Quantum Cohomology of Slices of the Affine Grassmannian

Home , Affine space, Grassmannian, Quantum cohomology

Quantum Cohomology of Slices of the Aﬃne Grassmannian

Ivan Danilenko

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2020 c 2020

Ivan Danilenko

Quantum Cohomology of Slices of the Aﬃne Grassmannian

Ivan Danilenko

The affine Grassmannian associated to a reductive group G is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[MO2] to write an explicit formula for this connection. In order to do this, we construct a recursive relation for the stable envelopes in the G = PSL2 case and compute the first-order correction in the general case. The computation of the purely quantum part of the multiplication is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[BMO]. For the case of simply-laced G, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group G∨. Contents

List of Figuresv

Acknowledgments vi

1 Introduction1 1.1 Overview...... 1 1.2 Overview of the main result...... 3 1.2.1 Stable basis and purely quantum multiplication...... 3 1.2.2 Classical multiplication...... 6 1.2.3 Identiﬁcation...... 7 1.3 Structure of the thesis...... 9

2 Slices of the aﬃne Grassmannian 10 2.1 Representation-theoretic notation...... 10 2.2 Classical constructions...... 11 2.2.1 Aﬃne Grassmannian...... 11 2.2.2 Transversal slices...... 15 2.3 Poisson structures and symplectic resolutions...... 16 2.3.1 Poisson structure...... 16 2.3.2 Symplectic resolutions...... 19 2.3.3 Resolutions of slices...... 22 2.4 Group action on resolutions...... 24

i 2.5 Beilinson-Drinfeld Grassmannian...... 25 2.6 Homology of resolutions...... 31

3 T-equivariant geometry 36 3.1 T-fixed points...... 36 3.2 Tangent weights at T-fixed points...... 38 3.2.1 Tangent spaces of the affine Grassmannian...... 38 3.2.2 Tangent spaces of the G (O)-orbits of the affine Grassmannian. 40 3.2.3 Tangent spaces of the resolutions of slices...... 41 3.2.4 Classical examples...... 49 3.2.5 Combinatorial descripltion of weight multiplicities...... 53 3.3 Behavior on walls in T ...... 58 3.3.1 Walls...... 58 3.3.2 Affine root subgroups...... 60 3.3.3 Invariant curves in Gr...... 61 3.3.4 Some T-invariant curves in the resolutions of slices...... 62 3.3.5 All T-invariant curves in the resolutions of slices...... 65 3.4 T-equivariant line bundles...... 67 3.5 More on Beilinson-Drinfeld Grassmannian...... 77 3.6 Behavior on walls in A ...... 79

4 Stable Envelopes 84 4.1 Preliminaries on Equivariant Cohomology...... 84 4.2 Stable Envelopes...... 86 4.2.1 Assumptions...... 86 4.2.2 Chambers...... 87 4.2.3 Attracting manifolds...... 87 4.2.4 Partial order...... 88 4.2.5 Support and degree in A ...... 88

ii 4.2.6 Polarizations...... 89 4.2.7 Deﬁnition of stable envelopes...... 90 4.2.8 First properties...... 91 4.2.9 Steinberg correspondences...... 92 4.2.10 Torus restriction...... 94

2 4.3 Stable Envelopes restrictions modulo ~ : A1 type...... 95

4.3.1 Steinberg correspondences in type A1 ...... 95 4.3.2 Recursion...... 101

4.3.3 Formulas for stable envelopes restrictions: type A1 ...... 104

4.4 Stable Envelopes restrictions modulo ~2: general case...... 110 4.4.1 Wall-crossing for stable envelopes...... 110 4.4.2 Formulas for stable envelopes restrictions: general case..... 113 4.5 Classical Multiplication...... 115 4.5.1 Stable envelopes and classical multiplication...... 115 4.5.2 Main result on classical multiplication...... 118

5 Quantum Cohomology 123 5.1 Preliminaries on quantum cohomology...... 124 5.2 Reduction to Picard rank 1...... 126 5.2.1 Two approaches to the reduced virtual class...... 126 5.2.2 Choice of basis...... 129 5.2.3 Reduction to walls...... 132 5.3 Computation of wall contributions...... 136 5.3.1 Wall deformations...... 136 5.3.2 Unbroken maps...... 138 5.3.3 Oﬀ-diagonal case...... 140 5.3.4 Diagonal case...... 150 5.3.5 Quantum multiplication...... 154

iii 6 Representation-Theoretic Reformulation 158 6.1 Trigonometric Knizhnik-Zamolodchikov connection...... 158 6.2 Quantum connection...... 163 6.3 Identiﬁcation...... 165

iv List of Figures

3.1 A typical path associated to a ﬁxed point...... 38 3.2 Tangent spaces...... 47 3.3 Left and Right Conditions...... 48 3.4 Incidence conditions on T α∨ and T α∨ ...... 49 Σi Σi+1 α∨ 3.5 A typical picture of Tp X at a ﬁxed point p...... 49

α∨ −α∨ 3.6 A typical picture of Tp X ⊕ Tp X at a ﬁxed point p...... 50

3.7 Tangent spaces at the T-ﬁxed points of the resolution of type A4 .... 51

∗ − 3.8 Tangent spaces at the T-ﬁxed points of T G/Pν ...... 52 3.9 Tangent spaces at the T-ﬁxed points of T ∗G/B ...... 53

1 ∨ 3.10 Typical pictures for four possible cases of k = n + intersecting with Γα . 54 2 p

3.11 Weight multiplicities in TpX for p represented by the red path...... 57 3.12 An example of two T-ﬁxed points connected by a curve...... 64

v Acknowledgments

First of all, I would like to thank my advisor Andrei Okounkov for his guidance in the last ﬁve years. He was the center of an incredible group of students and postdocs at Columbia, and his enthusiasm and knowledge was the driving force for our community. I would like to thank my fellow students at Columbia Noah Arbesfeld, Micah Gay, Yakov Kononov, Dmitrii Korb, Henry Liu, Renata Picciotto, Dmitrii Pirozhkov, Petr Pushkar, Andrey Smirnov, Shuai Wang and many others. I can’t overestimate the impact of such a welcoming environment on my life. My deepest gratitude goes to all people whose support and guidance helped me to love mathematics and go to a graduate school, especially Natalia Soprunova, Elena Bunina, Petr Sergeev, Evgeni Vyrodov, Alexei Morozov and Ivan Cherednik. I am thankful to Mina Aganagic, Joel Kamnitzer, Davesh Maulik, Andrei Negut, Alex Weekes and Changjian Su for the discussions related to this work. Last but not least, I would like to thank my family for being supportive.

