ such that hΨiZ ∩ Φ = Ψ. Ψ is complete if hΨiR ∩ Φ = Ψ. If Ψ ⊂ Φ is a root subsystem, we set Ψ+ = Ψ ∩ Φ+.
Φ The algebra Aell associated for a root system Φ is defined as follows,
Φ Definition 6.1.2. Let Aell be the Lie algebra generated by:
• a set of elements {tα}α∈Φ, such that tα = t−α, • two linear maps x : h → A, y : h → A, such that:
P 1. For any root subsystem Ψ of Φ, [tα, β∈Ψ+ tβ] = 0;
187 2.[ x(u), x(v)] = 0, [y(u), y(v)] = 0, for any u, v ∈ h;
3.[ y(u), x(v)] = Σγ∈Φ+ hv, γihu, γitγ.
4.[ tα, x(u)] = 0, [tα, y(u)] = 0, if hα, ui = 0.
Φ Note that algebra Aell is bigraded, with the grading deg(x(u)) = (1, 0), deg(y(v)) = (0, 1),
and deg(tα) = (1, 1).
Remark 6.1.3. In topology, there is a explicit construction of the holonomy Lie algebra,
see [?, Section 2]. By construction, the holonomy Lie algebra is a quadratic Lie algebra, that
is, it has a presentation with generators in degree 1 and relations in degree 2 only. While,
the Lie algebra Aell we defined here is not a quadratic Lie algebra. We call Aell a generalized
holonomy Lie algebra in the rest of the paper.
∨ Pn ∨ Pn Letα ˜ be the highest root in Ψ. Writeα ˜ = i=1 giαi , and let g(Ψ) := 1 + i=1 gi.
Lemma 6.1.4. Let Φ be of rank 2, and T ⊂ A the linear subspace spanned by the elements tα, α ∈ Φ+. Then the image of the map h ⊗ h → T given by u ⊗ v → [x(u), y(v)] is contained in the subspace of T given by
X X X { cγtγ| for anyΨ1 6= Ψ2 ⊆ Φrank 2 root systems, cγ/g(Ψ1) = cγ/g(Ψ2)}
γ∈Φ+ γ∈Ψ1+ γ∈Ψ2+ Proof. write [x(u), y(v)] = P c (u, v)t , where c (u, v) = hu, γihv, γi. Then, for any γ∈Φ+ γ γ γ P Ψ ⊂ Φ, we have γ∈Ψ cγ = g(Ψ)(u|v).(see Lemma 6.13.4 for this equality.)
6.2 The universal connection for arbitrary root sys-
tems
6.2.1 Principal bundles
Let Λτ := Z + Zτ, and Eτ := C/Λτ be the elliptic curve with modular parameter τ∈ / R.
188 Let
∨ T := P ⊗Z Eτ
be the simply-connected torus.
∗ ∨ For any root α ∈ Q ⊂ h , we have a natural map χα : P ⊗Z Eτ → Eτ . Denote kernel of S χα by Tα, which is a divisor of T . Denote Treg = T \ α∈Φ Tα.
∨ Φ Φ Consider the principal bundle h ×Λτ ⊗P exp(Acell) over T with structure group exp(Acell),
∨ ∨ where the action of Λτ ⊗ P on h = C ⊗ P is by translations, and the action on group
Φ exp(Acell) is given by:
∨ ∨ ∨ −2πix(λi ) λi (g) = g and τλi (g) = e g.
∨ Φ Denote by Pτ,n the restricting the bundle h ×Λτ ⊗P exp(Acell) over Treg.
∨ ∨ Equivalently, denote by π : C ⊗ P → Eτ ⊗ P the natural projection. Let C(E, τ) be
∨ ∨ the bundle on Eτ ⊗ P with sections on U ⊂ Eτ ⊗ P given by:
∨ −1 ∨ ∨ −2πix(λi ) {f : π (U) → exp(Acell) | f(z + λi ) = f(z), f(z + τλi ) = e f(z).}
Then, the bundle Pτ,n is the restricting of the bundle C(E, τ) under the inclusion
∨ reg ∨ (Eτ ⊗ P ) ,→ Eτ ⊗ P .
6.2.2 Some facts about theta functions
We recall some basic facts about theta functions in this section. Let q := e2πiτ . Denote
∞ n2 P 8 ϑ1(z, q) a classical Theta-function (See [100] Page 463-465.), that is, ϑ1(z, q) = 2 n=0 q sin(nz). We have: ∞ ∂ϑ1 1 Y n 3 (0, q) = 2q 8 (1 − q ) . ∂z n−1 Now consider the following theta function considered in [10]:
ϑ (πz, q) θ(z|τ) := 1 . ∂ϑ1 ∂z (0, q)
189 1/24 Q n 3 If η(τ) := q n≥1(1 − q ), then ϑ1(z, τ) = 2η(τ) θ(z|τ). Note that ϑ1(z, τ) satisfies the
2 1 ∂ ϑ1(z,τ) ∂ϑ1(z,τ) partial differential equation 4πi ∂z2 = ∂τ .
Let Λτ := Z + Zτ ⊂ C and H be the upper half plane, i.e. H := {z ∈ C | Im(z) > 0}.
The following properties of θ(z|τ) is straightforward from properties of ϑ1(z, q).
1. θ(z|τ) is a holomorphic function C × H → C, such that {z | θ(z|τ) = 0} = Λτ .
∂θ 2. ∂z (0|τ) = 1.
3. θ(z + 1|τ) = −θ(z|τ) = θ(−z|τ), and θ(z + τ|τ) = −e−πiτ e−2πizθ(z|τ) = θ(−z|τ).
4. θ(z|τ + 1) = θ(z|τ), while θ(−z/τ| − 1/τ) = −(1/τ)e(πi/τ)z2 θ(z|τ).
5. The following product formula holds:
1 Y s Y s −1 1 Y s −2 θ(z|τ) = u 2 (1 − q u) (1 − q u ) (1 − q ) , (6.2.1) 2πi s>0 s≥0 s>0
where u = e2πiz.
6.2.3 General form
Set θ(z + x|τ) 1 k(z, x|τ) := − . (6.2.2) θ(z|τ)θ(x|τ) x
Consider the Aell−valued connection on Hreg given by
n X xα∨ X ∇ = d − k(α, ad( )|τ)(t )dα + y(ui)du , (6.2.3) KZB 2 α i α∈Φ+ i=1 i ∗ where Φ+ ⊂ Φ is a chosen system of positive roots, {ui}, and {u } are dual basis of h and h respectively.
Note that the form is independent of the choice of Φ+, which follows from k(z, x|τ) =
−k(−z, −x|τ) using the theta function θ(z|τ) being an odd function.
190 Rewriting the connection (6.2.3) as the form
n ! X X xα∨ ∇ = d − (α, α )k(α, ad( )|τ)(t ) − y(α ) dλ∨ KZB i 2 α i i i=1 α∈Φ+ Lemma 6.2.1. For any i, the function
X xα∨ K := (α, α )k(α, ad( )|τ)(t ) − y(α ) i i 2 α i α∈Φ+ satisfies the conditions
∨ ∨ ∨ −2πix(λj ) Ki(z + λj ) = Ki(z) and Ki(z + τλj ) = e Ki(z).
Proof. We use the fact that for any integer m ∈ Z, k(z + m, x|τ) = k(z, x|τ) and e−2πimx − 1 k(z + τm, x|τ) = e−2πimxk(z, x|τ) + . x
∨ Then it’s obvious that Ki(α + λj ) = Ki(α). We have
∨ Ki(α + τλj )
X xα∨ = (α, α )k(α + τ(α, λ∨), ad( )|τ)(t ) − y(α ) i j 2 α i α∈Φ+ ∨ ! ∨ ∨ X ∨ x(α ) xα∨ exp −2πi(α, λj )xα /2 − 1 = (α, αi) exp −2πi(α, λj ) k(α, ad( )|τ) + (tα) − y(αi) 2 2 xα∨ /2 α∈Φ+ ∨ ! X xα∨ − exp 2πix(λj ) − 1 = (α, α ) exp −2πix(λ∨) k(α, ad( )|τ) + (α, λ∨) (t ) − y(α ) i j 2 x(λ∨) j α i α∈Φ+ j x(α∨) by the relation [−(α, λ∨) + x(λ∨), t ] = 0. j 2 j α ∨ X xα∨ exp −2πix(λj ) − 1 X = (α, α ) exp −2πix(λ∨) k(α, ad( )|τ)(t ) + ( (α, α )(α, λ∨)t ) − y(α ) i j 2 α x(λ∨) i j α i α∈Φ+ j α∈Φ+
X xα∨ = (α, α ) exp −2πix(λ∨) k(α, ad( )|τ)(t ) − (exp −2πix(λ∨) − 1)(y(α )) − y(α ) i j 2 α j i i α∈Φ+
∨ X ∨ by the relation [y(αi), x(λj )] = (α, αi)(α, λj )tα. α∈Φ+
X xα∨ = (α, α ) exp −2πix(λ∨) k(α, ad( )|τ)(t ) − exp 2πix(λ∨) (y(α )) i j 2 α j i α∈Φ+
∨ = exp(−2πix(λj ))Ki(α).
191 Theorem 6.2.2. The connection ∇KZB (6.2.3) is flat if and only if the relations in Aell hold.
P xα∨ Pn i Proof. Set A := α∈Φ+ k(α, ad( 2 )|τ)(tα)dα− i=1 y(u )dui. Since dA = 0, the connection (6.2.3) is flat if and only if the curvature Ω = A ∧ A = 0.
Write Ω = Ω1 − Ω2 + Ω3, where
1 X h xα∨ xβ∨ i Ω = k(α, ad( )|τ)(t ), k(β, ad( )|τ)(t ) dα ∧ dβ (6.2.4) 1 2 2 α 2 β α6=β
X h xα∨ i Ω = k(α, ad( )|τ)(t ), y(ui) dα ∧ du (6.2.5) 2 2 α i α,i
1 X Ω = y(ui), y(uj) du ∧ du (6.2.6) 3 2 i j i6=j
i j The relation [y(u ), y(u )] = 0 implies that Ω3 = 0. In order to show Ω = 0, we now need
to show Ω1 = Ω2.
Corollary 6.2.3. In the case g = sln, connection (6.2.3) can be rewritten as ∇KZB := Pn d − i=1 Ki(z|τ)dzi, where
X X Ki(z|τ) = −yi + Kij(zij|τ) := −yi + k(zij, ad xi|τ)(tij). j|j6=i j|j6=i
Which coincides with the connection constructed in [10].
Proof. In the case g = sln, write α = i − j ∈ Φ, where {1, . . . , n} is a standard orthogonal
n basis of C . Notice that we have Kij(zij|τ) = −Kji(zji|τ), since k(z, x|τ) = −k(−z, −x|τ).
Using the relation [xi + xj, tij] = 0, we get
x − x (ad x )k(t ) = (− ad x )k(t ) = (ad i j )k(t ), i ij j ij 2 ij
for any k > 0. The rest of the proof is simply by calculation.
192 6.3 Flatness of the universal connection, part 1
∗ ∨ 2u ∨ For two vector u ∦ v ∈ h , set u := (u,u) , now define a new element ω(u , v), which is spanned by u and v, such that
(u∨ + ω(u∨, v))⊥u∨, ω(u∨, v)⊥v.
∨ −2(v,v) 2(u,v) More explicitly, we have ω(u , v) = (u,u)(v,v)−(u,v)2 u+ (u,u)(v,v)−(u,v)2 v. The following identities, which follow from a direct calculation, will be used later.
Lemma 6.3.1. 1. ω(u∨, v) − ω(v∨, u) = ω(u∨, u + v);
∨ 1 ∨ 2. ω((au + bv) , cu) = b ω(v , u);
u∨+ω(u∨,v) u∨+ω(u∨,u+v) 3. (u∨,v) = (u∨,u+v) ;
u∨+ω(u∨,v) ω(v∨,u) 4. (u∨,v) = −2 .
Proposition 6.3.2. Modulo the relations (tx), (xx), the following identity holds for any
α, β ∈ Φ+:
h x ∨ x ∨ i x ∨ x ∨ k(α, ad( α )|τ)(t ), k(β, ad( β )|τ)(t ) = k(α, ad( ω(α ,β) )|τ)k(β, ad( ω(β ,α) )|τ)[t , t ]. 2 α 2 β −2 −2 α β
Proof. If α = β, both sides is obviously equal to 0.
If α 6= β, since k(z, x) is a formal power series in x, with coefficient in C[[z]], it suffices to show above identity replacing k(z, x) by xn for any n ∈ N. Now we have:
n m [(ad xα∨ ) (tα), (ad xβ∨ ) (tβ)]
n m = [(− ad xω(α∨,β)) (tα), (− ad xω(β∨,α)) (tβ)]
n m = (− ad xω(α∨,β)) (− ad xω(β∨,α)) [tα, tβ]
So the assertion follows.
193 Corollary 6.3.3. The curvature term Ω1 is equal to
1 X xω(α∨,β) xω(β∨,α) Ω = k(α, ad( )|τ)k(β, ad( )|τ)[t , t ]dα ∧ dβ (6.3.1) 1 2 −2 −2 α β α6=β
Proposition 6.3.4. Suppose u⊥α, modulo the relations (yt),(yx),(tx), (xx), the following identity holds for any α ∈ Φ+, u ∈ h∗:
x ∨ xα∨ ω(α ,γ) h xα∨ i X ∨ k(α, ad( 2 )|τ) − k(α, ad( −2 )|τ) y(u), k(α, ad( )|τ)(tα) = (α , γ)(u, γ) [tγ, tα]. 2 adxα∨ + adxω(α∨,γ) γ∈Φ+
Proof. since k(z, x) is a formal power series in x, with coefficient in C[[z]], it suffices to show above identity replacing k(z, x) by xn for any n ∈ N. Now we have:
n [y(u), (ad xα∨ ) (tα)] n−1 X s n−1−s = (ad xα∨ ) (ad[y(u), xα∨ ])(ad xα∨ ) (tα) s=0 n−1 X ∨ X s n−1−s = (α , γ)(u, γ) (ad xα∨ ) (ad tγ)(ad xα∨ ) (tα) γ∈Φ+ s=0 n−1 X ∨ X s n−1−s = (α , γ)(u, γ) (ad xα∨ ) (ad tγ)(− ad xω(α∨,γ)) (tα) γ∈Φ+ s=0 n−1 X ∨ X s n−1−s = (α , γ)(u, γ) (ad xα∨ ) (− ad xω(α∨,γ)) (ad tγ)(tα) γ∈Φ+ s=0
X ∨ = (α , γ)(u, γ)f(ad xα∨ , − ad xω(α∨,γ))([tγ, tα]), γ∈Φ+
un−vn where f(u, v) = u−v . So the assertion follows.
Corollary 6.3.5. The curvature term Ω2 is equal to
x ∨ xα∨ ω(α ,γ) X ∨ k(α, ad( 2 )|τ) − k(α, ad( −2 )|τ) Ω2 = (α , γ) [tα, tγ]dα ∧ dγ. adxα∨ + adxω(α∨,γ) α6=γ∈Φ+
Proof. By definition,
X X h xα∨ i Ω = k(α, ad( )|τ)(t ), y(ui) dα ∧ du 2 2 α i α i
194 ∗ Choose the basis {ui} ⊂ h to be such that u1 = α and ui⊥α for i ≥ 2. Then since dα ∧ du1 = 0, by Proposition 6.3.4, the above is equal to
x ∨ xα∨ ω(α ,γ) X X i ∨ k(α, ad( 2 )|τ) − k(α, ad( −2 )|τ) − (u , γ)(α , γ) [tγ, tα]dα ∧ dui adxα∨ + adxω(α∨,γ) α,i≥2 γ∈Φ+ x ∨ xα∨ ω(α ,γ) X X i ∨ k(α, ad( 2 )|τ) − k(α, ad( −2 )|τ) = − (u , γ)(α , γ) [tγ, tα]dα ∧ dui adxα∨ + adxω(α∨,γ) α,i≥1 γ∈Φ+
i which is equal to the claimed result since γ = (γ, u )ui.
6.4 Flatness of the universal connection, part 2
From last section, we have a expression of Ω1 and Ω2. In this section, we will show Ω1 −Ω2 = 0. Write
Ω1 − Ω2
x ∨ xα∨ ω(α ,β) ! X 1 xω(α∨,β) xω(β∨,α) k(α, ad( )) − k(α, ad( )) = k(α, ad( ))k(β, ad( )) − (α∨, β) 2 −2 2 −2 −2 adxα∨ + adxω(α∨,β) α6=β
[tα, tβ]dα ∧ dβ.
Let k(α, β)[tα, tβ] be the subtraction of coefficient of dα ∧ dβ and coefficient of dβ ∧ dα in
the formula of Ω1 − Ω2,that is:
x ∨ x ∨ k(α, β) = k(α, ad( ω(α ,β) )|τ)k(β, ad( ω(β ,α) )|τ) −2 −2 x ∨ k(α, ad( xα∨ )|τ) − k(α, ad( ω(α ,β) )|τ) − (α∨, β) 2 −2 adxα∨ + adxω(α∨,β) x k(β, ad( xβ∨ )|τ) − k(β, ad( ω(β∨,α) )|τ) − (β∨, α) 2 −2 adxβ∨ + adxω(β∨,α)
Proposition 6.4.1. Notations as above, for any α, β ∈ Φ, we have
k(α, β)dα ∧ dβ + k(α, α + β)d(α + β) ∧ dα + k(β, α + β)dβ ∧ d(α + β) = 0. (6.4.1)
195 Proof. We only need to show: −k(α, β) + k(α, α + β) + k(β, α + β) = 0. By definition,
− k(α, β) + k(α, α + β) + k(β, α + β)
x ∨ xα∨ ω(α ,β) x ∨ x ∨ k(α, ad( )|τ) − k(α, ad( )|τ) = −k(α, ad( ω(α ,β) )|τ)k(β, ad( ω(β ,α) )|τ) + (α∨, β) 2 −2 −2 −2 adxα∨ + adxω(α∨,β) x k(β, ad( xβ∨ )|τ) − k(β, ad( ω(β∨,α) )|τ) + (β∨, α) 2 −2 adxβ∨ + adxω(β∨,α)
x ∨ x ∨ + k(α, ad( ω(α ,α+β) )|τ)k(α + β, ad( ω((α+β) ,α) )|τ) −2 −2 x ∨ k(α, ad( xα∨ )|τ) − k(α, ad( ω(α ,α+β) )|τ) − (α∨, α + β) 2 −2 adxα∨ + adxω(α∨,α+β) x x k(α + β, ad( (α+β)∨ )|τ) − k(α + β, ad( ω((α+β)∨,α) )|τ) − ((α + β)∨, α) 2 −2 adx(α+β)∨ + adxω((α+β)∨,α)
x ∨ x ∨ + k(β, ad( ω(β ,α+β) )|τ)k(α + β, ad( ω((α+β) ,β) )|τ) −2 −2 x k(β, ad( xβ∨ )|τ) − k(β, ad( ω(β∨,α+β) )|τ) − (β∨, α + β) 2 −2 adxβ∨ + adxω(β∨,α+β) x x k(α + β, ad( (α+β)∨ )|τ) − k(α + β, ad( ω((α+β)∨,β) )|τ) − ((α + β)∨, β) 2 −2 adx(α+β)∨ + adxω((α+β)∨,β)
α∨+ω(α∨,β) α∨+ω(α∨,α+β) β∨+ω(β∨,α) Now use the identities in Lemma 6.3.1, We have: (α∨,β) = (α∨,α+β) , (β∨,α) = β∨+ω(β∨,α+β) (α+β)∨+ω((α+β)∨,α) (α+β)∨+ω((α+β)∨,β) (β∨,α+β) , ((α+β)∨,α) = − ((α+β)∨,β) . Thus, using the fact that x : h → Aell is a linear function, we are able to make some cancelations:
− k(α, β) + k(α, α + β) + k(β, α + β)
xω(α∨,α+β) xω(α∨,β) x ∨ x ∨ k(α, ad( )|τ) − k(α, ad( )|τ) = −k(α, ad( ω(α ,β) )|τ)k(β, ad( ω(β ,α) )|τ) + (α∨, β) −2 −2 −2 −2 adxα∨ + adxω(α∨,β) x x k(β, ad( ω(β∨,α+β) )|τ) − k(β, ad( ω(β∨,α) )|τ) + (β∨, α) −2 −2 adxβ∨ + adxω(β∨,α)
x ∨ x ∨ + k(α, ad( ω(α ,α+β) )|τ)k(α + β, ad( ω((α+β) ,α) )|τ) −2 −2 x x k(α + β, ad( ω((α+β)∨,β) )|τ) − k(α + β, ad( ω((α+β)∨,α) )|τ) − ((α + β)∨, α) −2 −2 adx(α+β)∨ + adxω((α+β)∨,α)
x ∨ x ∨ + k(β, ad( ω(β ,α+β) )|τ)k(α + β, ad( ω((α+β) ,β) )|τ) −2 −2
196 ω(α∨,β) ω(β∨,α+β) Now, set u := −2 and v := −2 . Using the identities from Lemma 6.3.1, we get:
ω(α∨,β) β∨+ω(β∨,α) ω((α+β)∨,β) ω(β∨,α+β) ω(α∨,α+β) (α+β)∨+ω((α+β)∨,α) 1. u = −2 = (β∨,α) = −2 ; v = −2 = − −2 = − ((α+β)∨,α) .