vi to my son Kirill

vii Chapter 1

Introduction

1.1 Overview

The central object in this work is the moduli space called the affine Grassmannian Gr. It is associated to a complex connected reductive group G, and one can think of it as an analogue of flag varieties for a corresponding Kac-Moody group. It is known to have deep connections with representation theory and Langlands duality [G, MV1, MV2]. More precisely, the geometric objects we study are transversal slices in affine Grass- mannians. To give an idea of what these slices are, recall that the affine Grassmannian Gr has a cell structure similar to the Schubert cells in the ordinary flag variety. A

λ λ transversal slice Grµ describes how one orbit is attached to another. The slices Grµ ﬁrst appeared in mathematical physics as Coulomb branches in supersymmetic theories [SW]. They are naturally algebraic Poisson varieties and, in some cases, admit a

λ λ smooth symplectic resolution Grµ → Grµ, which is a notion of independent mathematical interest [K2,K3,BK]. The goal of this work is to study the quantum cohomologies of these resolutions. One of our motivations is 3d mirror symmetry. It is known to exchange the Coulomb (X) and the Higgs branch (X∨) of vacuua in a 3d supersymmetric ﬁeld theory [IS,SW]. In [MO2] D. Maulik and A. Okounkov studied the quantum cohomology for Nakajima quiver varieties [N2,N1,N3], symplectic resolutions which naturally arise as the Higgs 1 branch of vacua X∨. Mirror symmetry is a source of conjectural relations between the enumerative geometry of X and X∨, and understanding the Coulomb side makes it possible to verify this relation directly. In this project the main algebraic structure which captures the enumerative geom-

∗ etry of X is the (equivariant) quantum cohomology QHT (X). It is a deformation of usual (equivariant) cohomology by curve counts called Gromov-Witten invariants. The

∗ work of Maulik and Okounkov [MO2] showed that QHT (X) of Nakajima quiver varieties admit a natural action of a quantum group called the Yangian. The crucial step in this construction is a special basis of geometric origin in the equivariant cohomology

∗ HT (X), called stable envelopes. It is deﬁned for an algebraic symplectic variety with some mild assumptions, but to perform computations it is useful to have an explicit formula for them, i.e. know the restrictions to the ﬁxed locus (see [S1, S2] for similar

λ computations). In the case of Grµ, we provide recursive relations which allows to compute these restrictions for G of type A1. By reduction via wall-crossing to this case, we provide a formula for the ﬁrst-order corrections in case of arbitrary reductive G.

∗ The multiplication in QHT (X) for symplectic resolutions X → X0 gives rise to 2 ∗ an interesting flat connection with base H (X) and fiber HT (X) called the quantum connection. It appears as a differential equation on important generating functions in eqnumerative geometry. The quantum connection was identified with well-known representation-theoretic connections for some symplectic resolutions. For the case of Nakajima quiver varieties this connection was computed in [MO2], for the Hilbert scheme or An resolution in [MO1], for the cotangent bundle of a flag variety in [S2]. Here we identify the quantum connection in the case of resolutions of slices of the

λ Aﬃne Grassmannian Grµ, with the trigonometric Knizhnik-Zamolodchikov connection coming from the representation theory of a reductive group G∨. Moreover, the group G in the deﬁnition of Gr and G∨ in the Knizhnik-Zamolodchikov equations turn out to be Langlands-dual, which is natural because of Langlands nature of the geometry of Gr.

2 1.2 Overview of the main result

Quantum (equivariant) cohomology is a deformation of ordinary equivariant cohomology by three-point Gromov-Witten invariants. We deﬁne it by the following formula

X d hγ1, γ2 ∗ γ3i = hγ1, γ2 ∪ γ3i + hγ1, γ2, γ3i0,3,d q eﬀ d∈H2 (X,Z)

∗ X for any γ1, γ2, γ3 ∈ HT (X). Here h−, −, −i0,3,d are genus zero three-point degree d equivariant Gromov-Witten invariants and q is a formal parameter. We refer to the extra terms in this formula as purely quantum part. In this work the spaces of interest are resolutions of slices of the aﬃne Grassmannian

λ X = Grµ which are parametrized by a reductive group G, a sequence of minuscule coweights λ = (λ1, . . . , λl) and a dominant coweight µ. The quantum connection requires the quantum multiplication of special kind, when

2 γ3 is a divisor, i.e. γ3 = D ∈ HT (X). To simplify the problem, we can apply the divisor equation [HKK+]:

X X hγ1, γ2,Di0,3,d = hD, di hγ1, γ2i0,2,d .

We compute the classical and purely quantum part separately. In both cases we rely on reduction to wall cases: to walls in the equivariant parameters (i.e. Lie T) for the classiacal part and to the walls in K¨ahlerparameters (i.e. Pic (X) ⊗ C). Let us outline what we use to do it.

1.2.1 Stable basis and purely quantum multiplication

∗ The ﬁrst step is to construct a basis in the equivariant cohomology HT (X) which is easy to work with. One of the choices naturally comes after localization, i.e. passing to a related algebra

∗ ∗ ∗ H (X) = H (X) ⊗ ∗ Frac H (pt) , T loc T HT(pt) T 3 ∗ where Frac HT (pt) is the fraction field. In all the spaces X we consider, the fixed locus XT is a finite set of points. This

∗ T allows us to define a basis by pushing forward the units 1p ∈ HT (p), p ∈ X under inclusions ip : p ,→ X. The localization Theorem of Atiyah-Bott[AB] implies that this is a basis, called the fixed point basis. The fixed point basis was used by M. Viscardi in his PhD thesis [V] to compute the purely quantum multiplication in the first nontrivial case, i.e. when λ is a sum of two minuscule coweights, and only in the leading order. This computation shows the limitation of this approach. Viscardi computed only the leading term in the purely quantum part of the multiplication and finding the correction terms looks extremely hard if even possible. The way how we overcome this obstacle and protect ourselves from corrections is to use non-localized classes and rely heavily on dimensional arguments to have an auto- matic vanishing of corrections. Using pushforward classes ip,∗1 is not enough because of their high degree, 2 dim X. Such a degree still allows our formulas to have barely

computable corrections. This is why we use the stable basis StabC (p) introduced in ∗ [MO2], which has degree dim X. It is also given by the images of 1p ∈ HT (p), but under a diﬀerent map, going in the ”wrong” way

∗ A ∗ StabC :HT X → HT (X) .

The main choice one has to make is the choice of attracting directions in a torus,

which is denoted here by C as a Weyl chamber. Informally, the classes StabC (p) are ”corrected” versions of the (Poincar´edual) fundamental classes of the attracting varieties to p. The notion of attracting variety clearly depends on a choice of C.

We use the classes StabC (p) for our computations. Strictly speaking, these form a basis only after localization, but we still refer to these classes as the stable basis. These classes behave well for symplectic resolutions. If we have a (T-equivariant)

4 symplectic resolution (X, ω)

X → X0

of an aﬃne Poisson variety X0, then it is known[KV] to have a universal family of deformations of a pair (X, ω)

X Xeu (1.1)

0 H2 (X) where the base can be identiﬁed with the second cohomology of X. Then the stable envelopes naturally extend to this family. In a certain sense they are invariant under such deformations. If X is a conical symplectic resolution, then we have two simpliﬁcations:

• The two-point Gromov-Witten invariants are divisible by ~:

X hγ1, γ2i0,2,d = 0 mod ~.

• The oﬀ-diagonal restrictions of the stable envelopes are divisible by ~:

∗ ιqStabC (p) = 0 mod ~.

If we take γ1 = Stab−C (q) and γ2 = StabC (p) then by dimension argument

X Stab−C (q) , StabC (p) 0,2,d ∈ ~Q

Then one can reduce the computation of these invariants to the localization with respect to subtorus A = ker ~. As said above, almost all restrictions of stable envelopes vanish and the computation becomes similar to the computation of the leading term in the ﬁxed point basis. In Chapter5 we do this computation following the method of A. Braverman, D. 5 Maulik and A. Okounkov[BMO] of reducing the enumerative problem to contributions of walls in K¨ahlerparameters, that is, in the base of the family (1.1). The following explicit formula

ei−ej 2(ei−ej ) T X q ij q ij c1 (Ei) ∗q StabC (p) = − ~ StabC Ωe st, (p) + Ωe lg, (p) 1 − qei−ej 1 − q2(ei−ej ) i

ej −ei 2(ej −ei) X q ji q ji + ~ StabC Ωe st, (p) + Ωe lg, (p) 1 − qej −ei 1 − q2(ej −ei) j

1.2.2 Classical multiplication

As we have seen before, picking the stable basis gives easier computations in purely quantum part. Unfortunately, this creates some complications in the classical multiplication. In particular, the multiplication by a divisor is no longer diagonal. Fortunately, it’s not that far from it. For a T-equivariant line bundle T we have

T ∗ T X L c1 (L) ∪ StabC (p) = ιpc1 (L) · StabC (p) + ~ cp,q StabC (q) q

L L for some cp,q ∈ Q. Moreover, these numbers cp,q can be easily recovered if one knows ∗ 2 ιqStabC (p) modulo ~ . ∗ We ﬁnd the restrictions ιqStabC (p) in two steps

∗ 2 • Use the wall-crossing behaviour of ιqStabC (p) modulo ~ to reduce the computation to a similar computation on a wall in Lie A, which turns out to be a slice for

G = PSL2, the case of A1-type. This is done in Theorem 4.4.1

• Find the restrictions for A1-case via the action of Steinberg correspondences. This computation ends with Theorem 4.3.12 and has a ﬂavor similar to the computation in C. Su’s thesis[S2].

6 These results lead to the main formula of Chapter4

" # T i X ji X ij c1 (Ei) ∪ StabC (p) = StabC H (p) + ~ Ω−C, (p) − ~ Ω−C, (p) j

1.2.3 Identiﬁcation

Combining the results on quantum multiplication, we get for a simply-laced G

" ei ij ej ij # T i X q ΩC, (p) + q Ω−C, (p) c1 (Ei) ∗ StabC (p) = StabC H (p) + ~ qei − qej j6=i " # ej ij ei ij X q K (p) X q K (p) − ~ StabC + , (1.2) qei − qej qei − qej ji

This formula is a power series via the expansion in the region chosen with respect to

eﬀ the eﬀective cone H2 (X) ⊂ H2 (X)

qe1 qe2 · · · qel .

The quantum connection is a connection on a vector bundle over an open part of

2 ∗ HT (X) with a ﬁber HT (X). The of sections of these bundle in a formal neighbourhood ∗ can be identiﬁed with QHT (X). Then the quantum connection is given by the following formula

Q ∇D = ∂D − D∗

2 for the tangent vector in the base D ∈ HT (X). Taking into account the formula (1.2), we see that it looks similar to the trigonometric Knizhnik-Zamolodchikov connection (see [EFK], for example). Explicitly one can write it as a family of commuting diﬀerential operators

ij ij KZ ∂ i X ziΩC + zjΩ−C ∇i = zi − H − ~ ∂zi zi − zj j6=i

7 on the space

C[~](z1, . . . , zl) ⊗ V1 ⊗ · · · ⊗ Vl or, more geometrically, a flat connection on sections of a certain vector bundle over a base with coordinates zi. The trigonometric Knizhnik-Zamolodchikov connection appeared in conformal field theory [KZ] and representation theory of affine Lie groups [EFK]. The terms with Kij in (1.2) can be removed by applying a gauge transform. Then

ij i ij the main thing remaining in the identiﬁcation of the operators ΩC, and H with ΩC and Hi. To do it we ﬁrst have to identify variables, spaces and parameters on both sides.

λ To make the identiﬁcation with the quantum connection of Grµ, we take the representations Vi to be Vλi , the irreducible ﬁnite dimensional representations with highest weight λi. Moreover, we restrict the tensor product to the subspace of weight µ

C[~](z1, . . . , zl) ⊗ Vλ1 ⊗ · · · ⊗ Vλl .

Then we identify similar objects on two sides:

Trigonometric KZ Quantum

ei zi q

Expansion region |zi| |zj| Eﬀective cone ei − ej > 0

Choice of C in ΩC Choice of C in StabC

Basis vµ1 ⊗ · · · ⊗ vµl Stable envelopes StabC (p)

These identiﬁcations are the main content of Chapter6. Finally, the Theorem 6.3.1 we prove that the trigonometric Knizhnik-Zamolodchikov connection matches with the

λ quantum connection of Grµ for a simply-laced G.

8 1.3 Structure of the thesis

The thesis is organized as follows. In Chapter2 we recall the key definitions and facts about the affine Grassmannian. Then in Chapter3 we give a description of certain equivariant structures, such as fixed points, invariant curves and equivariant line bundles. In Chapter4 we introduce the stable envelopes, derive formulas for their restriction and apply this to compute the classical multiplication in this basis. In Chapter5 we find the purely quantum part of the multiplication by a divisor. Finally, in Chapter6 we identify for a simply-laced G the quantum connection with the trigonometric Knizhnik-Zamolodchikov connection for G∨.

9 Chapter 2

Slices of the aﬃne Grassmannian

In this chapter we recall some facts about slices of the aﬃne Grassmannians.

2.1 Representation-theoretic notation

Let us present notation we use for common representation-theoretic objects. One major difference from standard notation is that we use non-checked notation for coobjects (coweights, coroots, etc) and checked for ordinary ones (weighs, roots, etc). This is common in the literature on affine Grassmannian since it simplifies notation. And unexpected side effect is that weights in equivariant cohomology will have checks. We hope this won’t cause confusion.

• G is a connected simple complex group unless stated otherwise,

• A ⊂ G is a maximal torus,

• B ⊃ A is a Borel subgroup,

• g is the Lie algebra of G

× • X∗(A) = Hom(C , A) is the cocharacter lattice,

• X∗(A)+ ⊂ X∗(A) the submonoid of dominant cocharacters.