ω(β∨,α) α∨+ω(α∨,β) ω((α+β)∨,α) 2. u + v = −2 = (α∨,β) = −2 ;
Rewrite the above formula, we get:
− k(α, β) + k(α, α + β) + k(β, α + β)
= −k(α, u)|τ)k(β, u + v|τ) + k(α, −v|τ)k(α + β, u + v|τ) + k(β, v|τ)k(α + β, u|τ) k(α, −v|τ) − k(α, u|τ) k(β, v|τ) − k(β, u + v|τ) k(α + β, u|τ) − k(α + β, u + v|τ) + + − , u + v u −v
which is 0 by the following theta function identity used in [10].
k(z, −v)k(z0, u + v) − k(z, u)k(z0 − z, u + v) + k(z0, u)k(z0 − z, v) k(z0 − z, v) − k(z0 − z, u + v) k(z0, u) − k(z0, u + v) k(z, u) − k(z, −v) + + + = 0 u v u + v
This ends the proof.
Lemma 6.4.2. Let Φ be a root system. For any two roots α, β ∈ Φ, then Ψ := (Zα+Zβ)∩Φ is a root subsystem of rank 2.
Proof. It’s straightforward to check all the axioms of root system hold for Ψ.
For any α, β ∈ Φ, let Ψα,β := (Zα + Zβ) ∩ Φ. Whenever we have Ψα0,β0 $ Ψα,β, delete
Ψα0,β0 ; for Ψα0,β0 = Ψα,β, just keep one set Ψα,β. Thus, we have a finite collection of rank 2 subsystems. List them as Ψ1,..., Ψm. We will show for any α, β ∈ Φ, there exist exactly one Ψk, such that α, β ∈ Ψk.
It’s obvious that for any α, β ∈ Φ, there exist one Ψk, such that α, β ∈ Ψk.
197 Suppose α, β ∈ Ψk1 , and α, β ∈ Ψk2 , and Ψα,β 6= Ψki , i = 1, 2, then Ψk1 , Ψk1 are both
of type B2, or G2, If Ψk1 6= Ψk2 , then there exists one root α, such that 2α ∈ Φ, which
contradicts with the definition of root system. Thus, Ψk1 = Ψk2 .
Pm P Ψi Thus, we are able to write: Ω1 − Ω2 = + (Ω1 − Ω2) , where i=1 α,β∈Ψi
Ψi (Ω1 − Ω2)
X 1 xω(α∨,β) xω(β∨,α) = k(α, ad( )|τ)k(β, ad( )|τ)[t , t ]dα ∧ dβ. 2 −2 −2 α β α6=β∈Ψ+ x ∨ xα∨ ω(α ,β) X ∨ k(α, ad( 2 )|τ) − k(α, ad( −2 )|τ) − (α , β) [tα, tβ]dα ∧ dβ. adxα∨ + adxω(α∨,β) α6=β∈Ψ+
Ψ So, in order to show Ω1 − Ω2 = 0, we need to show (Ω1 − Ω2) = 0, for Ψ rank 2 root system.
6.4.1 Case-A2
3 Let 1, 2, 3 be the standard orthogonal basis of R . Let α1 = 1 − 2, α2 = 2 − 3 be the
+ simple roots for A2. The root system Φ = {α1, α2, α1 + α2}. Now, by definition we have:
Ω1 − Ω2 = k(α1, α2)[tα1 , tα2 ]dα1 ∧ dα2 + k(α1, zα1+α2 )[tα1 , tα1+α2 ]dα1 ∧ d(α1 + α2)
+ k(α2, zα1+α2 )[tα2 , tα1+α2 ]dα2 ∧ d(α1 + α2)
Using Proposition 6.4.1, we can write
k(α1, α1 + α2)dα1 ∧ d(α1 + α2) = k(α1, α2)dα1 ∧ dα2 + k(α2, α1 + α2)dα2 ∧ d(α1 + α2).
Plug the above expression into the definition of Ω1 − Ω2, we have
Ω1 − Ω2 = k(α1, α2)[tα1 , tα2 + tα1+α2 ]dα1 ∧ dα2
+ k(α2, zα1+α2 )[tα1 + tα2 , tα1+α2 ]dα2 ∧ d(α1 + α2)
198 Using (tt)-relations, we have Ω1 − Ω2 = 0 in the case of A2. The above procedure will be represented as the graph:
(α1, α1 + α2) (6.4.2)
z & (α1, α2) (α2, α1 + α2)
6.4.2 Case-B2
2 Let 1, 2 be the standard orthogonal basis of R . Let α1 = 1 − 2, α2 = 2 be the simple
+ roots for B2. The root system Φ = {α1, α1 + α2, α1 + 2α2, α2}. Now, by definition we have:
Ω1 − Ω2 = k(α1, α2)[tα1 , tα2 ]dα1 ∧ dα2 + k(α1, zα1+α2 )[tα1 , tα1+α2 ]dα1 ∧ d(α1 + α2)
+ k(zα1+α2 , α2)[tα1+α2 , tα2 ]d(α1 + α2) ∧ dα2
+ k(α1, α1 + 2α2)[tα1 , tα1+2α2 ]dα1 ∧ d(α1 + 2α2)
+ k(zα1+α2 , α1 + 2α2)[tα1+α2 , tα1+2α2 ]d(α1 + α2) ∧ d(α1 + 2α2)
+ k(α1 + 2α2, α2)[tα1+2α2 , tα2 ]d(α1 + 2α2) ∧ dα2.
Using the graph
(α1, α2) (6.4.3) E
(α1, α1 + α2) / (α1 + α2, α2) E
(α1 + α2, α1 + 2α2) / (α1 + 2α2, α2)
We have:
Ω1 − Ω2 = k(α1, α2)[tα1 , tα2 + tα1+α2 ]dα1 ∧ dα2
+ k(α2, α1 + α2)[tα1 + tα2 + tα1+2α2 , tα1+α2 ]dα2 ∧ d(α1 + α2)
+ k(α2, α1 + 2α2)[tα2 + tα1+α2 , tα1+2α2 ]dα2 ∧ d(α1 + 2α2)
199 Note that each of the coefficient is of the form
X [tα, tβ], β∈Ψ+
where Ψ = B2 root system. By (tt)-relation, we have Ω1 − Ω2 = 0 in the case of B2.
Remark 6.4.3. Above is only one connected component of graph, the components are
labeled by the root subsystem of Φ. Here, there is another one corresponding to A1 × A1 and it’s just the one vertex graph: (α1, α1 + 2α2).
6.4.3 Case-G2
3 Let 1, 2, 3 be the standard orthogonal basis of R The simple roots α1 = 22 − 1 − 3, α2 =
+ 1 −2. The root systerm Φ = {α1, α2, α1 +α2, α1 +2α2, α1 +3α2, 2α1 +3α2}. The connected
component of the graph corresponding to G2 is:
(α1, α2) (6.4.4) F
(α1, α1 + α2) / (α1 + α2, α2) F
(α1 + α2, α1 + 2α2) / (α1 + 2α2, α2) F P
(α1 + α2, 2α1 + 3α2) / (2α1 + 3α2, α1 + 2α2)
(α1 + 2α2, α1 + 3α2) / (α1 + 3α2, α2)
200 Use the above graph, and Proposition 6.4.1, We have:
Ω1 − Ω2
= k(α1, α2)[tα1 , tα2 + tα1+α2 ]dα1 ∧ dα2 + k(α2, α1 + α2)[tα1 + tα2 , tα1+α2 ]dα2 ∧ d(α1 + α2)
+ k(α1 + 2α2, α2)([tα1+2α2 , tα2 + tα1+α2 + tα1+3α2 ] − [tα1+α2 , t2α1+3α2 ])d(α1 + 2α2) ∧ dα2
+ k(2α1 + 3α2, α1 + 2α2)[t2α1+3α2 , tα1+2α2 + tα1+α2 ]d(2α1 + 3α2) ∧ d(α1 + 2α2)
+ k(α1 + 3α2, α2)[tα1+3α2 , tα2 + tα1+2α2 ]d(α1 + 3α2) ∧ dα2
Note that this term
[tα1+2α2 , tα2 + tα1+α2 + tα1+3α2 ] − [tα1+α2 , t2α1+3α2 ]
=[tα1+2α2 , tα2 + tα1+α2 + tα1+3α2 + t2α1+3α2 ] − [tα1+α2 + tα1+2α2 , t2α1+3α2 ]
P and all the coefficients have the required form, that is, [tα, β∈Ψ+ tβ], where Ψ = G2 root system. Apply the (tt)-relation, we have Ω1 − Ω2 = 0 in the case of G2.
6.5 Equivariance under Weyl group action
Φ Let W be the Weyl group. Assume that W acts on the algebra Aell by the following way:
For any simple reflection si ∈ W corresponding to a simple root αi,
1. si(tα) = tsiα;
2. si(x(u)) = x(siu);
3. si(y(v)) = y(siv) for any α ∈ Φ, and u, v ∈ h.
Proposition 6.5.1. The connection ∇ is W -equvariant.
201 + xα∨ xα∨ Proof. Note that si permutes the set Φ \{αi}, and k(−α, − ad 2 ) = k(α, ad 2 ). We have
n−1 X sixα∨ X s∗∇ = d − k(s α, ad( )|τ)(s t )d(s α) + (s y(uj))d(s u ) i i 2 i α i i i j α∈Φ+ j=1 n−1 six −1 ∨ X (si β) X j = d − k(β, ad( )|τ)(sits−1β)dβ + (siy(u ))d(siuj) 2 i β∈Φ+ j=1 n−1 X xβ∨ X = d − k(β, ad( )|τ)(t )dβ + (y((s u)j))d(s u ) 2 β i i j β∈Φ+ j=1
= ∇
Thus, the connection is W -equivariant.
6.6 Degeneration of the elliptic connection
We show in this section that as τ → +i∞, the connection ∇ degenerates to a trigonometric connection of the form considered in [85].
6.6.1 The trigonometric connection
In [85], Toledano Laredo introduced the trigonometric connection, which we recall it here.
∗ Let H = HomZ(P, C ) be the complex algebraic torus with Lie algebra h and coordinate
ring given by the group algebra CP . We denote the function corresponding to λ ∈ P by
eλ ∈ C[H], and set
[ α Hreg = H \ {e = 1} (6.6.1) α∈Φ
Let Atrig be an algebra endowed with the following data:
• a set of elements {tα}α∈Φ ⊂ Atrig such that t−α = tα
• a linear map τ : h → Atrig
202 Consider the Atrig–valued connection on Hreg given by
X dα ∇ = d − t − du τ(ui) (6.6.2) eα − 1 α i α∈Φ+
i ∗ where Φ+ ⊂ Φ is a chosen system of positive roots, {ui} and {u } are dual bases of h and h respectively, the differentials dui are regarded as translation–invariant one–forms on H and the summation over i is implicit.
Theorem 6.6.1 (Toledano Laredo [85]).
(1) The connection (6.6.2) is flat if, and only if the following relations hold:
(tt) For any rank 2 root subsystem Ψ ⊂ Φ, and α ∈ Ψ,
X [tα, tβ] = 0.
β∈Ψ+
(ττ) For any u, v ∈ h,
[τ(u), τ(v)] = 0.
−1 (tτ) For any α ∈ Φ+, w ∈ W such that w α is a simple root and u ∈ h, such that α(u) = 0,
[tα, τw(u)] = 0,
where τ (u) = τ(u) − P β(u)t . w β∈Φ+∩wΦ− β
(2) Modulo the relations (tt), the relations (tτ) are equivalent to
(tY )
[tα,Y (v)] = 0,
for any α ∈ Φ and v ∈ h such that α(v) = 0.
Example 6.6.2. In the case of B2, the (tt) relations become:
X [tα1 , tα1+2α2 ] = 0, [tγ, tβ] = 0,
β∈Ψ+
203 while
[tα2 , tα1+α2 ] 6= 0.
Assume now that the algebra Atrig is acted upon by the Weyl group W of Φ.
Proposition 6.6.3.
(1) The connection ∇trig is W − equivariant if, and only if For any α ∈ Φ, simple reflection
si ∈ W and x ∈ h,
si(tα) = tsi(α), (6.6.3)
si(τ(x)) − τ(six) = (αi, x)tαi . (6.6.4)
(2) Modulo (6.6.3), the relation (6.6.4) is equivalent to W − equivariance of the linear map
Y : h → Atrig.
6.6.2
We study the degeneration of the connection ∇KZB (6.2.3) as the modular parameter τ changes as Im τ → +∞.
Let ℘(z) be the Weierstrass function with respect to the lattice Z + τZ,
1 X 1 1 ℘(z) = + − . z2 (z − mτ + n)2 (mτ + n)2 m,n Differentiating ℘(z) term by term, we get ℘0(z).
Let q = e2πiτ , the points (℘(z), ℘0(z)) lie on the curve defined by the equation
2 3 y = 4x − g2x − g3,
where ∞ ! (2πi)4 X n3qn g = 1 + 240 , 2 12 (1 − qn) n=1 and ∞ ! (2πi)6 X n5qn g = −1 + 504 . 3 63 (1 − qn) n=1
204 3 The cubic polynomial 4x − g2x − g3 has a discriminant given by
3 2 ∆ = g2 − 27g3.
The map
C/(Z + τZ) → E ⊂ P(C), given by z 7→ (1, ℘(z), ℘0(z)) is an isomorphism of complex Lie groups, where E is defined
2 3 by the equation y = 4x − g2x − g3. Let the the modular parameter τ changes as Im τ → +∞, then q → 0. The elliptic curve
E degenerates to E˜ defined by the equation
(2πi)4 (2πi)6 −4 y2 = 4x3 − x + = (2π2 − 3x)(3x + π2)2, 12 63 27
2π2 with discriminant ∆ = 0. It has a singular point at (x, y) = ( 3 , 0).
Removing the singular point x0 of the degenerating elliptic curve Eeτ , topologically, the
∗ open subset Eeτ \ x0 is homeomorphic to the complex torus C . Thus, we have a continues
∗ ∨ ∨ map on topological spaces φ : C ⊗ P → Eeτ ⊗ P .
Pulling-back the degenerating bundle Pgτ,n under the map φ, we get a trivial principal
∗ ∨ bundle on C ⊗ P with group exp(Acell). The section of the trivial bundle can be described as
−1 {f(z): π (U) → exp(Acell) | f(z + 1) = f(z)}, where U ⊂ C∗ ⊗ P ∨ and π = exp : C → C∗ be the natural exponential map.
6.6.3
We describe the degeneration of the connection (6.2.3) as Im τ → +∞ in this subsection.
We denote the Bernoulli numbers by Bn, which are given by the power series expansion
∞ z X Bn = zn. ez − 1 n! n=0
205 We have ∞ ∞ X (2π)2n X (2π)2n πz cot(πz) = B z2n = 1 + B z2n. (2n)! 2n (2n)! 2n n=0 n=1 Some special values of the Bernoulli numbers are given by
1 B = 1,B = − , and B = 0, for n ≥ 1. 0 1 2 2n+1
As Im τ → +∞, using the product formula (6.2.1), one can show that the theta function
θ(z|τ) tends to
1 −1 1 sin(πz) θ(z|τ) −→ u 2 (1 − u ) = . (6.6.5) 2πi π Thus, by (6.6.5),
xα∨ x ∨ sin(πα + π ad( )) 2 k(α, ad( α )|τ) −→π 2 − xα∨ ∨ 2 sin(πα) sin(π ad( 2 )) ad xα x ∨ 1 =π cot(πα) + π cot(π ad( α )) − 2 α∨ ad x( 2 ) ∞ 2n X (2π) xα∨ =π cot(πα) + B ad( )2n−1. (2n)! 2n 2 n=1
As Im τ → +∞, the connection ∇KZB (6.2.3) degenerates to a flat connection ∇e on the regular points
∗ ∨ 2πiα (C ⊗ P ) \ {∪α∈Φ+ e − 1}.
The degenerate connection ∇e is on the trivial principal bundle with fibers the group exp(Adtrig).
2i Using cot(πα) = e2πiα−1 + i, we get:
Proposition 6.6.4. The degenerate connection ∇e takes the following form:
∞ 2n n X X (2π) xα∨ 2n−1 X i ∇e =d − (π cot(πα) + B2n ad( ) )(tα)dα + y(u )dui (2n)! 2 α∈Φ+ n=1 i=1 ∞ 2n ! n X 2πitα X (2π) xα∨ X =d − + πit + B ad( )2n−1(t ) dα + y(ui)du . e2πiα − 1 α (2n)! 2n 2 α i α∈Φ+ n=1 i=1 We modify the trigonometric connection (6.6.2) slightly by pulling-back to
∗ ∨ [ 2πiα (C ⊗ P ) \ {e = 1}. α∈Φ
206 We then get an Atrig–valued flat trigonometric connection:
X 2πidα ∇ = d − t − du τ(ui). (6.6.6) e2πiα − 1 α i α∈Φ+
Theorem 6.6.1 and Proposition 6.6.3 are the same with the modified connection.
By universality of the trigonometric holonomy algebra Atrig, Proposition 6.6.4 gives rise to a map Atrig → Aell given by
tα 7→ tα
∞ 2n ! X X (2π) xα∨ τ(u) 7→ −y(u) + (α, u) πit + B ad( )2n−1(t ) . α (2n)! 2n 2 α α∈Φ+ n=1
Note that this map does not preserve the usual gradings of Atrig,Aell, but it preserves the
following gradings of Atrig,Aell. The grading on Atrig is given by: deg(tα) = deg(τ(u)) = 1.
The grading on Aell is given by : deg(tα) = deg(y(u)) = 1 and deg(x(u)) = 0.