• Λ ⊃ X∗(A) is the coweight lattice, Λ = X∗(A) if G is adjoint type. 10 • W = N(A)/A is the Weyl group of G,

L • g = h ⊕ gα∨ is the root decomposition of g. α∨ • ρ∨ is the halfsum of positive roots.

• K (•, •) Weyl-invariant scalar product on coweight space Λ ⊗Z R, normalized in such a way that for the shortest coroot the length squared is 2 (equivalently, the length squared of the longest root is 2).

We identify all weights with Lie algebra weights (in particular we use additive notation for the weight of a tensor product of weight subspaces).

2.2 Classical constructions

2.2.1 Aﬃne Grassmannian

Let O = C[[t]] and K = C((t)). We will refer to D = Spec O and D× = Spec K as the formal disk and the formal punctured disk respectively.

Let G be a connected reductive algebraic group over C. The aﬃne Grassmannian

GrG is the moduli space of

 

 P is a G-principle bundle over D,     ∼  (P, ϕ) ϕ: P | × −→ P| × is a trivialization of P (2.1) 0 D D    ×   over the punctured disk D  where P0 is the trivial principal G-bundle over D. It is representable by an ind-scheme.

One can give a less geometric deﬁnition of the C-points of the aﬃne Grassmannian

GrG(C) = G (K) /G (O) (2.2) where we use notation G (R) for R-points of the scheme G given a C-algebra R. We will use this point of view when we talk about points in GrG.

11 The see that these are exactly the C-points of GrG (i.e. the pairs (P, ϕ) as in (2.1)), note that on D any G-principle bundle is trivializable, so take one trivialization ∼ ψ : P0 −→ P. This gives a composition of isomorphisms

−1 ψ|D× ϕ P0|D× −−−→ P|D× −−→ P0|D× (2.3)

which is a section of P0|D× , i.e. an element of G (K). Change of the trivialization

ψ precomposes with a section of P0, in other words, precomposes with an element of G (O). This gives a quotient by G (O) from the left.

Given two complex reductive groups G and H, there’s a canonical isomorphism of ind-schemes ∼ GrG×H = GrG × GrH.

∼ × k If the reductive group is a torus G = (C ) , then GrG is a disjoint union of countably many copies of Spec C, one for each element of the cocharacter lattice. This allows us to restrict ourselves to the case of connected simple (possibly not simply-connected) G in what follows without loosing much generality. Restating the results to allow arbitrary connected complex reductive group is straightforward. From now on we ﬁx a connected simple complex group G. Since it can’t cause confusion, we omit later G in the notation GrG and just write Gr.

The group G (K) o C× naturally acts on Gr. In the coset formulation the G (K)- action is given by left multiplication and C×-acts by scaling variable t in K with weight 1. In the moduli space description G (K) acts by changes of section g·(P, ϕ) = (P, ϕg−1) (these two actions are the same action if one takes into account identiﬁcation (2.3)),

C× scales D such that the coordinate t has weight 1. We will later denote this C× as × when we want to emphasize that we use this algebraic group and its natural action. C~ We will need actions by subgroups of this group, namely A ⊂ G ⊂ G (O) ⊂ G (K). It will also be useful to consider extended torus T = A × ×, where × part comes C~ C~ from the second term in the product G (K) ×. o C~ 12 The canonical projection T → × gives a character of T which we call (this C~ ~ explains the subscript in the notation ×). Then the weight of coordinate t on D is ~ C~ by the construction of the ×-action. ~ C~ Remark. We call the maximal torus of G by A to have notation similar to [MO2], i.e. T is the maximal torus acting on the variety, and A is the subtorus of T preserving the symplectic form.

The partial flag variety G/P (where P is parabolic) has a well-known decomposition by orbits of B-action called Schubert cells. The affine Grassmannian has a similar feature. One can explicitly construct fixed points of the A-action. Given a cocharacter

λ: C× → T, one can construct a map using natural inclusions

× × λ D ,→ C −→ T ,→ G i.e. an element of G (K). Projecting it naturally to Gr we get an element [λ] ∈ Gr. These points are A-ﬁxed and even the following stronger statements are true.

Proposition 2.2.1.

1. GrA = F {[λ]} λ∈X∗(A) 2. Moreover, GrT = GrA

Using these elements we deﬁne

Grλ = G (O) · [λ] as their orbits. These have the following properties

Proposition 2.2.2.

1. For any w ∈ W one has Grwλ = Grλ.

2. Gr = F Grλ λ∈X∗(A)+ 13 3. Grλ = F Grµ µ≤λ µ∈X∗(A)+ 4. As a G-variety, Grλ is isomorphic to a the total space of a G-equivariant vector

− − bundle over a partial ﬂag variety G/Pλ with parabolic Pλ whose Lie algebra contains all roots α∨ such that hα∨, λi ≤ 0.

5. D-scaling × in T scales the ﬁbers of this vector bundle over G/P− with no zero C~ λ weights.

6. Grλ is the smooth part of Grλ. In particular, Grλ is smooth iﬀ λ is a minuscule coweight or zero.

Corollary 2.2.3. Gr admits a T-invariant cell structure.

− Proof. Each G/Pλ admits an A-invariant cell structure, namely Schubert cells. A- equivariant vector bundles over these cells give A-invariant cells for each of Grλ. Hence we have an A-invariant cell structure for Gr. Moreover this cell structure is invariant under D-scaling ×-action, because it just C~ scales the ﬁbers of the vector bundles. Finally, we get that this cell decomposition T-invariant.

Corollary 2.2.4. Orbits of diagonal G (K)-action on Gr × Gr are in bijection with dominant cocharacters.

Proof. First recall that for any group H and a subgroup K one has a bijection

    H-diagonal orbits of K-orbits of ' K\H/K '  H/K × H/K   H/K 

Applying this for H = G (K) and K = G (O) and that the rightmost set is in bijection with dominant cocharacters by Proposition 2.2.2 we get the statement.

λ We write L1 −→ L2 if (L1,L2) ∈ Gr×Gr is in the G (K)-orbit indexed by a dominant µ ≤ λ. 14 λ Remark. Explicitly, one says that L1 −→ L2 iﬀ picking representatives L1 = g1G (O),

L2 = g2G (O) with g1, g2 ∈ G (K) we have

−1 G (O) g1 g2G (O) ⊂ G (O)[µ] for µ ≤ λ or, equivalently,

−1 g1 g2G (O) ∈ G (O) · [λ]

Note that this independent on the choice of the representatives of L1, L2.

2.2.2 Transversal slices

As one can see, elements g(t) of G (O) ⊂ G can be characterized as the elements of g(t) ∈ G (K) such that lim g(t) ∈ G ⊂ G (K). t→0 We can use a similar way to deﬁne a subgroup transversal to G (O). Let

 

G1 = g(t) ∈ G lim g(t) = e t→∞

This is a subgroup of G (K).

G (O) and G1 are transversal in the sense that

TeG (K) = TeG (O) ⊕ TeG1 because

TeG (K) = g((t))

TeG (O) = g[[t]]

−1 −1 TeG1 = t g[t ]

One uses G1 to analyze how G (O)-orbits in Gr are attached to each other. The

15 ind-varieties

Grµ = G1 · [µ] is the slice transversal to Grµ at [µ]. We will use use the convention that µ is dominant. The main object of interest for us are the following varieties

λ λ Grµ = Gr ∩ Grµ

These are non-empty iﬀ µ ≤ λ.

Proposition 2.2.5.

λ 1. Grµ is a T-invariant subscheme of Gr;

λ 2. The only A-ﬁxed (and T-ﬁxed) point of Grµ is [µ];

3. The D-scaling ×-action contracts Grλ to [µ]. C~ µ

λ ∨ 4. Grµ is a normal aﬃne variety of dimension h2ρ , λ − µi.

2.3 Poisson structures and symplectic resolutions

2.3.1 Poisson structure

The aﬃne Grassmannian has a natural Poisson structure. Let us describe it using the Manin triples introduced by V. Drinfeld[D1,D2]. First recall the notion of a Poisson group. A Poisson group is an Lie (algebraic) group G equipped with a Poisson bivector

0 2 π ∈ H G, Λ TG which is compatible with the multiplication. One way to say it is that the multiplication is a Poisson map. Explicitly one can write it in the following way

π|g1g2 = Lg1,∗π|g2 + Rg2,∗π|g1 , 16 where Lg and Rg, g ∈ G are the standard left and right shifts

Lg,Rg : G → G

Lg(h) = gh

Rg(h) = hg.

0 2 The notion of H (G, Λ TG) depends on the category in which we work, and lead to the notion of Poisson-Lie groups, Poisson algebraic groups, Poisson ind-groups, etc. For our purposes we need Poisson ind-groups, but let us discuss ﬁrst the ﬁnite-dimensional case of Poisson-Lie groups for simplicity. As a Lie group G gives a Lie algebra g the tangent space at e, a Poisson-Lie group equips this space with a stronger structure. It is called a Lie bialgebra. This a the usual Lie bracket on g plus a compatible cobracket

δ : g → g ∧ g which satisﬁes the formally dual axioms for a cobracket. Here ”compatible” can be phrased in the following way: δ is a 1-cocycle.

0 2 The cobracket δ comes as a derivative of π is one trivializes H (G, Λ TG) via right translations. The category of connected simply connected Lie groups is equivalen to the category of Lie algebras. Drinfeld proved[D1] the following closely related fact

Theorem 2.3.1. (Drinfeld) The category of connected simply connected Poisson-Lie groups is equivalent to the category of ﬁnite-dimensional Lie bialgebras.

To construct Lie bialgebras we need the notion of Manin triples. A Manin triple

(g+, g−, g) , where g is a Lie algebra equipped with a nondegenerate invariant scalar product (•, •), 17 such that g+, g− are isotropic Lie subalgebras. In the ﬁnite dimensional case the form (•, •) automatically induces isomorphisms

∗ ∗ g+ ' g− and g− ' g+. In inﬁnite-dimensional case we have to require it (one even has to be even more careful and specify what duals we take), or to substiture it by another condition allowing us to deﬁne cobrackets as the duals of brackets. Let us skip this technical point. Then by dualizing the bracket maps

g+ ∧ g+ → g+,

g− ∧ g− → g− we get cobracket maps

δ− : g− → g− ∧ g−,

δ+ : g+ → g+ ∧ g+ and a cobracket

δ : g → g ∧ g,

δ = δ+ + δ−.

Then all three Lie algebras g, g− and g+ are Lie bialgebras. Now we show that G (K) is a Poisson ind-group. We have a Manin triple

g[[t]], t−1g[t−1], g((t)) , if we equip g((t)) with the standard invariant scalar product

n m (xt , yt ) = K (x, y) δn+m,0,

18 −1 −1 where δ•,• is the Kronecker delta. Then the subalgebras g[[t]], t g[t ] are isotropic as desired. Thus g((t)) is a Lie bialgebra and g[[t]] is a Lie subbialgebra. We have that G (K) is a Poisson ind-group and G (O) is its Poisson subgroup. By theorem of Drinfeld[D2] the quotient Gr = G (K) /G (O) is Poisson.

λ The slices Grµ ⊂ Gr are Poisson subvarieties. Moreover, their smooth parts are symplectic leaves, hovewer not all symplectic leaves have this form: they might be shifted by the G-action.

2.3.2 Symplectic resolutions

λ One of the reasons what motivates us to study varieties Grµ is that they provide a family of symplectic resolutions. Let us recall some basics on them. See [K2] for more details. A symplectic resolution is a smooth algebraic variety X equipped with a closed non-degenerate 2-form ω such that the canonical map

∗ X → X0 = Spec H (0) X, OX is a birational projective map. For our purposes we will require the resolution is conical, i.e. that there is a ×- C~ action on X such that the X is scaled to a point by the induced ×-action. We assume 0 C~ that the symplectic form is an eigenvector of this action, then its weight ~ is non-zero. Well-known examples of the symplectic resolutions are Du Val singularities, the Hilbert scheme points on a plane (or a Du Val singularity), cotangent bundles to ﬂag varieties, Nakajima quiver varieties.

X0 carries a natural structure of (possible singular) Poisson aﬃne scheme. One can

λ pose a question if aﬃne Poisson varieties Grµ we study admit a symplectic resolution, λ i.e. a smooth symplectic X with X0 = Grµ such that the assumptions above hold. A result in [KWWY] answers this question and we present their result later. Before it let us discuss properties of the symplectic resolutions in general. 19 One rich source of symplectic varieties is the symplectic reduction. Let (Z, ω) be a symplectic variety and let a reductive group G have a Hamiltonian action on it. I.e. let there be a map µ: Z → g∗,

where g = Lie G and µ defines the infinitesimal action of g by vector fields ∂ξ, ξ ∈ g, via the Hamilton equations

ω (∂ξ, •) = d hµ, ξi .

The standard extra condition is that µ is G-equivariant. The G-ﬁxed locus in g∗ is z∗ = [g, g]⊥ (in particular, if G is a torus, then z∗ = g∗). For any ζ ∈ z∗ we have a G-invariant subvariety µ−1 (ζ) ⊂ Z.

−1 If the quotient Xζ = µ (ζ) /G exists, then it is naturally symplectic. Thus we have constructed a family of varieties over z∗

X → z∗

which is fiberwise symplectic, with a fiber Xζ over ζ. Each fiber Xζ is called the symplectic reduction of Z.