6.7 Generalized Formality Isomorphism
The following definition can be found in [16, Definition 2.4]. See [5, ?] for more details.
Definition 6.7.1. A finitely generated group G is 1-formal if its Malcev Lie algebra is
isomorphic to the completion of its holonomy Lie algebra as filtered Lie algebras.
Remark 6.7.2. 1. By construction (see [16]), the holonomy Lie algebra is a quadratic
Lie algebra, that is, it has a presentation with generators in degree 1 and relations in
degree 2 only.
2. As showed in [5], the fundamental group G of configuration spaces of n points on a
smooth compact Riemann surface is 1-formal except when the genus g = 1, and n ≥ 3.
See also [16, Example 10.1] for the failing of 1-formality when g = 1, and n ≥ 3.
207 3. When g = 1, and n ≥ 3, there is a weaker statement in [5, 10] that the Malcev Lie
algebra of G is isomorphic to the completion of the generalized holonomy Lie algebra,
which is not a quadratic Lie algebra.
In this section, we follow the method in [10] and establish the weaker Formality Isomor- phism.
6.7.1 The double affine braid group
We first recall some results from [13] about the topological interpretation of the double affine braid group.
Let ι2 = −1, and set
n ∗ U = {z ∈ C | (α, z) ∈/ Z + ιZ, for any root α ∈ Φ.}, U¯ = C × U.
Let W¯ be the 2-extended Weyl group W¯ := W n (P ∨ ⊕ ιP ∨). For any elementw ¯ = ¯ ¯ (w, a, b) ∈ W , the action ofw ¯ onz ¯ = (z∗, z) ∈ U is giving by:
w¯(¯z) = ((−1)l0(w), w(z) + a + ιb), where l0(w) is the modified length of w.
0 0 0 0 0 We fix a base pointz ¯ = (z∗), z such that the real and imaginary parts of {z∗, zj , 1 ≤ j ≤ n} are positive and sufficiently small.
Note that the action of W¯ on U¯ is not free.
Definition 6.7.3. [13, Definition 2.3] The double affine braid group B is formed by the paths γ ⊂ U¯ joiningz ¯0 with points from {w¯(¯z0), w¯ ∈ W¯ } modulo the homotopy and the
action of W¯ .
We introduce the elements tj := tαj , xj, yj for 1 ≤ j ≤ n and c ∈ B by the paths, for 1 ≤ ψ ≤ 1: (z0, α )α t (ψ) = (exp(l0(j)πιψ)z0, z0 + (exp(πιψ) − 1) j j ), j ∗ 2 208 0 0 0 0 xj(ψ) = (z∗, z + ψbj), yj(ψ) = (z∗, z + ιψbj),
0 0 c(ψ) = (exp(−2πιψ)z∗, z ).
Theorem 6.7.4. [13, Theorem 2.4] The group B is generated by {ti, 1 ≤ i ≤ n}, the two
sets {xi}, {yi} of pairwise commutative elements and central c with the following relations:
1. titjti ··· = tjtitj ... , mij factors on each side.
2. t x t = x x−1c−1 and t−1y t−1 = y y−1c , i i i i ai i i i i i ai i
−1 −(bi,θ) −1 −1 3. prxipr = xjxi and prxr∗ pr = xr ,
l0(i) −1 −1 where ci = c , pr := yrtw for w = σr .
(ρ0,b) −(ρ0,b) d As is discussed in [13], the map Ti 7→ ti, Xb 7→ xbc and Yb 7→ ybc , δ 7→ c , where
0 P l0(α)α (ρ0,θ)+1 ∨ ρ := α∈Φ+ 2 , d = m ∈ Z, b ∈ P , identifies the double affine Hecke group (see [13] for the definition) with a subgroup of B. In particular, we have the relations:
TiXb = XbTi TiYb = YbTi. if (b, αi) = 0.
We work on the spaces of the regular points of the adjoint torus
Xreg = (P ∨ ⊗ E)reg/W = U/W.¯
Corollary 6.7.5. The fundamental group of the space Xreg is isomorphic to the double affine braid group B modulo the central element c.
6.7.2
We have the following short exact sequence
0 → P B → B → W → 0, (6.7.1)
where P B := π1(Xreg, xo).
209 Let γαi ∈ P B be the path around the hypertorus Hαi , then under the inclusion
P B ,→ B,
2 we gave γαi 7→ (ti) , Xi 7→ Xi, and Yi 7→ Yi, for any 1 6= i ≤ n. In particular, the following relations holds in π1(Xreg, xo):
1. γαX(u) = X(u)γα, if (α, u) = 0;
2. γαY (u) = Y (u)γα, if (α, u) = 0;
3. X(u)X(v) = X(v)X(u), and Y (u)Y (v) = Y (v)Y (u).
The flatness of the universal connection (6.2.3) gives rise to the monodromy map
ˆ µ : P B → exp Aell,
ˆ where Aell is the completion of Aell with respect to the grading deg(x(u)) = deg(y(u)) = 1
and deg(tα) = 2.
Let J ⊆ C[P B] be the augmentation ideal of group ring C[P B], and let C\[P B] be the completion with respect to the augmentation ideal J.
Let U(Aell) be the universal enveloping algebra of the generalized holonomy Lie algebra
Aell. The monodromy map µ induces the following map on the completions:
µˆ : C\[P B] → U\(Aell)
Theorem 6.7.6. The induced map µˆ : C\[P B] → U[(A) is an isomorphism of Hopf algebras.
Taking the primitive elements of the Hopf algebras C\[P B] and U\(Aell), we get an iso-
morphism of the Malcov Lie algebra of P B and the generalized holonomy Lie algebra Aell
defined in Definition 6.1.2.
Corollary 6.7.7. The defining relations of the generalized holonomy Lie algebra Aell are
necessary to make the connection (6.2.3) flat.
210 The rest of the section is devoted to prove the generalized formality isomorphism in
Theorem 6.7.6.
Proof. Since both C[[π1] and Acell is N−filtered. It suffices to show the associated graded
gr(ˆµ) : gr(C\[P B]) → gr(U\(Aell)) = U\(Aell)
is an isomorphism.
We construct a map:
p : U\(Aell) → gr(C[[π1]),
∨ ∨ by sending tα 7→ Tα, x(λi ) 7→ Xi, and y(λi ) 7→ Yi. We need to check the following:
1. p is a homomorphism, that is, p sends the relations in Aell to 0.
2. p is surjective.
3. gr(ˆµ) ◦ p is an isomorphism.
∨ Let T := HomZ(Q, E) = P ⊗Z E be the torus with adjoint type. Denote ker(χα) by
Tα ⊆ T .
Lemma 6.7.8. There exist a component (Tα ∩ Tβ)χ, such that (Tα ∩ Tβ)χ contained in a subtorus Tγ if and only if γ ∈ Zα + Zβ.
Proof. We can assume α is a simple root. Note Zα + Zβ may not be a primitive sublattice
in Q. Pick a vector e2 such that α, e2 generate the primitive closure Zα + Zβ of Zα + Zβ.
Write β = aα + be2. Use α, e2 generates a set of basis of Q, call it {e1 := α, e2, e3, . . . , en}.
∨ Choose a dual basis {f1, f2, f3, . . . , fn}, which is a set of basis of P . Since the coweight lattice P ∨ is dual to root lattice Q.
211 ∨ n Identify T = P ⊗Z E with (E) using the basis {f1, f2, f3, . . . , fn}. Then, the maps
χα, χβ are giving by:
n n χα :(E) → E, Σi=1zifi 7→ z1. and
n n χβ :(E) → E, Σi=1zifi 7→ az1 + bz2.
∼ n−1 n n Thus, Tα = {0} × (E) ⊆ (E) , and Tα ∩ Tβ = {(0, z2, . . . , zn) ⊆ (E) | bz2 = 0}. It’s clear
2 that the number of connected components of Tα ∩ Tβ is b .
1 n Pn Take the component of (Tα ∩ Tβ)χ = {(0, b , z3 . . . , zn) ⊆ (E) }. Write γ = i=1 miei, then,
1 (T ∩ T ) ⊆ T ⇐⇒ m + Σn m z ∈ + τ , ∀z ∈ , i = 3, . . . n. α β χ γ b 2 i=3 i i Z Z i C
⇐⇒ b | m2, m3 = ··· = mn = 0.
⇐⇒ γ ∈ Zα + Zβ.
∨ Let H := HomZ(P, E) = Q ⊗Z E be the torus with simply connected type. Denote ker(χα) by Hα ⊆ H. We can modify the above proof such that a similar lemma also holds for simply connected case.
Lemma 6.7.9. There exist a component (Hα ∩ Hβ)χ, such that (Hα ∩ Tβ)χ contained in a subtorus Hγ if and only if γ ∈ Zα + Zβ.
Proof. Use the same notation as in the proof of Lemma 6.7.8. We know the coroot lattice
Q∨ is a sublattice of the coweight lattice P ∨. Write the transition matrix of the basis
∨ ∨ ∨ ∨ {α1 , . . . , αn } of Q and the basis {f1, . . . , fn} of P by (aij)i=1,...,n,j=1,...,n, which is an integral matrix.
212 We do row transformations to the matrix (aij)i=1,...,n,j=1,...,n. From linear algebra, we can get an integral upper triangular matrix. With abuse of notation, we still denote the resulted matrix by (aij)i=1,...,n,j=1,...,n.
∨ The process is equivalent to say we can find a set of basis of Q , called it, {g1, . . . , gn}, such that, g1 = a11f1 + a12f2 + ··· + a1nfn, g2 = a22f2 + ··· + a2nfn, . . . , gn = annfn.
∨ n Identify H = Q ⊗Z E with (E) using the basis {g1, . . . , gn}. Then, the maps χα, χβ are giving by:
n n χα :(E) → E, Σi=1zigi 7→ a11z1. and
n n χβ :(E) → E, Σi=1zigi 7→ (aa11 + ba12)z1 + ba22z2.
1 n Take the component of (Hα ∩ Hβ)χ = {(0, , z3 . . . , zn) ⊆ (E) }. Then the rest of the ba22 proof is exactly the same as the proof of Lemma 6.7.8.
P Proposition 6.7.10. The (tt)-relation [tα, β∈Ψ+ tβ] = 0 holds in gr(C\[P B]).
Proof. Fix two roots α, β, take the component (Hα ∩ Hβ)χ as in Lemma 6.7.9. Choose a point x0 ∈ (Hα ∩ Hβ)χ, such that x0 ∈/ Hγ, for any Hγ satisfying (Hα ∩ Hβ)χ " Hγ.
Take an open disc neighborhood D of x0, such that D ∩ Hγ = ∅, for those Hγ, satisfying
(Hα ∩ Hβ)χ " Hγ. By Lemma 6.7.9, we have
[ [ D ∩ ( Hγ) = (D ∩ Hγ). γ∈Φ+ γ∈Zα+Zβ
Now using Lemma 1.45 in [86], we get the (tt) relation in gr(C\[P B]).
Proposition 6.7.11. The relation [y(u), x(v)] = Σγ∈Φ+ hv, γihu, γitγ holds in gr(C\[P B]).
∨ ∨ Proof. Note that the loop [y(λj ), x(λi )] of Hreg is a loop wrapping around the hypertori Hγ,
γ ∈ Φ+. How it wrap those hypertori maybe complicated. But in gr(C\[P B]), we have
∨ ∨ [y(λj ), x(λi )] = Σγ∈Φ+ Ci,j,γtγ.
213 ∨ Now let’s determine the coefficients Ci,j,γ: Consider the map χα : Q ⊗Z E → E, which induces a map
∨ χα :(P ⊗Z E) \ ker χα → E \ {0}.
Denote by ϕα the composition
∨ [ ∨ ϕα : Treg := (P ⊗Z E) \ ( ker χγ) ,→ (P ⊗Z E) \ ker χα → E \ {0} γ∈Φ+
∨ 1 1 ∨ ∨ 1 1 ∨ Write a loop x(λi ) = (t+ 2 + 2 τ)λi , where 0 ≤ t ≤ 1, and a loop y(λj ) = (sτ + 2 + 2 τ)λj , where 0 ≤ s ≤ 1.
Using the method in Corollary 5.1, in [6], it’s not hard to see in E\{0}, the following relation holds:
1 ∨ 1 ∨ [ ∨ ϕα(y(λj )), ∨ ϕα(x(λi ))] = tα, (λj , α) (λi , α)
where tα is the loop around the removed point {0}. The above relation can be rewritten as
[ϕα(y(u)), ϕα(x(v))] = (u, α)(v, α)tα.
∨ ∨ So we have Ci,j,γ = hλj , γihλi , γi.
The relations [x(u), x(v)] = 0, [y(u), y(v)] = 0, for any u, v ∈ h and the relations
[tα, x(u)] = 0, [tα, y(u)] = 0 are supposed to be easier.
The following proposition can be found in [8](Proposition A.1).
Proposition 6.7.12 (M. Brou´e,G. Malle, R. Rouquier). Let i be the injection of an ir-
reducible divisor D in a smooth connected complex variety Y and base point x0 ∈ Y − D. Then, the kernel of the morphism
π1(i): π1(Y − D, x0) → π1(Y, x0)
is generated by all the generators of the monodromy around D.
214 Sn Corollary 6.7.13. Suppose X = Y − i=1 Di, where Di are irreducible divisors of Y . Then, the fundamental group π1(X, x0) is generated by all the generators of the monodromy around the divisors Di and generators of π1(Y, x0).
Proof. By induction on n: From the above proposition, the kernel of the morphism
π1((Y \ D1) − (D2 \ D1), x0) → π1(Y − D1, x0)
is generated by all the generators of the monodromy around D2. Thus, π1(Y − D1 − D2, x0)
is generated by generators of the monodromy around D2 and generators of π1(Y − D1, x0).
Induction step implies that π1(Y − D1 − D2, x0) is generated by generators of the mon-
odromy around D1, D2 and generators of π1(Y, x0).
Remark 6.7.14. The presentation of C\[P B] can be described as follows: Suppose P B is
presented by generators g1, . . . , gn and relations Ri(g1, . . . , gn), i = 1, . . . , p. Then C\[P B]
is the quotient of the free Lie algebra generated by γ1, . . . , γn by the ideal generated by
γ1 γn log(Ri(e , . . . , e )), i = 1, . . . , p.
+ Thus, the generators of C\[P B] can be chosen as Tα, X1,...,Xn, Y1,...,Yn, where α ∈ Φ .
Let deg(Xi) = deg(Yi) = 1 and deg(Tα) = 2. From the definition of p, we know p
is surjective, since we can pick the generators of gr(C\[P B]) to be the projection of the
+ generators of C\[P B] Tα, X1,...,Xn, Y1,...,Yn, where α ∈ Φ .
Pn Lemma 6.7.15. Write the connection in the following form: ∇ = d − i=1 ki(α)dαi, where
X xα∨ dα ∨ ki(α) = k(α, ad( )|τ)(tα) − y(λi ). 2 dαi α∈Φ+
Then
−2πix(λ∨) ki(α + αj) = ki(α), ki(α + ταj) = e j ki(α).
−2πix e−2πix−1 Proof. Note we have k(z + τ, x|τ) = e k(z, x|τ) + x .
215 We will also use the following fact: if [x0, t] = 0, [x0, t] = 0, then e−2πi ad x0 k(z, x|τ)(t) = k(z, x|τ)(t). Now
xα∨ dα k(α + ταj, ad( )|τ)(tα) 2 dαi x −2πi ad( α∨ ) dα x ∨ 2 dα −2πi ad( α ) dα xα∨ e j − 1 dα = e 2 dαj k(α, ad( )|τ)(t ) + (t ) α xα∨ α 2 ad( 2 ) dαi
−2πi ad(x(λ∨)) ! ∨ xα∨ e j − 1 dα dα −2πi ad(x(λj )) = e k(α, ad( )|τ)(tα) + ∨ (tα ) , 2 ad(x(λj )) dαj dαi where the second equality follows from the fact (− α∨ dα + λ∨, α) = − dα + (λ∨, α) = 0. So 2 dαj j dαj j
ki(α + ταj)
X xα∨ dα ∨ = k(α + ταj, ad( )|τ)(tα) − y(λi ) 2 dαi α∈Φ+ −2πi ad(x(λ∨)) ∨ xα∨ dα e j − 1 dα dα −2πi ad(x(λj )) X X ∨ =e k(α, ad( )|τ)(tα) + ∨ ( tα ) − y(λi ) 2 dαi ad(x(λ )) dαj dαi α∈Φ+ j α∈Φ+ −2πi ad(x(λ∨)) ∨ xα∨ dα 1 − e j −2πi ad(x(λj )) X ∨ ∨ ∨ =e k(α, ad( )|τ)(tα) + ∨ [x(λj ), y(λi )] − y(λi ) 2 dαi ad(x(λ )) α∈Φ+ j
∨ xα∨ dα ∨ −2πi ad(x(λj )) X −2πi ad(x(λj )) ∨ =e k(α, ad( )|τ)(tα) − e y(λi ) 2 dαi α∈Φ+
−2πi ad(x(λ∨)) =e j ki(α).
Check: gr(ˆµ) ◦ p is isomorphism.
i Lemma 6.7.16. The map gr(ˆµ) : gr(C\[P B]) → U\(Aell) is given by: Xui 7→ −y(u ), Yui 7→
i i 2πix(u ) − τy(u ), and Tα 7→ 2πitα.
Proof. Recall the connection is
n X xα∨ X ∇ = d − k(α, ad( )|τ)(t )dα + y(ui)du , 2 α i α∈Φ+ i=1
xα∨ 1 where k(α, ad( 2 )|τ)(tα) = α tα+ terms of degree ≥ 3.
216 n 0 i 0 Let Fu (u) be the horizonal section. Then, log Fu0 (u) = −Σi=1(ui − ui )y(u )+ terms of degree ≥ 2.
Corollary 6.7.17. For some open sets U, V , then following holds:
U U i Fu0 (u + δi) = Fu0 (u)µu0 (x(u )),
and
V −2πix(ui) V i Fu0 (u + τδi) = e Fu0 (u)µu0 (y(u )).
Proof. This follows directly from Lemma 6.7.15.
i i i If so, then we have log µu0 (x(u )) = −y(u )+ terms of degree ≥ 2 and log µu0 (y(u )) = 2πix(ui) − τy(ui)+ terms of degree ≥ 2. So gr(ˆµ)(x(ui)) = −y(ui) and gr(ˆµ)(y(ui)) =
2πix(ui) − τy(ui).
Note that
Z 1 µˆ(Tα) = ( tα+ terms of degree ≥ 3 )dα = 2πitα+terms of degree ≥ 3. Tα α
So gr(ˆµ)(Tα) = 2πitα.
i Corollary 6.7.18. The composition gr(ˆµ) ◦ p : U\(Aell) → U\(Aell) is given by: x(u ) 7→
i i i i −y(u ), y(u ) 7→ 2πix(u ) − τy(u ), and tα 7→ 2πitα. In particular, it’s an isomorphism.
6.8 Some set S
Lemma 6.8.1. Let Φ be a root system of rank 2. Then, for any α ∈ Φ+, the sum
X [tα, tβ]
β∈Φ+:hα,βiZ=Φ
lies in the subspace spanned by the (tt) relations.
217 Proof. The statement is clearly true if Φ does not properly contain a root subsystem, for in
that case the sum ranges over all β ∈ Φ+ \{α}. In general, write
X X X 0 = [tα, tβ] = [tα, tβ] β6=α Ψ⊂Φ hα,βiZ=Ψ where Ψ ranges over all subsystems of Φ containing α. Write above as
X X X [tα, hα, βiZ = Φtβ] + [tα, hα, βiZ = Ψtβ] Ψ(Φ By induction on Ψ, the second summand is zero, and so the first one must be too.
(β) Let Φ be a rank 2 subsystem, for any root β ∈ Φ, we define a set SΨ by
(β) SΨ = {α ∈ Φ | hα, βiZ = Ψ},
where Ψ is a rank 2 subsystem of Φ.
(β) Lemma 6.8.2. The set {SΨ } is one to one correspondence to the set
{Ψ ⊆ Φ of rank 2 subsystem | β ∈ Ψ}
0 Proof. Sending Ψ in the second set to SΨ = (Ψ \{β}) \ ∪Ψ0(ΨΨ gives us the bijection.
∨ (β) Lemma 6.8.3. The element ω(α , β) is independent up of α,(modulo ±1) for any α ∈ SΨ .
(β) Proof. For ui ∈ S , i = 1, 2. Since hui, βi = Ψ, the transition matrix between the two Ψ Z 1 y integral basis {u1, β} and {u2, β} is of the form , thus x = ±1. Write u1 = 0 x ∨ ∨ ∨ ±u2 + yβ. Then, ω(u1 , β) = ω((±u2 + yβ) , β) = ±ω(u2 , β). So the assertion follows.
Φ 6.9 Derivation of the Lie algebra Aell
Let d be the Lie algebra with generators ∆0, d, X, and δ2m(m ≥ 1), and relations
[d, X] = 2X, [d, ∆0] = −2∆0, [X, ∆0] = d,
218 2m+1 [δ2m,X] = 0, [d, δ2m] = 2mδ2m, (ad ∆0) (δ2m) = 0.
Now, define a Lie algebra morphism d → Der(AΦ ), denoted by ξ 7→ ξ˜. Defined as follows, ell d˜(x(u)) = x(u); X˜(x(u)) = 0; ∆ (x(u)) = y(u); f0 ˜ ˜ d(y(u)) = −y(u); X(y(u)) = x(u); ∆f0(y(u)) = 0; ˜ ˜ d(tα) = 0. X(tα) = 0. ∆f0(tα) = 0. δ˜ (x(u)) = 0; 2m ˜ 1 P P x(α∨) p x(α∨) q δ2m(y(u)) = 2 α α(u) p+q=2m−1[(ad 2 ) (tα), (ad − 2 ) (tα)]; ˜ x(α∨) 2m δ2m(tα) = [tα, (ad 2 ) (tα)].
Φ Proposition 6.9.1. The above map d → Der(Aell) is a Lie algebra morphism.
˜ Φ Proof. To check the map is well defined, we need to check that ξ is a derivation of Aell, for ˜ ˜ any ξ being generators of d. The fact that d, X and ∆f0 are derivations are clear. It’s also clear the δg2m preserves the relations [x(u), x(v)] = 0 and [tα, x(u)] = 0, if (α, u) = 0. It ˜ remains to show that δ2m preserves the following relations
1.[ y(u), x(v)] = Σγ∈Φ+ (u, γ)(v, γ)tγ;
P 2.[ tα, β∈Ψ+ tβ];
3.[ tα, y(u)] = 0, if (α, u) = 0;
4.[ y(u), y(v)] = 0.
6.9.1 δ2m preserve the commutating relation of [y, x].
˜ ˜ ˜ We check [δ2m(y(u)), x(v)] + [y(u), δ2m(x(v))] = Σγ∈Φ+ (u, γ)(v, γ)δ2m(tγ) in this subsection. ˜ By definition of δ2m, we have
1 X X xγ∨ xγ∨ [δ˜ (y(u)), x(v)]+[y(u), δ˜ (x(v))] = [ γ(u) [(ad )p(t ), (ad − )q(t )], x(v)] 2m 2m 2 2 γ 2 γ γ p+q=2m−1
219 1 0 0 ∨ Notice that we can write x(v) = 2 (v, γ)xγ + x(v ), where (γ, v ) = 0, using the (tx)-relation, we have
x ∨ x ∨ [[(ad γ )p(t ), (ad − γ )q(t )], x(v0)] = 0 2 γ 2 γ
Thus,
˜ ˜ [δ2m(y(u)), x(v)] + [y(u), δ2m(x(v))]
1 X X xγ∨ xγ∨ xγ∨ =[ γ(u) [(ad )p(t ), (ad − )q(t )], (v, γ) ] 2 2 γ 2 γ 2 γ p+q=2m−1
1 X X xγ∨ xγ∨ xγ∨ xγ∨ = (v, γ)γ(u)[ ([(ad )p+1(t ), (ad − )q(t )] − [(ad )p(t ), (ad − )q+1(t )]) 2 2 γ 2 γ 2 γ 2 γ γ p+q=2m−1
X xγ∨ 2m = (v, γ)γ(u)[t , (ad − ) (t )] = Σ + (v, γ)(u, γ)δ˜ (t ). γ 2 γ γ∈Φ 2m γ γ
˜ P P ˜ 6.9.2 Check [δ2m(tα), β∈Ψ+ tβ] + [tα, β∈Ψ+ δ2m(tβ)] = 0
P Suppose we have a relation [tα, β∈Ψ+ tβ] = 0, where Ψ is a rank 2 system. We first claim that:
X xγ δ˜ (t ) = [t , (ad )2m(t )] 2m β β 2 γ γ∈Ψ+
P xγ∨ 2m Proof of the claim: It suffices to check that: γ∈Ψ+,γ6=β[tβ, (ad 2 ) (tγ)] = 0. By definition,
X xγ∨ X xω(γ∨,β) [t , (ad )2m(t )] = [t , (ad )2m(t )] β 2 γ β −2 γ {γ∈Ψ+,γ6=β} {γ∈Ψ+,γ6=β} (β) X x(cpsi0 ) 2m X = [tβ, (ad ) ( tγ)] = 0, 0 −2 Ψ ⊂Ψ γ∈S(β) ψ0
(β) ∨ (β) ∨ where cpsi0 := ω(γ , β), for γ ∈ Sψ0 . According to Lemma 6.8.3, we know ω(γ , β) is a (β) constant (up to a sign), so cψ0 is well-defined. (β) P The last equality follows from (tx) relation [tβ, x(cψ0 )] = 0, and (tt) relation [tβ, γ∈S(β) tγ] = ψ0 0. So that finish the proof of claim.
220 Now, we have,
˜ X X ˜ [δ2m(tα), tβ] + [tα, δ2m(tβ)] β∈Ψ+ β∈Ψ+
X xα∨ 2m X X xγ = − [t , [t , ad (t )]] + [t , [t , (ad )2m(t )]] α β 2 α α β 2 γ β∈Ψ+ β∈Ψ+ γ∈Ψ+
X X xγ∨ X X xω(γ∨,α) =[t , [t , (ad )2m(t )]] = [t , [t , (ad )2m(t )]] α β 2 γ α β 2 γ β∈Ψ+ {γ|γ6=α} β∈Ψ+ {γ|γ6=α} (α) X X X x(cψ0 ) X = [t , [ t , (ad )2m t ]] = 0, α β 2 γ 0 + Ψ ⊂Ψ β∈Ψ {γ|γ6=α} γ∈S(α) ψ0
(α) P The last equality follows from (tx) relation [tα, x(cψ0 )] = 0, and (tt) relations [tα, γ∈S(α) tγ] = ψ0 P 0, and [tα, β∈Ψ+ tβ] = 0.
˜ ˜ 6.9.3 Check that [y(u), δ2m(tα)] + [δ2m(y(u)), tα]=0, if (α, u) = 0
˜ + Now, let’s calculate the second term [δ2m(y(u)), tα] first. Write Φ = {α, γ1, ··· , γm}. The purpose for written like this is giving an order for the positive roots in Φ+ \{α}.
Lemma 6.9.2. Now we assume we are in the rank 2 situation, then
x ∨ xω(γ∨,γ ) ±ω(γ ,γ ) (ad i j )2m − (ad j i )2m ˜ 1 X X −2 2 [δ2m(y(u)), tα] = [tα, γi(u) x ∨ [tγ , tγ ]] xω(γ∨,γ ) ω(γ ,γ ) i j 2 i j j i Ψ⊂Φ (α) ∓ {j6=i|γi,γj ∈SΨ } −2 2 Proof. For fixed p, q, such that p + q = 2m − 1, note we have
x ∨ x ∨ [[(ad γ )p(t ), (ad − γ )q(t )], t ] 2 γ 2 γ α x ∨ x ∨ x ∨ x ∨ = [(ad ω(γ ,α) )q[t , t ], (ad γ )p(t )] − [(ad ω(γ ,α) )p[t , t ], (ad γ )q(t )]. 2 α γ 2 γ −2 α γ −2 γ
(α) Now for γi ∈ SΨ , pluging the (tt)-relation
X [tα, tγi ] = −[tα, tγj ] (6.9.1) (α) {j6=i|γj ∈SΨ }
221 ˜ into definition of [δ2m(y(u)), tα], we get:
X X xγ∨ xγ∨ [ γ(u) [(ad )p(t ), (ad − )q(t )], t ] 2 γ 2 γ α γ p+q=2m−1 X X X xω(γ∨,α) X xγ∨ = − γ (u)( [(ad i )q[t , t ], (ad i )p(t )] i 2 α γj 2 γi Ψ⊂Φ i p+q=2m−1 (α) {j6=i|γj ∈SΨ } xω(γ∨,α) X xγ∨ − [(ad i )p[t , t ], (ad i )q(t )]) −2 α γj −2 γi (α) {j6=i|γj ∈SΨ } X X X xω(γ∨,α) xγ∨ = − γ (u)[(ad i )q[t , t ], (ad i )p(t )] i 2 α γj 2 γi Ψ⊂Φ (α) p+q=2m−1 {j6=i|γi,γj ∈SΨ } xω(γ∨,α) xγ∨ − γ (u)[(ad j )p[t , t ], (ad j )q(t )] j −2 α γi −2 γj x ∨ ∨ X X X ±γj xγ = − γ (u)[[t , (ad )qt ], (ad i )p(t )] i α −2 γj 2 γi Ψ⊂Φ (α) p+q=2m−1 {j6=i|γi,γj ∈SΨ } xγ∨ x±γ∨ − γ (u)[[t , (ad i )pt ], (ad j )q(t )] i α 2 γi −2 γj ∨ x ∨ X X X xγ ±γj = − γ (u)[(ad i )p(t ), (ad )q(t )], t ] i 2 γi −2 γj α Ψ⊂Φ (α) p+q=2m−1 {j6=i|γi,γj ∈SΨ } x ∨ xω(γ∨,γ ) ±ω(γ ,γ ) i j 2m j i 2m X X (ad −2 ) − (ad 2 ) = − γi(u)[ x ∨ [tγ , tγ ], tα], xω(γ∨,γ ) ω(γ ,γ ) i j i j j i Ψ⊂Φ (α) ∓ {j6=i|γi,γj ∈SΨ } −2 2
where ± means: if ω(γj, α) = ω(γi, α), we use +; if ω(γj, α) = −ω(γi, α), we use −.
The following corollary of Lemma 6.9.2 will be used later(to prove Claim 6.9.4).
(α) Corollary 6.9.3. For any p, q ∈ N, such that p + q = 2m − 1, for two different sets SΨ , (α) SΨ0 , we have:
∨ x ∨ X X xγ ±γj [ [(ad i )p(t ), (ad )q(t )], t ] = 0 2 γi −2 γj α γ ∈S(α) γ ∈S(α) i Ψ j Ψ0
(α) Proof. For any γi ∈ SΨ , in the proof of Lemma 6.9.2, instead of using the (tt)-relation
222 (6.9.1), we use X X [tα, tγi ] = −[tα, tγi0 ] − C[tα, tγj ], (α) 0 {j|γj ∈SΨ0 } {i 6=i|γi0 ∈SΨ }
(α) (α) (α) lcm(n(S ),n(S )) n(S ) (α) (α) where C = Ψ Ψ0 ( Ψ )q and n(S ) := Index(Ψ ⊆ Φ ), in particular, n(S ) = n(S(α)) n(S(α)) Ψ Z Z Φ Ψ Ψ0 1. The benefit of choosing this constant C is that we are able to write
(α) (α) ∨ ∨ xω(γ ,α) lcm(n(S ), n(S 0 )) x±γ γ (u)(ad i )qC[t , t ] = Ψ Ψ γ (u)[t , (ad j )qt ] i 2 α γj (α) i α −2 γj n(SΨ )
(α) n(SΨ ) ∨ Above equality follows from γi = ± γj + kα, for some k ∈ . So we get ω(γ , α) = n(S(α)) Z i Ψ0 n(S(α)) Ψ0 ∨ (α) ω(±γj , α). Now do the same thing as in the proof of Lemma 6.9.2, we can get: n(SΨ )
X xγ∨ xγ∨ [ γ(u)[(ad )p(t ), (ad − )q(t )], t ] 2 γ 2 γ α γ
∨ x ∨ X X xγ ±γj = − γ (u)[(ad i )p(t ), (ad )q(t )], t ] i 2 γi −2 γj α k {j6=i|γi,γj ∈Sψ}
0 x ∨ x±γ∨ X X lcm(n(SΨ), n(SΨ )) γi p j q −[ γi(u)[(ad ) (tγi ), (ad ) (tγj )], tα], n(SΨ) 2 −2 γi∈SΨ γj ∈SΨ0
lcm(n(S(α)),n(S(α))) Ψ Ψ0 Note that (α) γi(u) is a constant. Now compare it with the result in Lemma n(SΨ ) 6.9.2, the conclusion follows.
˜ Let’s calculate the first term [y(u), δ2m(tα)], since [y(u), tα] = 0, we have
˜ [y(u), δ2m(tα)]
x ∨ 2m =[t , [y(u), ad α (t )]] α 2 α 2m 1 X X xα∨ s xω(α∨,γ) =[t , (α∨, γ)(u, γ) ad (ad )2m−1−s[t , t ]] α 2 2 −2 γ α γ6=α s=0 x ∨ 2m xω(α∨,γ) 2m X ad α − (ad ) =[t , (u, γ) 2 −2 [t , t ]] α xω(γ∨,α) γ α γ6=α ad −2
+ Let Φ = {α, γ1, ··· , γm}. The following claim gives the formula for the first term.