The symplectic reduction over a torus can be in certain sense inverted [K2]. That is, one can reconstruct the family from a ﬁber if it is a symplectic resolution. The contruction is called twistor deformation. Let (X,L) be a pair containing a symplectic resolution X and a line bundle L over it.

Fix a connected subscheme S ⊂ A1 containing 0 ∈ A1. Then a twistor deformation Z over the base S is the pair X,e Le over S with the ﬁber (X,L) over 0 and a symplectic

× × form ωZ on the principal C -bundle Z → Xe associated to Le, such that C -invariant, and the projection Z → Xe → S is (the base change from Lie C× ' A1 to S of) the moment maps for the C×-action on Z. In this case X is the symplectic reduction of Z

20 as a ﬁber over 0. A theorem by Kaledin [K1], this deformation exists for complex varieties (even more generally, in characteristic 0 case) formally locally, that is for S = Spec C[[z]]. It can be extended to the whole line A1.

The fiber Xes over a point s ∈ S gets a symplectic form ωs whose class in de Rham 2 2 cohomology [ωs] ∈ H Xes is [ω0] + s · c1 (L) ∈ [ωs] ∈ H (X) under identification ∗ ∗ H Xes ' H (X) via Guass-Manin connection. One can do twistor deformations with respect to different line bundles. This gives the universal family of deformations [KV] of X as a symplectic variety:

X Xeu

0 Bu

over the base which is a subscheme of the aﬃne space associated to Pic (X) ⊗Z C ' 2 H (X, C). Bu contains at least the formal neighborhood of 0. In particular, the space of deformations of (X, ω) as a symplectic variety is ﬁnite-dimensional in contrary to the deformations of X as a variety.

The map Xeu → Bu is called the period map. Similarly to the case of 1-parametric

u deformation, it sends the ﬁber Xes with a symplectic structure ωs the de Rham class 2 [ωs] ∈ H (X). Now we are ready to describe the wall structure in H2 (X). Fix a point s ∈ H2 (X)

u and look at the symplectic variety in the ﬁber Xes with the natural algebraic symplectic

u form ωs. Let C ⊂ Xes be a smooth (algebraic, not topologic) curve. Then the restriction of ωs to C as a real manifold is trivial because ωs is C-linear and the tangent space to C is C-dependent. This gives

Z 0 = ωs = h[C] , si , C where the last expression is the pairing of a homology class with a cohomology class.

21 This gives that a class d ∈ H2 (X, Z) can be represented by an algebraic curve only in the ﬁbers over such s ∈ H2 (X) that

hd, si = 0, that is s must lie on a certain hyperplane.

2 u This implies that for a generic s ∈ H (X) the ﬁber Xes has no algebraic curves. This wall structure will be the key ingredient in the computation of the purely quantum part of the multiplication.

2.3.3 Resolutions of slices

λ We want to study the geometry of the symplectic resolutions of Grµ. Unfortunately, these spaces do not always possess a symplectic resolution. We’ll follow [KWWY] to construct resolution in special cases. For construction of resolutions it is convenient to assume that the cocharacter lattice is as large as possible, that is, equal to the coweight lattice. This happens when G is of adjoint type (the other extreme from being semi-simple). This assumption doesn’t change the space we are allowed to consider: If Ge → G is a finite cover, then there is a natural inclusion Gr ,→ Gr with the image being a collection of connected G Ge components of Gr . Since the slices are connected, they belong only to one component Ge and can be thought as defined for Ge . Last but not least the T-equivariant structures are compatible if one takes into account that the torus for G is a finite cover over the torus for Ge . From now on we assume that G is a connected simple group of adjoint type (so the center is trivial) and make no distinction between cocharacters and coweights unless overwise is stated.

Consider a sequence of dominant weights λ = (λ1, . . . λl) Deﬁne the (closed) convolution product:

22 n λ o λ ×l λ1 λ2 l−1 λl Gr = (L1,...,Ll) ∈ Gr e · G (O) −→ L1 −→ ... −−→ Ll−1 −→ Ll

There’s a map

0 λ mλ : Gr → Gr,

(L1,...,Ll) 7→ Ll.

0 λ P One can show that the image of mλ is inside Gr , where λ = i λi, so we can actually make a map

λ λ mλ : Gr → Gr .

Then let us deﬁne

λ −1 λ Grµ = mλ Grµ .

These spaces and maps are useful to construct symplectic resolutions of singular

λ aﬃne Grµ. Let all λi in λ be fundamental coweights (which means this is a maximal P sequence which splits λ = i λi into nonzero dominant coweights). We will also need the corresponding irreducible highest weight representations V (λk) of the Langlands dual group G∨. Then there’s the following result (see [KWWY], Theorem 2.9)

Theorem 2.3.2. The following are equivalent

λ 1. Grµ possesses a symplectic resolution;

λ λ 2. Grµ is smooth and thus mλ gives a symplectic resolution of singularities of Grµ.

P 3. There do not exist coweights ν1, . . . , νl such that i νi = µ, for all k, νk is a weight

of V (λk) and for some k , νk is not an extremal weight of V (λk) (that is, not in

the Weyl orbit of λk).

Corollary 2.3.3. If all λi are minuscule, then all three statements of the Theorem 2.3.2 hold.

23 Corollary 2.3.4. If G = PSLn, then all fundamental coweights are minuscule and for all dominant coweights µ ≤ λ all three statements of the Theorem 2.3.2 hold.

Corollary 2.3.5. If G is not PSLn, then there exist dominant coweights µ ≤ λ such that all three statements of the Theorem 2.3.2 are not satisﬁed.

From this time on we assume that the pair of dominant coweights µ ≤ λ is such that the statements of the Theorem 2.3.2 hold. There are important special cases of these symplectic resolutions

Example 2.3.1. Let G = PSL2, then the cocharacter and coroot lattice are character

(or, equivalently, weight) and root lattices of SL2 respectively. For any integer n ≥ 1

λ by setting λ = (n + 1)ω and µ = (n − 1)ω we get that Grµ is An-singularity and the (ω,...,ω) λ map Grµ → Grµ is a symplectic resolution of singularity.

Example 2.3.2. Let ν be a minuscule coweight of G and ι is the Cartan involution

(ν,ιν) ν+ιν and set λ = ν + ιν, µ = 0. Then we get a resolution Gr0 → Gr0 is the Springer (ν,ιν) ∼ ∗ − resolution. Here Gr0 = T (G/Pν ) (the cotangent bundle of a partial ﬂag variety),

− − where Pν is the maximal parabolic corresponding to ν (i.e. the Lie algebra of Pν has

∨ ∨ ν+ιν all roots α of G satisfying hν, α i ≤ 0). Gr0 is the aﬃnization.

Example 2.3.3. Let G = PSLn, ω1 be highest weight of the deﬁning representation of SLn (so it is a PSLn-coweight). Set also λ = nω1, λ = (ω1, . . . , ω1) and µ = 0.