223 Claim 6.9.4. We claim that in rank 2 case, we have the following equality:
x ∨ x ω(α ,γj ) 2m ω(α∨,γ ) 2m X X (ad ) − (ad i ) [y(u), δ˜ (t )] = −[t , (u, γ ) −2 −2 [t , t ]]. 2m α α i xω(γ∨,α) γi γj i Ψ≤Φ (α) ad −2 {i Proof. Lemma 6.9.5. The following (tt)-relations hold: (α) P 1. For γi ∈ SΦ , then:[tγi , α] = − j6=i[tγi , tγj ]. (α) (α) P 1 P 2. For γi ∈ S 6= S , then,[tγ , α] = − (α) [tγ , tγ ] − (α) (α) [tγ , tγ ]. Ψ Φ i {γj ∈S } i j 2 γj ∈/S i j Ψ n(SΨ ) Ψ P Proof. The relation (1) follows from the relation [tγi , α + j6=i tγj ] = 0. (α) (α) For γi ∈ SΨ , where Ψ is a root subsystem. In this case, SΨ ∪ {α} is the root system P Ψ provided that Ψ 6= Φ. The relation [tγi , α + (α) tγj ] = 0 implies [tγi , α] = {γj ∈SΨ ,j6=i} P − (α) [tγi , tγj ]. {γj ∈SΨ } P P Note that we also have [tγi , α + j6=i tγj ] = 0, which implies (α) [tγi , tγj ] = 0. Thus, γj ∈/SΨ the relation (2) follows. If we denote the coefficient in (1) and (2)(in Lemma 6.9.5) by ij, that is: [tγi , tα] = P (u,γi) (u,γj ) − ij[tγi , tγj ]. Then, the following equality holds: ∨ ij = ∨ ji, for any i, j, for j6=i ω(γi ,α) ω(γj ,α) (α) (α, u) = 0. If γi, γj ∈ SΨ , then ij = 1, the equality is obvious. (α) (α) 0 1 1 If γi ∈ S , γj ∈ S 0 ,Φ 6= Ψ , then ij = , and ji = . The equality follows Ψ Ψ n(S(α))2 n(S(α))2 Ψ Ψ0 (u,γi) (α) 2 (u,γ0) (α) from the fact that ∨ = n(SΨ ) ∨ , for some γ0 ∈ SΦ . ω(γi ,α) ω(γ0 ,α) Plugging relation (1) and (2) into the equation above Claim 6.9.4, we get: x ∨ 2m xω(α∨,γ) 2m X ad α − (ad ) [t , (u, γ) 2 −2 [t , t ]] α xω(γ∨,α) γ α γ6=α ad −2 ∨ ∨ X (u, γi) xω(α ,γj ) xω(α ,γi) = − [t , ((ad )2m − (ad )2m)[t , t ]] α ij xω(γ∨,α) γi γj i −2 −2 i 224 Suppose aijγi + ajiγj = sijα, for some integers aij, aji, sij having no common divisor. By calculation, we have ∨ ∨ X (u, γi) xω(α ,γj ) xω(α ,γi) [t , ((ad )2m − (ad )2m)[t , t ]] α ij xω(γ∨,α) γi γj i −2 −2 i 2m (α) (α) ij sij (u,γi) Notice that γi ∈ S , γj ∈ S 0 with [tγ , tγ ] 6= 0, then the term p+1 q is constant. Ψ Ψ i j (aij ) (±aji) By corollary 6.9.3, the above summation for γi, γj coming from different sets SΨ, SΨ0 is 0. Thus, the conclusion of claim 6.9.4 follows. Now, we compare the results in Lemma 6.9.2 and in Claim 6.9.4, Assume that γi = ±γj + α, then: ∨ ∨ ∨ ∨ • ω(γi , γj) = ω(α , γj), ω(γj , γi) = ∓ω(α , γi); ∨ ∨ ∨ • ω(α , γj) − ω(α , γi) = ω(γi , α). ˜ ˜ Thus, under this assumption, we have [y(u), δ2m(tα)] + [δ2m(y(u)), tα] = 0. The above assumption doesn’t hold only in the case G2, when α is the short root. In which case we have: γi = ±γj + 3α. Thus, in this case, the lemma can be rewritten as 1 X X xγ∨ xγ∨ [ γ(u) [(ad )p(t ), (ad − )q(t )], t ] 2 2 γ 2 γ α γ p+q=2m−1 x ω(3α,γj ) 2m x±ω(3α,γ ) 2m 1 X X (ad ) − (ad i ) = [t , γ (u) −2 2 [t , t ]] α i xω(γ∨,α) γi γj 2 i k (α) −2 {j6=i|γi,γj ∈SΨ } γi(u) Notice that, ∨ is a constant, for γi are three long roots in G2. We can use the (tt)- ω(γi ,α) relation for the three long roots to show that above summation is 0. Similarly, the summation in Claim 6.9.4 that involving the three long roots is 0. 225 ˜ ˜ 6.9.4 Check [y(u), δ2m(y(v))] + [δ2m(y(u)), y(v)] = 0 By definition, we have ˜ ˜ [y(u), δ2m(y(v))] + [δ2m(y(u)), y(v)] 1 X X xγ∨ xγ∨ =[y(u), γ(v) [(ad )p(t ), (ad − )q(t )]] 2 2 γ 2 γ γ p+q=2m−1 1 X X xγ∨ xγ∨ − [y(v), γ(u) [(ad )p(t ), (ad − )q(t )]] 2 2 γ 2 γ γ p+q=2m−1 X 1 X xγ∨ xγ∨ = [y , [(ad )p(t ), (ad − )q(t )]] η 2 2 γ 2 γ γ p+q=2m−1 X X xγ∨ xγ∨ = [[y , (ad )p(t )], (ad − )q(t )]] η 2 γ 2 γ γ p+q=2m−1 where yη = γ(v)y(u) − γ(u)y(v). Notice that [yη, tγ] = 0, thus, ˜ ˜ [δ2m(y(u)), y(v)] + [y(u), δ2m(y(v))] X X X X xγ∨ xω(γ∨,β) xγ∨ = [ det(β, γ) (β, γ∨) (ad )s(ad )p−1−s[t , t ], (ad − )q(t )], ij 2 2 β γ 2 γ γ p+q=2m−1 β s where det(β, γ)ij = β(u)γ(v) − β(v)γ(u) Therefore, we have to show xγ∨ p xω(γ∨,β) p X X X (ad 2 ) − (ad −2 ) xγ∨ q [ det(β, γ)ij [tβ, tγ], (ad − ) (tγ)] = 0 (6.9.2) xω(β∨,γ) 2 γ p+q=2m−1 β ad −2 We are in the situation of rank 2 case. (βl) Lemma 6.9.6. For three fixed roots, β, γ, βl ∈ Φ. Assume β ∈ SΨ , for some root subsystem 0 (βl) Ψ of Φ. For any β ∈ SΨ . If we have aβl + bβ + cγ = 0, for some a, b, c ∈ Z, with gcd(a, b, c) = 1, then, there exits an integral a0 ∈ Z, such that the following equation holds 0 0 a βl ± bβ + cγ = 0. 226 Remark 6.9.7. Based on the above lemma, the integer c is independent of the elements in (βl) (βl) SΨ . So we can write c(SΨ ). + For fixed root γ ∈ Φ, where Φ = {γ, β1, ··· , βm}. Now, similar as Lemma 6.9.5, we use (γ) the following (tt)-relation. For βl ∈ SΦ , then: X [tβl , γ] = − [tβl , tβs ] (6.9.3) s6=l (γ) (γ) For βl ∈ SΨ 6= SΦ , 1 X X (βl) 2m−2 X [tβl , γ] = − [tβl , tβs ] − c(SΨ0 ) [tβl , tβs ] (6.9.4) n(S(γ))2 (γ) Ψ {Ψ0 Φ,Ψ06=Ψ} (βl) {βs∈S } β ∈S Ψ s Ψ0 (βl) Proof. The above equation is obtained by using the fact that SΨ0 is a root subsystem, provided that Ψ0 Φ, Ψ0 6= Ψ. Claim 6.9.8. Assume that aslβs + blsβl + cslγ = 0, then, x x ∨ γ∨ p ω(γ ,βl) p X X (ad ) − (ad ) xγ∨ [det(β , γ) 2 −2 [t , t ], (ad − )q(t )] l ij xω(β∨,γ) βl γ γ l 2 l p+q=2m−1 ad −2 ∨ ∨ ∨ X X x(b β ) x(a βs) x(c γ) = C [[(ad ls l )r(t ), (ad sl )n(t )], (ad sl )q(t )], l,s,γ 2 βl 2 βs 2 γ l Lemma 6.9.9. Cl,s,γ = Cs,γ,l = Cγ,l,s. Proof. It suffices to show that Cl,s,γ = −Cs,l,γ, and Cl,s,γ = −Cγ,s,l. (γ) (γ) (γ) Proof of: Cl,s,γ = −Cs,l,γ: Let βl ∈ SΨ , and βs ∈ SΨ0 , then βl = n(SΨ )β0 + klγ, and ±n(S(γ)) (γ) −b Ψ0 βs = ±n(SΨ0 )β0 + ksγ, for some β0. The fact a = (γ) implies the equality. n(SΨ ) 227 Proof of:Cl,s,γ = −Cγ,s,l: First notice that det(βl, γ)ij = − det(γ, βl)ij. (γ) (βl) Case 1: βs, βl ∈ SΨ and βs, γ ∈ SΨ . The equality holds. (γ) (βl) (βl) (γ) Case 2: βs, βl ∈ SΨ , βs ∈ SΨ0 or βs, γ ∈ SΨ , βs ∈ SΨ0 . This case can only happen γ (βl) in root system G2, when hγ, βliZ = Φ, thus, n(SΦ) = n(SΦ ) = 1. Therefore, the equality holds. γ γ (βl) Case 3: βl ∈ SΨ, βs ∈ SΨ0 , γ ∈ SΨ00 . (γ) (βl) The equality holds in this case simply because n(SΨ ) = n(SΨ ), since hβl, γiZ = Ψ. Finally, take the summation over all positive roots γ, there will be two terms of the following: x ∨ x ∨ x ∨ C [[(ad (aslβs) )n(t ), (ad (cslγ) )q(t )], (ad (blsβl) )r(t )] s,γ,l 2 βs 2 γ 2 βl and x ∨ x ∨ x ∨ C [[(ad (cslγ) )q(t ), (ad (blsβl) )r(t )], (ad (aslβs) )n(t )] γ,l,s 2 γ 2 βl 2 βs Using Jacobi identity and Lemma 6.9.9, we know the summation of the three terms is 0. Thus, the equation (6.9.2) that we are trying to show is 0. 6.10 Bundle on the moduli space M1,n Let e, f, h be the standard basis of sl2. Then, there is a Lie algebra morphism d → sl2 defined by δ2m 7→ 0, d 7→ h, X 7→ e, ∆0 7→ f. We denote by d+ ⊂ d its kernel. Since there is a section of the morphism d → sl2(given by e, f, g 7→ X, ∆0, d), we have a semidirect decomposition d = d+ o sl2. Then, we have Aell o d = (Aell o d+) o sl2. The following lemma is similar to Lemma 3.2 in [10]. Lemma 6.10.1. Aell o d+ is positively graded. 228 2 Proof. Define Z −grading of d and Aell by deg(∆0) = (−1, 1), deg(d) = (0, 0), deg(X) = (1, −1), deg(δ2m) = (2m + 1, 1), deg(x(u)) = (1, 0), deg(y(u)) = (0, 1) and deg(tα) = (1, 1). We can form the following semidirect products Gn := exp(A\ell o d+) o SL2(C). ∨ 2 The semidirect product (P ) o SL2(Z) acts on h × H by (n, m) ∗ (z, τ) := (z + n + τm, τ) ∨ 2 a b z aτ+b a b for (n, m) ∈ (P ) and ( c d ) ∗ (z, τ) := ( cτ+d , cτ+d ), for ( c d ) ∈ SL2(Z). ∨ ˜ Let χα,τ : P ×Z Eτ → Eτ be the natural morphism. We define Hα,τ to be ˜ Hα,τ = {(z, τ) ∈ h × H | α(z) ∈ Λτ }. ∨ 2 S ˜ Lemma 6.10.2. The group (P ) o SL2(Z) preserves h × H \ α,τ Hα,τ . S ∨ Proof. Take (z, τ) ∈ h × H \ α Hα, that is, for any α ∈ Φ, χα,τ (z) ∈/ P × Λτ . ∨ It’s obvious that χα,τ (z + n + τm) ∈/ P × Λτ , for any α ∈ Φ. z ∨ Suppose χ aτ+b ( ) ∈ P × Λ aτ+b . α, cτ+d cτ+d cτ+d Case 1, if c = 0, then, d = ±1. we have χα,τ (z) ∈ Z + Z(aτ + b). It’s a contradiction. z aτ+b Case 2, if c 6= 0, then, χα,τ (z) = (cτ + d)χ aτ+b ( ) ∈ (cτ + d) ( ) ⊂ + τ. It’s α, cτ+d cτ+d Z cτ+d Z Z a contradiction. We can identify the quotient with the moduli space M1,n of our space X. S Denote by π : h × H \ α(Hα) → M1,n the projection map. We will define a principal Gn−bundle with flat connection (Pn, ∇Pn ) over M1,n. × d u 0 vX 1 v For u ∈ C , u := 0 u−1 ∈ SL2(C) ⊂ Gn and for v ∈ C, e := ( 0 1 ) ∈ SL2(C) ⊂ Gn. Proposition 6.10.3. There exists a unique principal Gn−bundle Pn over M1,n, such that −1 ∨ a section of U ⊂ M1,n is a function f : π (U) → Gn, such that f(z + αi |τ) = f(z|τ), ∨ −2πixα∨ z 1 d 2πi P f(z + τα |τ) = e i f(z|τ), f(z|τ + 1) = f(z|τ) and f( | − ) = τ exp( ( z x ∨ + i τ τ τ i i αi X))f(z|τ). 229 Proof. In [10], Calaque, Enriquez and Etingof showed there exists a principal bundle over n 2 h × H/((Z ) o SL2(Z)) with the structure group Gn. Pulling back of this bundle along the n 2 map M1,n ,→ h × H/((Z ) o SL2(Z)) gives us the desired vector bundle Pn. 6.11 Flat connection on the moduli space M1,n n Denote Diagn := {(z, τ) ∈ C ×H | for some i 6= j, zi − zj ∈ Λτ .} Set g(z, x|τ) := kx(z, x|τ) = ∂k(z,x|τ) ∂x . We have g(z, x|τ) ∈ Hol((C × H) − Diag1)[[x]]. P 2n P P For ψ(x) = n≥1 b2nx , we set δψ := n≥1 b2nδ2n, ∆ψ := ∆0 + n≥1 b2nδ2n. If we set ϕ(x) = g(0, 0|τ) − g(0, x|τ), P 2n P then, ϕ(x) = n≥1 a2nE2n+2(τ)x , Denote δϕ := n≥1 a2nE2n+2(τ)δ2n, We have ∆ϕ = ∆0 + δϕ. For u ∈ h, define 1 1 X 1 X xβ∨ ∆ = ∆(u, τ) := − ∆ − a E (τ)δ + g(β, ad |τ)(t ) 2πi 0 2πi 2n 2n+2 2n 2πi 2 β n≥1 β∈Φ+ 1 1 X xβ∨ = − ∆ + g(β, ad |τ)(t ), 2πi ϕ 2πi 2 β β∈Φ+ 2n+1 where a2n = −(2n + 1)B2n+1(2iπ) /(2n + 1)! and Bn are the Bernoulli numbers given by x P r n ex−1 = r≥0(Br/r!)x . This is a a meromorphic function C × H → (Aell o d+) o n+ ⊂ S Lie(Gn), (where n+ = C∆0 ⊂ sl2) with only poles at α∈Φ+ Hfα. Theorem 6.11.1. The following Aell o d-valued connection on M1,n is flat. n X xα∨ X ∇ = d − ∆dτ − k(α, ad |τ)(t )dα + y(ui)du . (6.11.1) 2 α i α∈Φ+ i=1 230 P xα∨ Note we have already known the flatness of ∇τ = d − α∈Φ+ k(α, ad 2 |τ)(tα)dα + Pn i i=1 y(u )dui. Set n X xα∨ X A := ∆dτ + k(α, ad |τ)(t )dα − y(ui)du . 2 α i α∈Φ+ i=1 The rest of this section is devoted to show the Theorem 6.11.1. By definition of the flatness of the connection ∇, we need to show: dA + A ∧ A = 0. 6.11.1 Check that dA = 0 n ∂A X ∂A dA = dτ + dα ∂τ ∂α i i=1 i X ∂ xα∨ X ∂ = k(α, ad |τ)(tα)dτ ∧ dα + ∆(α, τ)dαi ∧ dτ ∂τ 2 ∂αi α∈Φ+ i X ∂ xα∨ 1 X X ∂ xα∨ = k(α, ad |τ)(tα) − g(β, ad |τ)(tα)dτ ∧ dαi ∂τ 2 2πi ∂αi 2 α∈Φ+ i β X ∂ xα∨ 1 ∂ xα∨ = ( k(α, ad |τ)(t ) − g(α, ad |τ)(t ))dτ ∧ dα ∂τ 2 α 2πi ∂α 2 α α∈Φ+ 1 which is 0 by the following identity (∂τ k)(z, x|τ) = 2πi (∂zg)(z, x|τ), which can be found in [10], on Page 190. 231 6.11.2 Check that A ∧ A = 0 By definition, we have, X xα∨ X A ∧ A = [∆(α, τ), k(α, ad |τ)(tα)]dτ ∧ dα − [∆(α|τ), yλ∨ ]dτ ∧ dαi 2 i α∈Φ+ i −1 X X xβ∨ xα∨ = ( [∆ − g(β, ad |τ)(t ), τ), k(α, ad |τ)(t )]dτ ∧ dα 2πi ϕ 2 β 2 α α∈Φ+ β∈Φ+ X X xβ∨ − [∆ϕ − g(β, ad |τ)(tβ), yλ∨ ]dτ ∧ dαi) 2 i i β∈Φ+ −1 X xα∨ = ( [∆ , k(α, ad |τ)(t )]dτ ∧ dα 2πi ϕ 2 α α∈Φ+ X X xβ∨ xα∨ − [ g(β, ad |τ)(t ), τ), k(α, ad |τ)(t )]dτ ∧ dα 2 β 2 α α∈Φ+ β∈Φ+ X X X xβ∨ − [∆ϕ, yλ∨ ]dτ ∧ dαi + [ g(β, ad |τ)(tβ), yλ∨ ]dτ ∧ dαi) i 2 i i i β∈Φ+ Now let’s calculate each terms in the above expression of A ∧ A. Lemma 6.11.2. Modulo the relations (tx), (xx), the following identity holds for any α 6= β ∈ Φ+: x ∨ x ∨ x ∨ x ∨ [g(β, ad β |τ)(t ), k(α, ad α |τ)(t )] = g(β, (ad ω(β ,α) )|τ)k(α, (ad ω(α ,β) )|τ)[t , t ] 2 β 2 α −2 −2 β α Lemma 6.11.3. Modulo the relations (yt),(yx),(tx), (xx), the following identity holds: ∨ X xβ∨ X β xβ∨ [y(ui), g(β, ad |τ)(t )]dτ ∧ du − [y( ), g(β, ad |τ)(t )]dτ ∧ dβ 2 β i 2 2 β i β∈Φ+ xβ∨ xω(β∨,γ) X g(β, ad 2 |τ) − g(β, (ad −2 )|τ) = x [tγ, tβ]dτ ∧ dγ ad ω(γ∨,β) γ6=β∈Φ+ −2 x x ∨ ∨ β∨ ω(β ,γ) X γ(β ) g(β, ad 2 |τ) − g(β, (ad −2 )|τ) − x [tγ, tβ]dτ ∧ dβ 2 ad ω(γ∨,β) γ6=β∈Φ+ −2 232 Proof. If β(ui) = 0, we have: xβ∨ xω(β∨,γ) i xβ∨ X ∨ i g(β, ad 2 |τ) − g(β, (ad −2 )|τ) [y(u ), g(β, ad |τ)(tβ)] = (β , γ)γ(u ) [tγ, tβ] 2 adxβ∨ + adxω(β∨,γ) γ∈Φ+ i For a fixed root β, choose {u1, . . . , un}, such that u1 = β. Then, β(u ) = 0, for i 6= 1. Then, we have: n X xβ∨ [y(ui), g(β, ad |τ)(t )]dτ ∧ du 2 β i i=1 n xβ∨ xω(β∨,γ) X X ∨ i g(β, ad 2 |τ) − g(β, (ad −2 )|τ) = (β , γ)γ(u ) [tγ, tβ]dτ ∧ dui adxβ∨ + adxω(β∨,γ) i=1 γ∈Φ+ x x ∨ ∨ β∨ ω(β ,γ) X ∨ β g(β, ad 2 |τ) − g(β, (ad −2 )|τ) − (β , γ)γ( ) [tγ, tβ]dτ ∧ dβ 2 adxβ∨ + adxω(β∨,γ) γ∈Φ+ ∨ β x ∨ + [y( ), g(β, ad β |τ)(t )]dτ ∧ dβ 2 2 β ∨ β x ∨ =[y( ), g(β, ad β |τ)(t )]dτ ∧ dβ 2 2 β xβ∨ xω(β∨,γ) X ∨ g(β, ad 2 |τ) − g(β, (ad −2 )|τ) + (β , γ) [tγ, tβ]dτ ∧ dγ adxβ∨ + adxω(β∨,γ) γ6=β∈Φ+ x x ∨ ∨ β∨ ω(β ,γ) X ∨ β g(β, ad 2 |τ) − g(β, (ad −2 )|τ) − (β , γ)γ( ) [tγ, tβ]dτ ∧ dβ 2 adxβ∨ + adxω(β∨,γ) γ∈Φ+ x x ∨ ∨ β∨ ω(β ,γ) β xβ∨ X g(β, ad 2 |τ) − g(β, (ad −2 )|τ) =[y( ), g(β, ad |τ)(tβ)]dτ ∧ dβ + x [tγ, tβ]dτ ∧ dγ 2 2 ad ω(γ∨,β) γ6=β∈Φ+ −2 x x ∨ ∨ β∨ ω(β ,γ) X γ(β ) g(β, ad 2 |τ) − g(β, (ad −2 )|τ) − x [tγ, tβ]dτ ∧ dβ 2 ad ω(γ∨,β) γ∈Φ+ −2 Thus, the conclusion follows. Lemma 6.11.4. Modulo the relation (δy), we have: X X X xα∨ xα∨ [∆ , y(ui)]du = [f ad (t ), g (− ad )(t )]dτ ∧ dα, ϕ i s 2 α s 2 α i α∈Φ+ s 1 ϕ(u)−ϕ(v) P where 2 u−v = s fs(u)gs(v). 233 Proof. By definition, we have 1 X X xα∨ p xα∨ [δ , y(ui)] = α(ui) [ad (t ), (ad − )q(t )], 2m 2 2 α 2 α α p+q=2m−1 Then: X X xα∨ xα∨ [∆ , y(ui)] = α(ui) [f ad (t ), g (− ad )(t )]. ϕ s 2 α s 2 α α s Lemma 6.11.5. We have: xα∨ X xα∨ xα∨ [δ , k(α, ad |τ)(t )] = [lα(ad )(t ), mα(ad )(t )], ϕ 2 α s 2 α s 2 α s P α α where k(α, u + v)ϕ(v) = s ls (u)ms (v). Proof. Lemma 6.11.6. Modulo the relation [∆0, x], [∆0, y], [∆0, t], and (yx), we have: xα∨ yα∨ xα∨ X xα∨ xα∨ [∆ , k(α, ad |τ)(t )] = [ , g(α, ad |τ)(t )] − [h (ad (t ), k (ad (t )] 0 2 α 2 2 α s 2 α s 2 α s x ∨ x ∨ x ∨ xα∨ ω(α ,γ) xα∨ ω(α ,γ) ω(α ,γ) X ∨ 2 k(α, ad 2 − k(α, (ad −2 ) − (ad 2 + ad 2 )kx(α, (ad −2 )) − (α , γ) 2 [tγ, tα] (ad xα∨ + ad xω(α∨,γ)) γ6=α 1 k(α,u+v)−k(α,u)−vkx(α,u) k(α,u+v)−k(α,v)−ukx(α,v) P where 2 ( v2 − u2 ) = − s hs(u)ks(v). Proof. By definition [∆0, xα∨ ] = yα∨ , [∆0, yα] = 0 and [∆0, tα] = 0. Since [∆0, tα] = 0 and 234 P ∨ 2 [xα∨ , yα∨ ] = − γ∈Φ+ (α , γ) tγ, we have: n [∆0, (ad xα∨ ) (tα)] n−1 n−1 X s n−1−s X s n−1−s = (ad xα∨ ) (ad[∆0, xα∨ ])(ad xα∨ ) (tα) = (ad xα∨ ) (ad yα∨ )(ad xα∨ ) (tα) s=0 s=0 n−1 s−1 n−1 X X i n−2−i = n ad yα∨ (ad xα∨ ) (tα) + (ad xα∨ ) ad[xα∨ , yα∨ ](ad xα∨ ) (tα) s=1 i=0 n−1 s−1 n−1 X X X ∨ 2 i n−2−i = n ad yα∨ (ad xα∨ ) (tα) − (α , γ) (ad xα∨ ) ad tγ(ad xα∨ ) (tα) s=1 i=0 γ∈Φ+ n−1 s−1 n−1 X X i n−2−i = n ad yα∨ (ad xα∨ ) (tα) − 4(ad xα∨ ) ad tα(ad xα∨ ) (tα) s=1 i=0 n−1 s−1 X X X ∨ 2 i n−2−i − (α , γ) (ad xα∨ ) ad tγ(ad xα∨ ) (tα) s=1 i=0 γ6=α The third equality follows from: s n−1−s (ad xα∨ ) (ad yα∨ )(ad xα∨ ) (tα) s−1 n−1−s s−1 n−s = (ad xα∨ ) (ad[xα∨ , yα∨ ])(ad xα∨ ) (tα) + (ad xα∨ ) (ad yα∨ )(ad xα∨ ) (tα) The conclusion follows from the following identity which can be showed by induction: n−1 s−1 X X an − bn − nbn−1(a − b) aibn−2−i = . (a − b)2 s=1 i=0 Now plug all the above lemmas into the formula of A ∧ A, we get: X X xα xα X X −2πiA ∧ A = [F α(ad ),Gα(ad )]dτ ∧ dα + ( H(α, γ)[t , t ])dτ ∧ dα, s 2 s 2 γ α α∈Φ+ s α∈Φ+ γ6=α P α α where s Fs(u) Gs (v) = −L(α, u, v), and 1 ϕ(u) − ϕ(v) 1 L(z, u, v) = + k(z, u + v)(ϕ(u) − ϕ(v)) 2 u + v 2 1 + (g(z, u)k(z, v) − k(z, u)g(z, v)) 2 1 k(z, u + v) − k(z, u) − vg(z, u) k(z, u + v) − k(z, v) − vg(z, v) − ( − ). 2 v2 u2 235 As showed in [10] that, L(z, u, v) = 0. We have x x ∨ k(α, ad α∨ ) − k(α, ad ω(α ,γ) ) ∨ xα∨ 2 −2 γ(α ) g(α, ad 2 ) H(α, γ) = − x + x ω(γ∨,α) 2 2 ω(γ∨,α) (ad −2 ) ad −2 xγ∨ xω(γ∨,α) g(γ, ad ) − g(γ, ad ) x ∨ x ∨ + 2 −2 − g(γ, ad ω(γ ,α) )k(α, ad ω(α ,γ) ). xω(α∨,γ) −2 −2 ad −2 Proposition 6.11.7. The following identity holds: H(α, γ) − H(α, α + γ) − H(γ + α, γ) + H(γ + α, α) ≡ 0 Proof. By calculation, we have: H(α, γ) − H(α, α + γ) − H(γ + α, γ) + H(γ + α, α) xω(α∨,α+γ) xω(α∨,γ) xω(α∨,α+γ ) k(α, ad −2 ) − k(α, ad −2 ) g(α, ad −2 ) = − x − x ω(γ∨,α) 2 ω(γ∨,α) (ad −2 ) ad −2 xω((α+γ)∨,α) xω((α+γ)∨,γ) xω((α+γ)∨,α) k(α + γ, ad −2 ) − k(α + γ, ad −2 ) g(α + γ, ad −2 ) + x + x ω(α∨,α+γ) 2 ω(α∨,α+γ) (ad −2 ) ad −2 x x g(γ, ad ω(γ∨,α+γ) ) − g(γ, ad ω(γ∨,α) ) + −2 −2 xω(α∨,γ) ad −2 x ∨ x ∨ x ∨ x ∨ − g(γ, ad ω(γ ,α) )k(α, ad ω(α ,γ) ) + g(α + γ, ad ω((α+γ) ,α) )k(α, ad ω(α ,α+γ) ) −2 −2 −2 −2 x ∨ x ∨ x ∨ x ∨ + g(γ, ad ω(γ ,α+γ) )k(α + γ, ad ω((α+γ) ,γ) ) − g(α, ad ω(α ,α+γ) )k(α + γ, ad ω((α+γ) ,α) ). −2 −2 −2 −2 Now use the fact (showed in [10]) that k(z, u + v) − k(z, u) − vg(z, u) k(z0, u + v) − k(z0, v) − ug(z0, v) H(z, z0, u, v) = − v2 u2 g(z0 − z, −u) − g(z0 − z, v) + − g(−z0, −v)k(−z, −u) + g(−z, −u)k(−z0, −v) u + v − g(z − z0, −v)k(z, u + v) + g(z0 − z, −u)k(z0, u + v) ≡ 0 Note that g(z, x) = g(−z, −x). Thus, the conclusion holds. Now similar as the argument in §6.4, it suffices to check the flatness for Φ is a rank 2 root systems. We will check the flatness case by case. 236 6.11.3 Case A2 (α , α + α ) 1 1 2 Use the graph (α1, α2) → (α1 + α2, α1) , meaning write (α1 + α2, α2) H(α1, α2) = H(α1, α1 + α2) − H(α1 + α2, α1) − H(α1 + α2, α2) according to Proposition 6.11.7. P P Then, in the expression of α∈Φ+ ( γ6=α H(α, γ)[tα, tγ])dτ ∧dα, the coefficient of dτ ∧dα1 becomes: H(α1, α1 + α2)[tα1 , tα1+α2 + tα2 ] + H(α1 + α2, α1)[tα1+α2 + tα2 , tα1 ] + H(α1 + α2, α2)[tα1+α2 + tα1 , tα2 ] = 0. (α , α + α ) 2 1 2 Use the graph (α2, α1) → (α1 + α2, α2) , then, the coefficient of dτ ∧ dα2 becomes: (α1 + α2, α1) H(α2, α1 + α2)[tα2 , tα1+α2 + tα1 ] + H(α1 + α2, α1)[tα1+α2 + tα2 , tα1 ] + H(α1 + α2, α2)[tα1+α2 + tα1 , tα2 ] = 0. Thus, the connection is flat in the case of A2. 237 6.11.4 Case B2 (α , α + α ) (α + 2α , α + α ) 1 1 2 1 2 1 2 Use the graph (α1, α2) → (α1 + α2, α1) , and (α1 + 2α2, α2) → (α1 + α2, α1 + 2α2) , (α1 + α2, α2) (α1 + α2, α2) then the coefficient of dτ ∧ dα1 is: H(α1, α1 + α2)[tα1 , tα1+α2 + tα2 ] + H(α1 + α2, α1)[tα1+α2 + tα2 , tα1 ] +H(α1 + α2, α1 + 2α2)[tα1+α2 + tα2 , tα1+2α2 ] + H(α1 + α2, α2)[tα1+α2 + tα1 + tα1+2α2 , tα2 ] +H(α1 + 2α2, α1 + α2)[tα1+2α2 , tα1+α2 + tα2 ] = 0. (α , α + α ) (α + 2α , α ) 2 1 2 1 2 2 Use the graph (α2, α1) → (α1 + α2, α2) ,(α2, α1 + 2α2) → (α1 + 2α2, α1 + α2) , (α1 + α2, α1) (α2, α1 + α2) (α + 2α , α + α ) 1 2 1 2 and (α1 + α2, α1 + 2α2) → (α1 + 2α2, α2) , then, the coefficient of dτ ∧ dα2 is: (α1 + α2, α2) H(α2, α1 + α2)[tα2 , tα1+α2 + tα1 + tα1+2α2 ] + H(α1 + 2α2, α1 + α2)[tα1+2α2 , tα1+α2 + tα2 ] +2H(α1 + 2α2, α2)[tα1+2α2 , tα2 + tα1+α2 ] + H(α1 + α2, α1)[tα1+α2 + tα2 , tα1 ] +H(α1 + α2, α2)[tα1+2α2 + tα1+α2 + tα1 , tα2 ] = 0. 6.11.5 Case G2 Use the following graphs in order: (α , 2α + 3α ) (α , α + α ) 1 1 2 1 1 2 (α1, α1 + 3α2) → (2α1 + 3α2, α1) ,(α1, α2) → (α1 + α2, α1) , (2α1 + 3α2, α1 + 3α2) (α1 + α2, α2) 238 (α + 3α , 2α + 3α ) (α + 3α , α + 2α ) 1 2 1 2 1 2 1 2 (α1 + 3α2, α1) → (2α1 + 3α2, α1) ,(α1 + 3α2, α2) → (α1 + 2α2, α2) , (2α1 + 3α2, α1 + 3α2) (α1 + 2α2, α1 + 3α2) (α + α , α + 2α ) (α + 2α , α + α ) 1 2 1 2 1 2 1 2 (α1+α2, 2α1+3α2) → (2α1 + 3α2, α1 + 2α2) ,(α1+2α2, 2α1+3α2) → (2α1 + 3α2, α1 + 2α2) , (2α1 + 3α2, α1 + α2) (2α1 + 3α2, α1 + α2) (α + α , α + 2α ) 1 2 1 2 (α1 + α2, α2) → (α1 + 2α2, α2) . (α1 + 2α2, α1 + α2) Then the coefficient of dτ ∧ dα1 is H(α1, α1 + α2)[tα1 , tα1+α2 + tα2 ] + H(α1, 2α1 + 3α2)[tα1 , t2α1+3α2 + tα1+3α2 ] +H(α1 + α2, α1)[tα1+α2 + tα2 , tα1 ] + H(α1 + α2, α1 + 2α2)[tα1+α2 , tα1+2α2 + t2α1+3α2 + tα1 + tα2 ] +H(α1 + 2α2, α2)[tα1+2α2 + tα1+α2 + tα1 + tα1+3α2 , tα2 ] + H(α1 + 2α2, α1 + α2)[tα1+2α2 + t2α1+3α2 + tα1 + tα2 , tα1+α2 ] +H(α1 + 2α2, α1 + 3α2)[tα1+2α2 + tα2 , tα1+3α2 ] + H(α1 + 3α2, α1 + 2α2)[tα1+3α2 , tα1+2α2 + tα2 ] +H(α1 + 3α2, 2α1 + 3α2)[tα1+3α2 , t2α1+3α2 + tα1 ] + 2H(2α1 + 3α2, α1)[tα1+3α2 + t2α1+3α2 , tα1 ] +H(2α1 + 3α2, α1 + α2)[t2α1+3α2 , tα1+α2 + tα1+2α2 ] + H(2α1 + 3α2, α1 + 2α2)[t2α1+3α2 , tα1+α2 + tα1+2α2 ] +2H(2α1 + 3α2, α1 + 3α2)[tα1 + t2α1+3α2 , tα1+3α2 ] = 0 Use the following graphs in order: 3(α + 3α , 2α + 3α ) (α , α + α ) 1 2 1 2 2 1 2 3(α1 + 3α2, α1) → 3(α1 + 3α2, α1) ,(α2, α1) → (α1 + α2, α2) , 3(α1 + 3α2, α1 + 3α2) (α1 + α2, α1) 239 1(2α + 3α , α + 2α ) 2(2α + 3α , α + 2α ) 1 2 1 2 1 2 1 2 1(2α1+3α2, α1+α2) → 1(α1 + α2, α1 + 2α2) ,2(2α1+3α2, α1+α2) → 2(α1 + 2α2, α1 + α2) , 1(α1 + α2, 2α1 + 3α2) 2(α1 + 2α2, 2α1 + 3α2) 1(α + 2α , α + α ) 1(α + 2α , α + α ) 1 2 1 2 1 2 1 2 1(α1 + 2α2, α2) → 1(α1 + α2, α2) ,1(α1 + 2α2, α2) → 1(α2, α1 + α2) , 1(α1 + α2, α1 + 2α2) 1(α2, α1 + 2α2) (α + 3α , α + 2α ) (α + α , α + 2α ) 1 2 1 2 1 2 1 2 (α1 + 3α2, α2) → (α1 + 2α2, α2) ,(α1 + α2, α2) → (α1 + 2α2, α2) , (α1 + 2α2, α1 + 3α2) (α1 + 2α2, α1 + α2) (α , α + 2α ) (α + 2α , α + α ) 2 1 2 1 2 1 2 (α2, α1 +3α2) → (α1 + 2α2, α1 + 3α2) ,(α2, α1 +α2) → (α1 + 2α2, α2) , then, (α1 + 2α2, α2) (α2, α1 + 2α2) the coefficient of dτ ∧ dα2 is: H(α2, α1 + 2α2)[tα2 , tα1+2α2 − tα1+2α2 + tα1+3α2 − tα1+3α2 ] + H(α1 + α2, α1)[tα1+α2 + tα2 , tα1 ] +H(α1 + α2, α1 + 2α2)[tα1+α2 + t2α1+3α2 + tα2 + tα1+3α2 , tα1+2α2 ] + H(α1 + α2, 2α1 + 3α2)[tα1+α2 , t2α1+3α2 − t2α1+3α2 ] +2H(α1 + 2α2, α1 + α2)[tα1+2α2 , tα1+α2 + t2α1+3α2 + tα2 + tα1+3α2 ] + 2H(α1 + 2α2, α1 + 3α2)[tα1+2α2 + tα2 , tα1+3α2 ] +2H(α1 + 2α2, 2α1 + 3α2)[tα1+2α2 + tα1+α2 , t2α1+3α2 ] +3H(α1 + 3α2, α1 + 2α2)[tα1+3α2 , tα1+2α2 + tα2 ] + 3H(α1 + 3α2, 2α1 + 3α2)[tα1+3α2 , t2α1+3α2 + tα1 ] +3H(2α1 + 3α2, α1)[t2α1+3α2 + tα1+3α2 , tα1 ] + 3H(2α1 + 3α2, α1 + 2α2)[t2α1+3α2 , tα1+2α2 + tα1+α2 ] +3H(2α1 + 3α2, α1 + 3α2)[t2α1+3α2 + tα1 , tα1+3α2 ] = 0. Thus, the connection is flat in the case of G2. 240 6.12 Simply connected type and adjoint type Let β ∈ Φ be a root, and sβ ∈ W be the corresponding element in the Weyl group. The action of W on Q∨ is given by s : Q∨ → Q∨ , sending y 7→ y − y(β)β∨. Q β Q Q The induced action of W on X = Q∨ ⊗ E is X ∨ X ∨ X ∨ ∨ ziαi 7→ ziαi − ziβ(αi )β , i i i for sβ ∈ W, zi ∈ C. Thus, X ∨ X ∨ sβ( ziαi ) = ziαi i i X ∨ ∨ ∨ ⇐⇒ ziβ(αi )β ∈ Q ⊗ Λτ i X ∨ ⇐⇒β( ziαi ) ∈ Λτ i X ∨ ⇐⇒χβ( ziαi ) = 0 i From the above, we know, the hyperplanes Hβ consisting of sβ fixed points. Proposition 6.12.1. The Weyl group action on the regular points of the simpliy-connected torus T = h/(Q∨ ⊕ τQ∨) = Q∨ ⊗ E is free. Proof. To show the action is free, we need to prove for every regular point z of T = h/(Q∨ ⊕ τQ∨), the stabilizer of z in W {w ∈ W | wz = z} consists of the identity. 241 We have h = E ⊕ τE, where E is the real vector space. Letz ˜ = x + yτ ∈ h be a lifting ∨ ∨ of z ∈ T , then w(z) = z implies that w(˜z) =z ˜ + t, for some t = t1 + τt2 ∈ Q ⊕ τQ , which is equivalent to w(x) = x + t1, and w(y) = y + t2. Since z is regular, thus, either x is regular or y is regular. We can assume x is regular, which by definition [ x ∈ E \{ Hα,k}, where Hα,k = {λ ∈ E | (α, λ) = k.} α∈Φ,k∈Z ∨ Let Wa := W o Q be the affine Weyl group, then the element wa = (w, −t1) can be ∨ viewed as an element in Wa = W o Q . The fact w(x) = x + t1 implies that wa(x) = x. Let A be the set of alcoves in E. We know that the action Wa on A is simply transitive. See [43], chapter 4, Theorem 4.5, page 93. Let x ∈ A, for some alcove A, then, wa(x) = x ⇒ wa(A) ∩ A 6= ∅ ⇒ wa(A) = A ⇒ wa = id ⇒ w = id Thus, the action of W is free on T reg. The following example shows for adjoint torus, the action of W on Xreg is not free. Example 6.12.2. Consider A1 case, let α be the unique positive root. For adjoint type, ∨ ∼ ∨ α∨ X = P ⊗ E = E, where P = Z 2 be the coweight lattice. In this case α∨ α∨ H = {z | α(z ) ∈ Λ } α 2 2 τ α∨ = {z | z ∈ Λ } 2 τ ∼ Thus, X \ Hα = E\ pt. α∨ ∨ ∨ 1 W ∼ For sα fixed point z 2 , we have zα ∈ P ⊗ Λτ , which implies z ∈ 2 Λτ . Thus, X \ X = E\ 4pts. So if we consider adjoint type, the action of W on Xreg is not free. 242 6.13 The Homomorphism from Aell to rational Chered- nik algebras 6.13.1 The rational Cherednik algebra ∗ Let W be a Weyl group, for any reflection s ∈ W , fix αs ∈ h , such that s(αs) = −αs. Write S for the collection of linear functions {±αs | s is reflections in W }. Let t ∈ C, and c : S → C, s 7→ cs be a W -invariant function. ∗ Definition 6.13.1. The algebra H~,c is the quotient of the algebra CW nT (h⊕h )[~] (where T denotes the tensor algebra) by the ideal generated by the relations 0 0 X ∨ [x, x ] = 0; [y, y ] = 0; [y, x] = ~hy, xi − cshαs, yihαs , xis, s∈S where x, x0 ∈ h∗, y, y0 ∈ h. We recall some facts of the grading element of the rational Cherednik algebras. Let yi ∗ be the basis of h, and xi be the corresponding dual basis of h . Let X 1 X h := x y + dim h − c s, i i 2 s i s∈S and 1 X 1 X E := − x2, F := y2. 