(ω1,...,ω1) ∼ ∗ nω1 Then Gr0 = T (G/B) is the cotangent bundle of full ﬂag variety, Gr0 is the

(ω1,...,ω1) nω1 nilpotent cone and Gr0 → Gr0 is the Springer resolution.

2.4 Group action on resolutions

Recall that G (K) × acts on Gr, so it also acts diagonally on Gr×l for any l. o C~

Proposition 2.4.1.

1. For any sequence λ the subscheme Grλ ⊂ Gr×l is G (O) ×-invariant o C~

24 λ λ 2. The resolution of singularities mλ : Grµ → Grµ is T-equivariant.

Proof.

1. Let us ﬁrst prove that Grλ is G (O)-invariant. For any g ∈ G (O) we have

g · hG (O) = (gh)G (O) = (Adg−1 h)G (O)

Then we have

λ h1G (O) −→ h2G (O)

−1 ⇔ G (O) h1 h2G (O) ⊂ G (O)[µ] for some µ ≤ λ

−1 −1 −1 ⇔ G (O) Adg h1 Adg (h2) G (O) ⊂ G (O)[µ] for some µ ≤ λ

λ ⇔ g · h1G (O) −→ g · h2G (O)

A similar computation holds for C×.

2. Follows from the action being diagonal and the deﬁnition of mλ.

2.5 Beilinson-Drinfeld Grassmannian

By definition, the affine Grassmannian is the moduli space of local data in a formal neighbourhood of a point. One can allow this point to vary and even have several such points. This gives rise to a moduli space called the Beilinson-Drinfeld affine Grassmannian [BD].

l 1 Let (t1, . . . , tl) ∈ A be l-tuple of points in A . Then we can deﬁne the following moduli space similar to (2.1).

 

 P is a G-principle bundle on A1,     ∼  Grf (t1,...,t ) = (P, ϕ) ϕ: P | 1 −→ P| 1 l 0 A \{t1,...,tl} A \{t1,...,tl}    1   is a trivialization of P over A \{t1, . . . , tl} 25 l Varying (t1, . . . , tl) we get a family over A

Grf (t1,...,tl) Grf

l (t1, . . . , tl) A

with a ﬁber Grf (t1,...,tl). The family Grf is called the Beilinson-Drinfeld Grassmannian.

If we set t1 = ··· = tl = 0, then Grf (0,...,0) ' Gr, by the Beauville-Laszlo Theo- rem[BL] which allows us to pass to the formal neighborhoods. So the Beilinson-Drinfeld

Grassmannian Grf is a deformation of the usual affine Grassmannian Gr. The family Grf is fiberwise Poisson. The Poisson structure again comes from consideration of Manin triples and can be found in [EK]. Here we discuss several deformations of familiar spaces associated to the affine Grass- mannian Gr. All of them will be deformations over Al obtained by varying points l t1, . . . , tl. We will use notation of form Xe for the family over A and Xe(t1,...,tl) for the

ﬁber over (t1, . . . , tl).

Now let us deﬁne a deformation of Grµ. It was deﬁned as on orbit of [µ] under the action of G1 ⊂ G (K). In geometric terms the action of G (K) meant a change of trivialization ϕ by precomposing with a automorphisms of P0|D× , the subgroup G1 is the subgroup which doesn’t change the behaviour of ϕ at ∞. Informally, Grµ is the moduli space of (P, ϕ) such that the trivialization ϕ behaves like tµ as t → ∞. The analogue of G (K) is G R(t1,...,tl) , where

 1 1 R(t1,...,tl) = C t, ,..., t − t1 t − tl

1 are the global functions on A \{t1, . . . , tl}. Then the analogue if G1 is G1,(t1,...,tl), the subgroup of all elements in G R(t1,...,tl) whose limit as t → ∞ exists and is e:

 G1,(t1,...,t ) = g(t) ∈ G R(t1,...,t ) lim g(t) = e . l l t→∞

26 µ The analogue of [µ] is the equivalence class t − t1 ∈ G R(t1,...,tl) under an identiﬁ- cation similar to G (K) /G (O) for Gr. Note that the G1,(t1,...,tl) doesn’t change if we

µ µ µ −µ choose t − ti instead of t − t1 , because t − ti t − t1 ∈ G1,(t1,...,tl). The G1,(t1,...,tl)-

µ orbit of t − t1 deﬁnes Grf µ,(t1,...,tl) ⊂ Grf (t1,...,tl). This gives a family

l Grf µ → A

with Grf µ,(0,...,0) ' Grµ, so it is a deformation of Grµ. Now we can discuss how Grλ deforms in this way. It was a condition that at t = 0 the singularity is ”not worse” than tλ. For every point t ∈ A1 we can do the natural restriction of (P, ϕ) to the formal disks about t

ρt : Grf (t1,...,tl) → Gr

By deﬁnition this is trivial if t 6= ti for all i, 1 ≤ i ≤ l. For t = ti this allows to look at

the local behavior of L ∈ Grf (t1,...,tl) by

λ ρti (L) ∈ Gr .

In this case we say that L has Hecke type ≤ λ at ti.

l One feature of Grf is that points ti can split and merge as we move along A . We want the Hecke type to add whenever points merge. So we assign to each i, 1 ≤ i ≤ l, a P dominant weight λi and consider a formal linear combination i λiti with coeﬃcients in cocharacters X∗(A). One can think of this as a divisor colored in cocharacters

X∗(A). This divisor is ”effective”, i.e. all cocharacters are dominant. For G of rank 1 this will be indeed an effective divisor. We can sum the coefficients of the same point:

X λ(t) = λi.

i, ti=t

27 and write X X λiti = λ(t)t. i t∈A1 Then similarly to the case of usual Gr we write

P i λiti L1 −−−−→ L2

for L1,L2 ∈ Grf if λ(t) ρt (L1) −−→ ρt (L2) for all t ∈ A1. P λ t Then this deﬁnes Gr i i i ⊂ Gr as f (t1,...,tl) f (t1,...,tl)

P λ t P i i i i λiti Grf = L ∈ Grf (t1,...,t ) L0 −−−−→ L , (t1,...,tl) l

where L0 ∈ Grf (t1,...,tl) is the trivial element, the one with trivial G-bundle and the identity as a trivialization.

As previously, one can check that it deﬁnes a ﬂat family over Al

P λ t i i i l Grf → A

P which is a deformation of Gr λi . P λ t The ﬁbers Gr i i i split into a product f (t1,...,tl)

P i λiti Y Gr ' Grλ(t). (2.4) f (t1,...,tl) t∈A1

All but ﬁnitely many factors are points, so this space is well-deﬁned.