2 i 2 i i i Proposition 6.13.2 ([25], Proposition 3.18, 3.19, Page 17-18.). We have (i) For any x ∈ h∗, y ∈ h, [h, x] = x, [h, y] = −y; P xiyi+yixi (ii) h = i 2 ; (iii) h, E, F form a sl2–triple. 243 6.13.2 The homomorphism from Aell to the rational Cherednik al- gebra ∨ Pn ∨ Pn Letα ˜ be the highest root of g. Writeα ˜ = i=1 giαi , and let g := 1 + i=1 gi. In case all roots have the same length, we have g is the Coxeter number. Proposition 6.13.3. For any a, b ∈ C, we have a homomorphism of Lie algebras ξa,b : Aell → H~,c, defined as follows x(v) 7→ aπ(v), y(u) 7→ bu 2c t 7→ ab ~ − s s , γ g (γ|γ) γ where π : h → h∗ be the isomorphism induced by the non-degenerate bilinear form (·|·) on h. and sγ is the reflection which sending γ to −γ, and fixes the hyperplane ker(γ). For simplicity and without lost of generality we can assume a = b = 1. Note that in Aell, the generators x(v), y(u) are labeled by u, v ∈ h. We have the following relation of Aell: [y(u), x(v)] = Σγ∈Φ+ hv, γihu, γitγ. We show that the map ξ preserve the above relation of Aell. Lemma 6.13.4. The following equality holds: 1 X (·|·) = h·, γih·, γi. g γ∈Φ+ Proof. We have [y(u), x(v)] 7→[u, π(v)] X ∨ =~hu, π(v)i − cshαs, uihαs , π(v)is s∈S X ∨ =~(u|v) − cshγ, uihγ , π(v)isγ γ∈Φ+ 244 X X 2cs hv, γihu, γit 7→ hv, γihu, γi(k − s ) γ ~ (γ|γ) γ γ∈Φ+ γ∈Φ+ X X ∨ = hv, γihu, γik~ − cshπ(v), γ ihu, γisγ γ∈Φ+ γ∈Φ+ P Now (·|·): h × h → C, and γ∈Φ+ h·, γih·, γi : h × h → C are two symmetric bilinear forms on h, both are positive definite, and invariant under the Weyl group W . Thus, X (·|·) = k h·, γih·, γi, γ∈Φ+ for some constant k ∈ C. 1 It remains to show the constant k is equal to g , which follows immediately from the following lemma. [See [57], Lemma 1.2.] P ∨ ∨ Lemma 6.13.5. γ∈Φ+ hα˜ , γihα˜ , γi = 2g. Remark 6.13.6. In [57], the summation is over the set of γ ∈ Φ, thus the result is X hα˜∨, γihα˜∨, γi = 4g. γ∈Φ We take the summation over γ ∈ Φ+, so the result is half of 4g. ∨ ∨ 4 2 Since (˜α |α˜ ) = (˜α|α˜) . We can pick k := g·(˜α|α˜) . We normalize the inner product such that (˜α|α˜) = 2, then the constant k can be chosen 1 1 to be k := g . Note, in the special case of type An, we have k = n+1 , which coincide with the constant in [10]. 6.13.3 The extension of the homomorphism Proposition 6.13.7. Let yi be an orthogonal basis of h, and let xi be the corresponding dual ∗ basis of h . Then, The homomorphism ξa,b can be extended to the homomorphism ˜ ξa,b : U(Aell o d) o W → H~=1,c 245 by the following formulas w 7→ w, −1 −1 d 7→ h,X 7→ ab E, ∆0 7→ ba F, 2 1 2m−1 −1 X 4mcα 2m δ 7→ − a b (x ∨ ) . 2m 2 (α, α) α α∈Φ+ Proof. For simplicity, we assume that a = b = 1. We first show the above homomorphism preserve the relations of d. From Proposition 6.13.2, we know the triple h, E, F form a ˜ sl2-triple. Thus, the homomorphism ξ preserve the relations of the triple d, X, ∆0. ˜ ˜ It’s obvious that ξ preserve the relation [δ2m,X] = 0. The fact that [h, x] = x implies ξ preserves [d, δ2m] = 2mδ2m. ˜ 2m+1 Now we check ξ preserve the relation (ad ∆0) (δ2m) = 0. We have the following relation: [F, xα] = 2~yα, for any j. Since X 2 [F, xj] = [ yi , xj] i X = (yi[yi, xj] + [yi, xj]yi) i X X ∨ = 2~yj − cs(yi, αs)(xj, αs )(yis + syi) i s X X ∨ = 2~yj − cshxi, αsi(xj, αs )(yis + syi) by (yi, αs) = hxi, αsi i s ! X ∨ X = 2~yj − cs(xj, αs ) hxi, αsi(yis + syi) α∈Φ+ i X ∨ = 2~yj − cs(xj, αs )(αsα + sαα) α∈Φ+ = 2~yj by sαα = −αsα 246 Using above relation we obtain: 2m+1 2m (ad F) xα∨ 2m 2m =(ad F) [F, xα∨ ] 2m 2m−1 2m−1 =2~(ad F) xα∨ yα∨ + yα∨ xα∨ 2m 2m−1 2m 2m−1 =2~ (ad F) xα∨ yα∨ + yα∨ (ad F) xα∨ =0 by induction. ˜ Now it remains to show the that the derivation d → Der(Aell) ζ 7→ ζ corresponding to the ˜ ˜ operators ad(ξa,b(ζ)), for ξa,b(ζ) ∈ H ,c. Proposition 6.13.2 together with [h, w] = 0, for any ~ ˜ d(x(u)) = x(u); ˜ w ∈ W , implies that ξa,b(d) preserves the relations d˜(y(u)) = −y(u); We check that ˜ d(tα) = 0. ∗ [h, w] = 0, for any w ∈ W . Let ui be another basis of h , and vi be the corresponding dual basis of h. We have ! X X X xiyi = hxi, vkiuk yi i i k ! X X X = uk hxi, vkiyi = ukvk. k i k P P ∗ Similarly, i yixi = k vkuk. For any w ∈ W , w(xi) are a basis of h , and w(yi) are the corresponding dual basis of h. Thus, h = w(h). 247 Check [F, w] = 0, for any w ∈ W . X 2 X [yi , w] = ([yi, w]yi + yi[yi, w]) i i X = (yi − w(yi))wyi + yi(yi − w(yi))w) i X = (yi − w(yi))w(yi) + yi(yi − w(yi)) w i X = yiyi − w(yi)w(yi) w i = 0 {yi} are orthogonal basis. Since E and F are symmetric. The relations involving E can be proved using the same method. ˜ We prove ξa,b preserves relation x(α∨) δ˜ (t ) = [t , (ad )2m(t )] 2m α α 2 α x(α∨) 2m Check [sα, (ad 2 ) (sα)] = 0. x(α∨) [s , (ad )2m(s )] α 2 α x(α∨) =[s , x(α∨)2ms ] by [ , s ] = x(α∨)s α α 2 α α ∨ 2m ∨ 2m =sαx(α ) sα − x(α ) ∨ 2 =0 by [sαx(α ) ] = 0 P 2m Check [ α∈Φ+ (xα) , w] = 0, for w ∈ W . Note that w permute the root system Φ and preserve the killing form, thus we have X 2m 1 X 2m (x ∨ ) = (x ∨ ) α 2 α α∈Φ+ α∈Φ 1 X 2m X 2m = (x ∨ ) = w( (x ∨ ) ) 2 wα α α∈Φ α∈Φ+ ˜ Finally, we show that ξa,b preserves relation 1 X X x(α∨) x(α∨) δ˜ (y(u)) = α(u) [(ad )p(t ), (ad − )q(t )] (6.13.1) 2m 2 2 α 2 α α p+q=2m−1 248 On one hand, we have X x(α∨) x(α∨) [(ad )p(s ), (ad − )q(s )] 2 α 2 α p+q=2m−1 X x(α∨) = (−1)q[x(α∨)ps , x(α∨)qs ] by [ , s ] = x(α∨)s α α 2 α α p+q=2m−1 X q ∨ p ∨ q ∨ q ∨ p = (−1) x(α ) sαx(α ) sα − x(α )) sαx(α ) sα p+q=2m−1 X q q p ∨ 2m−1 ∨ ∨ = (−1) (−1) − (−1) x(α ) by sαx(α ) = −x(α )sα p+q=2m−1 =4mx(α∨)2m−1. Thus, the right hand side of (6.13.1) is 2 2 1 X ∨ 2m−1 2cα X cα ∨ 2m−1 α(u)4mx(α ) = 4m α (u)x ∨ 2 (α, α) (α, α) α α α On the other hand, X 2m X 2m−1 2m−1 [ (xα∨ ) , y(u)] = [xα∨ , y(u)]xα∨ + xα∨ [xα∨ , y(u)] α∈Φ+ α∈Φ+ X ∨ X ∨ ∨ 2m−1 = −~hu, α i + cγhγ , uihγ, α isγ xα∨ α∈Φ+ γ∈Φ+ 2m−1 ∨ X ∨ ∨ + xα∨ −~hu, α i + cγhγ , uihγ, α isγ γ∈Φ+ X ∨ 2m−1 = −2~α (u)xα∨ . α∈Φ+ The last equality follows from the following calculation. X ∨ 2m−1 2m−1 hγ, α i sγxα∨ + xα∨ sγ α∈Φ X ∨ 2m−1 ∨ 2m−1 = hγ, α i(sγxα∨ ) sγ + hγ, sγ(α )i(sγxα∨ ) sγ α∈Φ X ∨ ∨ ∨ 2m−1 = (hγ, 2α − hγ, α iγ i)(sγxα∨ ) sγ α∈Φ =0. 249 Thus, the left hand side of (6.13.1) is also 2 X cα ∨ 2m−1 4m α (u)x ∨ (α, α) α α 6.14 The Homomorphism from Aell to the deformed double current algebras 6.14.1 The deformed double current algebras The deformed double current algebras are introduced by Guay in [34][35], which is a defor- mation of the universal central extension of U(g[u, v]), for a semisimple Lie algebra g. The double loop presentations of the deformed double current algebras are established in [34] for g = sln, n ≥ 4 and in [36] for any simple Lie algebra g. We briefly recall the results here. Let g be a finite–dimensional, simple Lie algebra over C, in this section, we assume that g 6= sl2, sl3. Let (·, ·) be the Killing form on g and let ± + − Xi ,Hi, 1 ≤ i ≤ N be the Chevalley generators of g normalized so that (Xi ,Xi ) = 1 + − ± and [Xi ,Xi ] = Hi. For each positive root α, we choose generators Xα of g±α such that (X+,X−) = 1 and X± = X±. If α > 0, set X = X+; if α < 0, set X = X− . α α αi i α α α −α Theorem 6.14.1. [34, 36] The deformed double current algebra Dλ(g) is generated by ele- ments X,K(X),Q(X),P (X),X ∈ g, such that 1. K(X),X ∈ g generate a subalgebra which is an image of g⊗C C[u] with X ⊗u 7→ K(X); 2. Q(X),X ∈ g generate a subalgebra which is an image of g⊗C C[v] with X ⊗v 7→ Q(X); 3. P (X) is linear in X, and for any X,X0 ∈ g, [P (X),X0] = P [X,X0]. 250 and the following relations hold for all root vectors Xβ1 ,Xβ2 ∈ g with β1 6= −β2: (β1, β2) λ X [K(X ),Q(X )] = P ([X ,X ]) − λ S(X ,X ) + S([X ,X ], [X ,X ]), β1 β2 β1 β2 4 β1 β2 4 β1 α −α β2 α∈∆ (6.14.1) where S(a1, a2) = a1a2 + a2a1 ∈ U(g). For any root β of g, set 1 X Z(β) := [K(H ),Q(H )] − S([H ,X ], [X ,H ]) ∈ D (g). β β 4 β α −α β λ α∈∆ Theorem 6.14.2. [36] In the deformed double current algebra Dλ(g), we have 1. Z(β) = Z(γ), for any two roots β, γ of g. 2. The element Z(β) is central in Dλ(g). Let QDλ(g) be the quotient algebra of the deformed double current algebras Dλ(g) modulo the central element Z(β). The presentation of the algebra QDλ(g) has a nice form, which is given as follows. Proposition 6.14.3. [36] The quotient algebra QDλ(g) is generated by elements X,K(X), Q(X),P (X), X ∈ g, such that relations (1), (2), (3) hold as in Theorem 6.14.1, and the relations (6.14.1) hold for all vectors Xβ1 ,Xβ2 ∈ g, with β1, β2 ∈ Φ ∪ {0}. In particular, for any h, h0 ∈ g, we have: λ X λ X [K(h),Q(h0)] = S([h, X ], [X , h]) = (h, α)(h0, α)κ , 4 α −α 2 α α∈Φ α∈Φ+ where κα = XαX−α + X−αXα is the truncated Casimir operator. 251 6.14.2 The homomorphism to the deformed double current alge- bras a a 11 12 Proposition 6.14.4. There is a right action of SL2(C) on D. For A = a21 a22 ∈ SL2(C), the action is given by z 7→ z, K(z) 7→ a11K(z) + a12Q(z),Q(z) 7→ a22Q(z) + a21K(z), P (z) 7→ (a11a22 + a12a21)P (z) + a11a21[K(y),K(w)] + a12a22[Q(y),Q(w)], where z = [y, w], and for any z ∈ g. In particular, there is a well defined automorphism of D by K(z) 7→ −Q(z); Q(z) 7→ K(z); P (z) 7→ −P (z); and z 7→ z, for z ∈ g, which is of order 4. Remark 6.14.5. We can make the right action of SL2(C) to a left action of SL2(C) by T T defining Az˙ := zA , where A is the transpose of A. Thus, there is a left action of SL2(C) a a 11 12 on D. For A = ∈ SL2(C), the action is given by a21 a22 z 7→ z, K(z) 7→ a11K(z) + a21Q(z),Q(z) 7→ a22Q(z) + a12K(z), P (z) 7→ (a11a22 + a21a12)P (z) + a11a12[K(y),K(w)] + a21a22[Q(y),Q(w)], where z = [y, w], and for any z ∈ g. Proof. We first check that under the action of A preserves the defining relations of D(g). We only check it preserves relation (6.14.1), other relations are obvious. We set 1 1 X C(X ,X ) := − (β , β )S(X ,X ) + S([X ,X ], [X ,X ]), β1 β2 4 1 2 β1 β2 4 β1 α −α β2 α∈∆ The property S(a1, a2) = S(a2, a1) implies that C(Xβ1 ,Xβ2 ) = C(Xβ2 ,Xβ1 ). The relation (6.14.1) can be rewritten as [K(x),Q(y)] = P ([x, y]) + C(x, y). 252 Now under the action of A ∈ SL2(C), [K(x),Q(y)] 7→[a11K(x) + a12Q(x), a22Q(y) + a21K(y)] = a11a22[K(x),Q(y)] + a12a21[Q(x),K(y)] + a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)] = a11a22(P ([x, y]) + C(x, y)) − a12a21(P ([y, x]) + C(y, x)) + a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)] = (a11a22 + a12a21)P ([x, y]) + a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)] + C(x, y). and by definition P ([x, y]) + C(x, y) 7→ (a11a22 + a12a21)P ([x, y]) + a11a21[K(x),K(y)] + a12a22[Q(x),Q(y)] + C(x, y). Thus, the action of A preserves the relation (6.14.1). We then check it defines a right action of SL2. It’s obvious that z·Id = z, for any z ∈ D(g), where Id is the 2 × 2–identity matrix. For any A, B ∈ SL2(C), and for any z ∈ U(g), it’s a direct calculation to show (z·A)·B = z·(AB). We show the less obvious case when z = P ([x, y]) as follows. a a b b 11 12 11 12 Set A = and B = . It’s a direct computation to show that a21 a22 b21 b22 (P ([x, y])·A)·B = P ([x, y])·(AB). Corollary 6.14.6. The action of SL2(C) on D induces a Lie algebra sl2(C) action on D. x11 x12 sl For X = ∈ 2(C), the action is giving by: zX = 0, x21 x22 K(z)X = x11K(z) + x21Q(z), Q(z)X = x22Q(z) + x12K(z), P (z)X = x21[K(y),K(w)] + x21[Q(y),Q(w)], where z = [y, w]. 253 1 0 g In particular, h = , for any z ∈ , then: hz = 0, hP (z) = 0, hK(z) = K(z), 0 −1 and hQ(z) = −Q(z). Proposition 6.14.7. There is a map from Aell → QDλ(g), giving by x(u) 7→ Q(u), y(v) 7→ λ + − − + K(v), and tα 7→ 2 κα, where κα := Xα Xα + Xα Xα is the truncated Casimir operator. Proof. It follows from the explicit relations of QDλ(g) in Proposition 6.14.3. 6.15 The extension of the elliptic Casimir connection 6.15.1 Recall in Section 6.9 we introduced the derivation d of generalized holonomy Lie algebra Aell. The generators of d are denoted by ∆0, d, X, and δ2m(m ≥ 1). In this section, we use the following action of sl2(C) on D(g). For any z ∈ g, 1. hz = 0, hK(z) = −K(z) and hQ(z) = Q(z). 2. ez = 0, eK(z) = Q(z), eQ(z) = 0, 3. fz = 0, fK(z) = 0, fQ(z) = K(z). Conjecture 6.15.1. The map Aell → D(g) can be extended to the following map Aell o d → D(g) o sl2, which is given by d 7→ h, X 7→ e, ∆0 7→ f, and 2m X 2m X δ 7→ (−1)q (h ⊗ up)(hi ⊗ u2m−p), 2m p i p=0 i i where hi, h are dual basis of h and h ⊗ u := Q(u) ∈ g[u]. 254 6.15.2 In this subsection, we show that for any X ∈ g, the necessary condition is 2m " # X 2m X δ (X) = (−1)q (h ⊗ up)(hi ⊗ u2m−p),X . 2m p i p=0 i In order to define the action of d on D(g) which commutes with the map Aell → D(g), we must have λ Q(α∨) δ S(X+,X−)) = [S(X+,X−), (ad )2m(S(X+,X−))]. 2m α α 2 α α 2 α α δ2m is a derivation. Then + − + − + − δ2mS(Xα ,Xα )) =S(δ2m(Xα ),Xα )) + S(Xα , δ2m(Xα ))). On the other hand, we have Q(α∨) [S(X+,X−), (ad )2m(S(X+,X−))] α α 2 α α h Q(α∨) i h Q(α∨) i =S X+, (ad )2m(S(X+,X−)) ,X− + S X+, X−, (ad )2m(S(X+,X−)) α 2 α α α α α 2 α α Thus, λh Q(α∨) i δ (X+) = X+, (ad )2m(S(X+,X−)) 2m α 2 α 2 α α λh X 2m i = X+, (−1)q S(X+ ⊗ up,X− ⊗ uq) 2 α p α α p+q=2m λ X 2m = (−1)q S(X+ ⊗ up, [X+,X−] ⊗ uq) 2 p α α α p+q=2m λ X 2m = (−1)q S(X+ ⊗ up,H ⊗ uq) 2 p α α p+q=2m 255 Similarly, λh Q(α∨) i δ (X−) = X−, (ad )2m(S(X+,X−)) 2m α 2 α 2 α α λh X 2m i = X−, (−1)q S(X+ ⊗ up,X− ⊗ uq) 2 α p α α p+q=2m λ X 2m = (−1)q S([X−,X+] ⊗ up,X− ⊗ uq) 2 p α α α p+q=2m λ X 2m = (−1)q S(−H ⊗ up,X− ⊗ uq) 2 p α α p+q=2m Now + − δ2m(Hα) =δ2m([Xα ,Xα ]) + − + − =[δ2mXα ,Xα ] + [Xα , δ2mXα ] λ h Q(α∨) i λ h Q(α∨) i = [ X+, (ad )2m(S(X+,X−)) ,X−] + [X+, X−, (ad )2m(S(X+,X−)) ] 2 α 2 α α α 2 α α 2 α α λh Q(α∨) i = H+, (ad )2m(S(X+,X−)) 2 α 2 α α λ Q(α∨) = (ad )2m([H ,S(X+,X−)]) 2 2 α α α =0 Thus, δ2m(h) = 0, for any h ∈ h. 6.15.3 In this subsection, we show that for any X ∈ g, we have: 2m " # X 2m X δ (Q(X)) = (−1)q (h ⊗ up)(hi ⊗ u2m−p),Q(X) . 2m p i p=0 i 256 + It’s obvious that δ2m(Q(h)) = 0, for h ∈ h. Thus, for Xα ∈ g, We can define Q(α∨) δ (Q(X+)) := [ , δ (X )] 2m α 2 2m α λ Q(α∨) X 2m = [ , (−1)q S(X+ ⊗ up,H ⊗ uq)] 2 2 p α α p+q=2m λ X 2m = (−1)q S(X+ ⊗ up+1,H ⊗ uq) 2 p α α p+q=2m Similarly, one has: λ X 2m δ (Q(X+)) = (−1)q S(−H ⊗ up,X− ⊗ uq+1) 2m α 2 p α α p+q=2m 6.15.4 In this subsection, we show that for any h ∈ h, then 2m " # X 2m X δ (K(h)) = (−1)q (h ⊗ up)(hi ⊗ u2m−p),K(h) , 2m p i p=0 i under the assumption of the following conjecture: Conjecture 6.15.2. Let h ⊗ u := Q(h) ∈ g[u] ⊂ QDλ(g), and h ⊗ v := K(h) ∈ g[v] ⊂ QDλ(g), then the following relation holds in QDλ(g), 1 X X [h0 ⊗ v, h ⊗ un] = S([h0,X ] ⊗ up, [X , h] ⊗ uq) 4 α −α α p+q=n−1 1 X X = (h, α)(h0, α) S(X ⊗ up,X ⊗ uq) 4 α −α α p+q=n−1 In particular, we have 1 X [h0 ⊗ v, h ⊗ u2] = S([h0,X ] ⊗ u, [X , h]) + S([h0,X ], [X , h] ⊗ u) 4 α −α α −α α 1 X = (h, α)(h0, α) S(X ⊗ u, X ) + S(X ,X ⊗ u) 4 α −α α −α α 1 X = (h, α)(h0, α) S(X ⊗ u, X ) 2 α −α α 257 Theorem 6.15.3. The Conjecture 6.15.2 implies the following: h X X 2m i δ (K(h)) = (−1)q (h ⊗ up)(hi ⊗ uq),K(h) 2m p i i p+q=2m Proof. On one hand, using the conjecture 6.15.2, we have: X X 2m [h ⊗ v, (−1)q (h ⊗ up)(hi ⊗ uq)] p i i p+q=2m X X 2m = (−1)q h ⊗ up[h ⊗ v, hi ⊗ uq] + [h ⊗ v, h ⊗ up]hi ⊗ uq p i i i p+q=2m 1 X 2m X X = (−1)q (α, h) H ⊗ up S(X ⊗ us,X ⊗ ut) 4 p α α −α p+q=2m α s+t=q−1 X s t q + S(Xα ⊗ u ,X−α ⊗ u )Hα ⊗ u s+t=p−1 1 X X 2m X = (α, h) (−1)q H ⊗ up S(X ⊗ us,X ⊗ ut) 4 p α α −α α p+q=2m s+t=q−1 X s t p + S(Xα ⊗ u ,X−α ⊗ u )Hα ⊗ u s+t=q−1 1 X X 2m X = (α, h) (−1)q S(H ⊗ up, S(X ⊗ us,X ⊗ ut)) 4 p α α −α α p+q=2m s+t=q−1 On the other hand, 258 We have for h ∈ h, δ2m(K(h)) λ2 X X Q(β∨) Q(β∨) = β(h) (−1)q[(ad )p(S(X+,X−)), (ad )q(S(X+,X−))] 8 2 β β 2 β β β p+q=2m−1 λ2 X X X p = β(h) (−1)q[ (−1)t S(X+ ⊗ us,X− ⊗ ut), 8 s β β β p+q=2m−1 s+t=p X q (−1)j (S(X+ ⊗ uk,X− ⊗ uj))] k β β k+j=q λ2 X X X X pq = β(h) (−1)q (−1)t+j 8 s k β p+q=2m−1 s+t=p k+j=q h + s − t + k − j i S(Xβ ⊗ u ,Xβ ⊗ u ),S(Xβ ⊗ u ,Xβ ⊗ u ) λ2 X X X X pq = β(h) (−1)q (−1)t+j 8 s k β p+q=2m−1 s+t=p k+j=q + k s+j − t + s t+k − j S(S(Xβ ⊗ u ,Hβ ⊗ u ),Xβ ⊗ u ) − S(Xβ ⊗ u ,S(Hβ ⊗ u ,Xβ ⊗ u )) We compute the term as follows: + k s+j − t + s t+k − j S(S(Xβ ⊗ u ,Hβ ⊗ u ),Xβ ⊗ u ) − S(Xβ ⊗ u ,S(Hβ ⊗ u ,Xβ ⊗ u )) − t + k s+j + k+s+j =Xβ ⊗ u 2(Xβ ⊗ u )(Hβ ⊗ u ) + (β, β)Xβ ⊗ u s+j + k + k+s+j − t + 2(Hβ ⊗ u )(Xβ ⊗ u ) − (β, β)Xβ ⊗ u Xβ ⊗ u + s − j t+k − j+t+k − Xβ ⊗ u 2(Xβ ⊗ u )Hβ ⊗ u − (β, β)Xβ ⊗ u t+k − j − j+t+k + s − 2(Hβ ⊗ u )(Xβ ⊗ u ) + (β, β)Xβ ⊗ u Xβ ⊗ u − t + k s+j s+j + k − t =2(Xβ ⊗ u )(Xβ ⊗ u )Hβ ⊗ u + 2Hβ ⊗ u (Xβ ⊗ u )(Xβ ⊗ u ) + s − j t+k t+k − j + s − 2(Xβ ⊗ u )(Xβ ⊗ u )Hβ ⊗ u − 2Hβ ⊗ u (Xβ ⊗ u )(Xβ ⊗ u ) Now fix the root β, then note that X X X pq X s + tk + j (−1)q (−1)t+j = (−1)k+t s k s k p+q=2m−1 s+t=p k+j=q s+t+k+j=2m−1 259 k+t s+t k+j If we switch the pair(s, j) with (t, k), then the coeffient (−1) s k is changed by a negative sign. Thus, we have two summards of the following form: − t + k s+j s+j + k − t 2(Xβ ⊗ u )(Xβ ⊗ u )Hβ ⊗ u + 2Hβ ⊗ u (Xβ ⊗ u )(Xβ ⊗ u ) + s − j t+k t+k − j + s − 2(Xβ ⊗ u )(Xβ ⊗ u )Hβ ⊗ u − 2Hβ ⊗ u (Xβ ⊗ u )(Xβ ⊗ u ) and − s + j k+t k+t + j − s − 2(Xβ ⊗ u )(Xβ ⊗ u )Hβ ⊗ u − 2Hβ ⊗ u (Xβ ⊗ u )(Xβ ⊗ u ) + t − k s+j s+j − k + t + 2(Xβ ⊗ u )(Xβ ⊗ u )Hβ ⊗ u + 2Hβ ⊗ u (Xβ ⊗ u )(Xβ ⊗ u ) Combine the above two summands with (s, t, k, j) and (t, s, j, k), note the coeffient is s + tk + j (−1)k+t , s k we get − t + k + t − k s+j 2(Xβ ⊗ u )(Xβ ⊗ u ) + 2(Xβ ⊗ u )(Xβ ⊗ u ) Hβ ⊗ u s+j + k − t − k + t + Hβ ⊗ u 2(Xβ ⊗ u )(Xβ ⊗ u ) + 2(Xβ ⊗ u )(Xβ ⊗ u ) + s − j − s + j t+k − 2(Xβ ⊗ u )(Xβ ⊗ u ) + 2(Xβ ⊗ u )(Xβ ⊗ u ) Hβ ⊗ u t+k − j + s + j − s − Hβ ⊗ u 2(Xβ ⊗ u )(Xβ ⊗ u ) + 2(Xβ ⊗ u )(Xβ ⊗ u ) − t + k + t − k s+j = S(Xβ ⊗ u ,Xβ ⊗ u ) + S(Xβ ⊗ u ,Xβ ⊗ u ) Hβ ⊗ u s+j + k − t − k + t + Hβ ⊗ u S(Xβ ⊗ u ,Xβ ⊗ u ) + S(Xβ ⊗ u ,Xβ ⊗ u ) + s − j − s + j t+k − S(Xβ ⊗ u ,Xβ ⊗ u ) + S(Xβ ⊗ u ,Xβ ⊗ u ) Hβ ⊗ u t+k − j + s + j − s − Hβ ⊗ u S(Xβ ⊗ u ,Xβ ⊗ u ) + S(Xβ ⊗ u ,Xβ ⊗ u ) − t + k s+j + t − k s+j =S S(Xβ ⊗ u ,Xβ ⊗ u ),Hβ ⊗ u + S S(Xβ ⊗ u ,Xβ ⊗ u ),Hβ ⊗ u + s − j t+k + j − s t+k − S S(Xβ ⊗ u ,Xβ ⊗ u ),Hβ ⊗ u − S S(Xβ ⊗ u ,Xβ ⊗ u ),Hβ ⊗ u 260 We have a identity of the binormial coefficients for any integers j, k, and n satisfying 0 ≤ j ≤ k ≤ n, then n X mn − m n + 1 = j k − j k + 1 m=0 Now let fix the index k, and t, and fix the sum s + j = χ, we change the index p = 0,..., 2m − 1, then the index s, j, q is changing according to p. The coefficient of the term − t + k s+j S S(Xβ ⊗ u ,Xβ ⊗ u ),Hβ ⊗ u is 2m−1 X p 2m − 1 − p 2m (−1)k+t = −(−1)χ . t 2m − 1 − χ − t χ p=0 Thus 2m λ2 X 2m X X δ (k(h)) = − (−1)χ (β, h)S H ⊗ uχ, S(X− ⊗ ut,X+ ⊗ uk) 2m 4 χ β β β χ=0 β t+k=2m−1−χ 6.16 Proof of Conjecture 6.15.2 in special cases In this section, we show the Conjecture 6.15.2 when (h, h0) = 0. Let m : U(g[u]) ⊗C U(g[u]) → U(g[u]) be the multiplication map. The following obser- 261 vation will be useful below: X p q S([Xβ1 ,Xα] ⊗ u , [X−α,Xβ2 ] ⊗ u ) α∈Φ ! X p q = m [Xβ1 ⊗ 1,Xα(u ) ⊗ X−α(u )], 1 ⊗ Xβ2 α∈Φ ! X q p +m [1 ⊗ Xβ1 ,X−α(u ) ⊗ Xα(u )],Xβ2 ⊗ 1 α∈Φ N ! X i p q = m [Xβ1 ⊗ 1, Ωp,q], 1 ⊗ Xβ2 − [Xβ1 ⊗ 1, eh (u ) ⊗ ehi(u )], 1 ⊗ Xβ2 i=1 N ! X i q p +m [1 ⊗ Xβ1 , Ωq,p],Xβ2 ⊗ 1 − [1 ⊗ Xβ1 , eh (u ) ⊗ ehi(u )],Xβ2 ⊗ 1 i=1 where we view X p q X i p q Ωp,q := Xα(u ) ⊗ X−α(u ) + eh (u ) ⊗ ehi(u ) α∈Φ i as an element of U(g[u]) ⊗C U(g[u]). Consequently, Lemma 6.16.1. For any X ∈ g, we have [Ωp,q,X ⊗1+1⊗X] = 0. But the above statement r r r is not true for X(u ). That is, we might have [Ωp,q,X(u ) ⊗ 1 + 1 ⊗ X(u )] 6= 0. Then we have: X p q X p q [S([Xβ1 ,Xα(u )], [X−α(u ),Xβ2 ]),Xγ] − S([[Xβ1 ,Xγ],Xα(u )], [X−α(u ),Xβ2 ]) α∈Φ α∈Φ X p q − [S([Xβ1 ,Xα(u )], [X−α(u ), [Xβ2 ,Xγ]]) (6.16.1) α∈Φ p q p q = − (γ, β2)S([Xβ1 ,Xγ] ⊗ u ,Xβ2 ⊗ u ) − (γ, β1)S(Xβ1 ⊗ u , [Xβ2 ,Xγ] ⊗ u ), (6.16.2) here we write X = Xγ ∈ gγ, for γ ∈ Φ ∪ {0}. The above equality follows from the following computation: 262 N X h i p q i (6.16.1) = − m Xβ1 ⊗ 1, [eh (u ) ⊗ ehi(u ),Xγ ⊗ 1 + 1 ⊗ Xγ] , 1 ⊗ Xβ2 i=1 N X h i q p i − m 1 ⊗ Xβ1 , [eh (u ) ⊗ ehi(u ),Xγ ⊗ 1 + 1 ⊗ Xγ] ,Xβ2 ⊗ 1 i=1 N X h i p q i p q i = − m Xβ1 ⊗ 1, (eh ,Hγ)Xγ(u ) ⊗ ehi(u ) + (ehi,Hγ)eh (u ) ⊗ Xγ(u ) , 1 ⊗ Xβ2 i=1 N X h i q p i q p i − m 1 ⊗ Xβ1 , (eh ,Hγ)Xγ(u ) ⊗ ehi(u ) + (ehi,Hγ)eh (u ) ⊗ Xγ(u ) ,Xβ2 ⊗ 1 i=1 h p q p q i = − m Xβ1 ⊗ 1,Xγ(u ) ⊗ Hγ(u ) + Hγ(u ) ⊗ Xγ(u ) , 1 ⊗ Xβ2 h q p q p i − m 1 ⊗ Xβ1 ,Xγ(u ) ⊗ Hγ(u ) + Hγ(u ) ⊗ Xγ(u ) ,Xβ2 ⊗ 1 p q p q = − m ([Xβ1 ,Xγ](u ) ⊗ [Hγ,Xβ2 ](u ) + [Xβ1 ,Hγ](u ) ⊗ [Xγ,Xβ2 ](u )) q p q p − m ([Xγ,Xβ2 ](u ) ⊗ [Xβ1 ,Hγ](u ) + [Hγ,Xβ2 ](u ) ⊗ [Xβ1 ,Xγ](u )) p q p q = − m ((γ, β2)[Xβ1 ,Xγ](u ) ⊗ Xβ2 (u ) + (γ, β1)Xβ1 (u ) ⊗ [Xβ2 ,Xγ](u )) q p q p − m ((γ, β1)[Xβ2 ,Xγ](u ) ⊗ Xβ1 (u ) + (γ, β2)Xβ2 (u ) ⊗ [Xβ1 ,Xγ](u )) p q p q = − (γ, β2)S([Xβ1 ,Xγ] ⊗ u ,Xβ2 ⊗ u ) − (γ, β1)S(Xβ1 ⊗ u , [Xβ2 ,Xγ] ⊗ u ) = (6.16.2). X p q [S [Xβ1 ,Xα(u )], [X−α(u ),Xβ2 ] , h(u)] p+q=n X p+1 q X p q+1 = S [[Xβ1 , h],Xα](u ), [X−α,Xβ2 ](u ) + S [Xβ1 ,Xα](u ), [X−α, [Xβ2 , h]](u ) p+q=n p+q=n n+1 n+1 + (h, α)S [Xβ1 ,Xα], [X−α,Xβ2 ](u ) − (h, α)S [Xβ1 ,Xα](u ), [X−α,Xβ2 ] We would like to show in the case (h, γ) = 0, then: n X X p q [h ⊗ v, Hγ ⊗ u ] = S([h, Xα] ⊗ u , [X−α,Hγ] ⊗ u ) p+q=n−1 α∈Φ 263 n [h ⊗ v, Hγ ⊗ u ] n =[h ⊗ v, [Xγ,X−γ] ⊗ u ] n−1 n−1 =[[h ⊗ v, Xγ ⊗ u],X−γ ⊗ u ] + [Xγ ⊗ u, [h ⊗ v, X−γ ⊗ u ]] 1 =[[h ⊗ v, X ⊗ u],X ⊗ un−1] − [X ⊗ u, [[h ⊗ v, H ⊗ un−1],X ]] γ −γ (γ, γ) γ γ −γ =A + B. where n−1 A =[[h ⊗ v, Xγ ⊗ u],X−γ ⊗ u ] X n−1 = [S([h, Xα], [X−α,Xγ]),X−γ ⊗ u ] α and 1 B = − [X ⊗ u, [[h ⊗ v, H ⊗ un−1],X ]] (γ, γ) γ γ −γ 1 X X = − [X ⊗ u, [ S([h, X ] ⊗ up, [X ,H ] ⊗ uq),X ]] (γ, γ) γ α −α γ −γ p+q=n−2 α∈Φ X X p q = [Xγ ⊗ u, [S([h, Xα] ⊗ u , [X−α,X−γ] ⊗ u )] p+q=n−2 α∈Φ X X p q+1 = [S([h, Xα] ⊗ u , [X−α,Hγ] ⊗ u ) α∈Φ p+q=n−2 X X p+1 q + S([h, [Xγ,Xα]] ⊗ u , [X−α,X−γ] ⊗ u ) p+q=n−2 α∈Φ X X p q+1 + S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ] ⊗ u ) p+q=n−2 α∈Φ 264 For the last two terms above, we use the fact that [Xγ, Ωp,q] = 0, then we have 0 = [Xγ, Ωp,q] X p q X p i q = [Xγ,Xα](u )X−α(u ) + [Xγ, hi](u )h ⊗ u α i X p q X p i q + Xα(u )[Xγ,X−α](u ) + hi(u )[Xγ, h ](u ) α i X p q X p q = [Xγ,Xα](u )X−α(u ) − Xγ(u )Hγ(u ) α i X p q X p q + Xα(u )[Xγ,X−α](u ) − Hγ(u )Xγ(u ) α i Thus, X p q [Xγ,Xα](u ) ⊗ X−α(u ) α X p q X p q X p q = − Xα(u ) ⊗ [Xγ,X−α](u ) + Xγ(u ) ⊗ Hγ(u ) + Hγ(u ) ⊗ Xγ(u ) α i i So, under the assumption of (h, γ) = 0, the last two term of B can be simplified as follows: X X p+1 q S([h, [Xγ,Xα]] ⊗ u , [X−α,X−γ] ⊗ u ) p+q=n−2 α∈Φ X X p q+1 + S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ] ⊗ u ) p+q=n−2 α∈Φ X X p+1 q = − S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ] ⊗ u ) p+q=n−2 α∈Φ X X p q+1 + S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ] ⊗ u ) p+q=n−2 α∈Φ X n−1 X n−1 = − S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ]) + S([h, Xα], [[Xγ,X−α],X−γ] ⊗ u ) α∈Φ α∈Φ 265 Thus, X n−1 X n−1 [S([h, Xα], [X−α,Xγ]),X−γ ⊗ u ] − S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ]) α α∈Φ X n−1 + S([h, Xα], [[Xγ,X−α],X−γ] ⊗ u ) α∈Φ X n−1 X n−1 = S([h, Xα], [X−α, [Xγ,X−γ]] ⊗ u ) + S([h, Xα], [[Xγ,X−α],X−γ] ⊗ u ) α α∈Φ X n−1 + S([h, Xα], [[X−α,X−γ],Xγ] ⊗ u ) α X n−1 X n−1 − S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ]) + S([h, [Xα,X−γ]] ⊗ u , [X−α,Xγ]) α∈Φ α X n−1 X n−1 = − S([h, Xα] ⊗ u , [[Xγ,X−α],X−γ]) − S([h, Xα] ⊗ u , [[X−α,X−γ],Xγ]) α∈Φ α X n−1 = S([h, Xα] ⊗ u , [X−α,H−γ]). α∈Φ Thus, when (h, γ) = 0, we have X n−1 X X p q A + B = S([h, Xα] ⊗ u , [X−α,H−γ]) + S([h, Xα] ⊗ u , [X−α,H−γ] ⊗ u ) α∈Φ p+q=n−2 α∈Φ X X p q = S([h, Xα] ⊗ u , [X−α,H−γ] ⊗ u ). p+q=n−1 α∈Φ 266 Bibliography [1] J. 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