λ We are interested in the transversal slices Grµ. Their deformation is

P P i λiti i λiti Grf µ = Grf ∩ Grf µ (2.5)

for any splitting of λ into a sum of dominant weights λi. 28 λ The fibers of (2.5) have properties, similar to the ones of Grµ: they are affine, normal, Poisson. If all λi are minuscule (or zero), then a generic fiber of (2.5) is

λ smooth. The singular ﬁbers (in particular, Grµ) may appear only when ti = tj for some i and j. In other words, we have walls

Hij = {ti = tj} over which the ﬁbers of (2.5) might be singular.

λ We can make a simultaneous resolution of (2.5) similar to resolutions of Grµ. Let

λ1, . . . , λl be an l-tuple of dominant minuscule weights which sum to λ (assuming it exists). Look at the moduli space

(λ1,...,λl) n ×l λ1t1 λ2t2 λl−1tl−1 λltl o Gr = (L1,...,Ll) ∈ Gr L0 −−→ L1 −−→ ... −−−−−→ Ll−1 −−→ Ll f (t1,...,tl) f (t1,...,tl)

The family is

(λ1,...,λl) l Grf → A is a deformation of Gr(λ1,...,λl). All ﬁbers are smooth. (λ ,...,λ ) The ﬁbers Gr 1 l have a product structure similar to (2.4): f (t1,...,tl)

(λ1,...,λl) Y Gr ' Grλ(t), (2.6) f (t1,...,tl) t∈A1 where λ(t) is the subsequence of (λ1, . . . , λl) of all λi such that ti = t. In particular, over a generic point of a wall Hij (i.e. when ti = tj and all other tk are distinct) we have the following ﬁber

(λ1,...,λl) Y Gr = Grλi,λj × Grλk f (t1,...,tl) k6=i k6=j

29 There is a natural map which takes the last element Ll

(λ ,...,λ ) P λ t m → Gr 1 l → Gr i i i . (2.7) e (λ1,...,λl) f f

Restricting it to a subspace

(λ1,...,λl) −1 Gr = m Grµ f µ e (λ1,...,λl) f we get a natural morphism P (λ1,...,λl) i λiti Grf µ → Grf µ (2.8) is the ﬁberwise aﬃnization. So it’s proper, birational and surjective and deforms the familiar resolution of slices P (λ1,...,λl) λi Grµ → Grµ .

P λ t Since any fiber Gr i i i over the complement of walls H is smooth, the map is f µ,(t1,...,tl) ij an isomorphism in fiber over a generic point. So, outside of the planes Hij the spaces (λ ,...,λ ) Gr 1 l don’t have any curves. f µ,(t1,...,tl) These constructions can be equipped with T action. A acts on these spaces by changing the trivializations by global multiplication by A. So it’s a fiberwise action. Let × scale all t simultaneously with weight . This acts on the whole families and C~ i ~ Al, making all morphisms coming from

l Grf → A

×-equivariant. C~ These A-action and ×-action commute, so we can say that we have a T-action. C~ It specializes to the one deﬁned before over (t , . . . , t ) = (0,..., 0), the only ×-ﬁxed 1 l C~ point of Al.

30 2.6 Homology of resolutions

By the end of the day we give a representation-theoretic description of the quantum connection. The tight connection between the geometry of the affine Grassmannian of a reductive group G and the representation theory of the Langlands dual group G∨ is explicit in the geometric Satake correspondence. Here we present a quick overview of it, the reader can find more details in [G,MV1,MV2]. The affine Grassmannian Gr has a natural (Whitney) stratification S. This allows one to define DS (Gr, C) the bounded derived category of S-constructable C-sheaves with respect to this stratification. There is a full subcategory of perverse sheaves

PervS (Gr, C). This category is abelian, and even is a heart of certain t-structure, see [BBD]. In what follows we don’t change the stratification S, so we omit it in the notation: Perv (Gr, C). The perverse sheaves were introduced by P. Deligne to give a sheaf-theoretic approach to the intersection cohomology, see the change form [GM1] to [GM2]. For a closure of strata S ∈ S we can assign a perverse sheaf IC S supported on S called the intersection cohomology sheaf. One can think of this sheaf as a correction of the constant sheaf on S on the lower stratum to preserve the features like Poincaréduality in the singular case. If S is smooth, then IC S is just the constant sheaf, possibly up to cohomological shift. For the affine Grassmannian the distinguished perverse sheaves are IC Grλ which we denote later as ICλ. These are simple objects in Perv (Gr, C). There is a standard sheaf cohomology functor

∗ H : Perv (Gr, C) → VectC M F 7→ Hi (Gr, F) i to the abelian category VectC of ﬁnite dimensional complex vector spaces.

31 The sheaf cohomology of the intersection cohomology sheaf

IH∗ S := H∗ IC S

give the dual space to intersection homology IH∗ S of S. In one of the simplest versions, the geometric Satake correspondence states that the category of complex perverse sheaves on Gr is equivalent

∼ ∨ Perv (Gr, C) −→ Rep (G , C) to the abelian category of ﬁnite dimensional complex representations of the Langlands dual group G∨. This equivalence moreover

• respects the ﬁber functors, that is the triangle

∼ Perv (Gr, C) Rep (G∨, C)

H∗ F

VectC

is commutative. The functor F here is the forgetful functor, which assigns the underlying vector space of the representation.

• respects the tensor structure of these categories. On Rep (G∨, C) the tensor product is the usual tensor product ⊗ of representations. The tensor structure ∗ on

Perv (Gr, C) is harder to describe and is called the convolution. It is deﬁned via natural operations with the convolution Grassmannian.

These properties immediately imply the isomorphism

∗ ∗ ∗ H (Gr, F1 ∗ F2) ' H (Gr, F1) ⊗ H (Gr, F2) as C-vector spaces. Under the geometric Satake equivalence simple objects must go to simple objects.

32 As we said above, the simple objects are the intersection cohomology sheaves ICλ of the closed cells parametrized by dominant coweight λ. In Rep (G∨, C) the simple objects are irreducible representations Vλ which are parametrized by dominant highest weight λ. A coweight of G is a weight of G∨, so these parametrizations naturally identify. The equivalence says that these

ICλ ↔ Vλ correspond to each other. By the construction of the convolution product we have the intersection cohomology sheaf of the convolution Grassmannian Gr(λ1,...,λl) pushforwards along the map

Gr(λ1,...,λl) → Gr

to the convolution ICλ1 ∗ · · · ∗ ICλl . Since the sheaf cohomology is the pushforward to the point, we get

 ∗ (λ1,...,λl) ∗ IH Gr ' H (Gr, ICλ1 ∗ · · · ∗ ICλl )

∗ ∗ ' H (Gr, ICλ1 ) ⊗ · · · ⊗ H (Gr, ICλl )

= IH∗ Grλ1 ⊗ · · · ⊗ IH∗ Grλl

' Vλ1 ⊗ · · · ⊗ Vλl

If all λi are minuscule, then the intersection cohomology is the same as ordinary cohomology, so ∗ (λ1,...,λl) H Gr ' Vλ1 ⊗ · · · ⊗ Vλl .

(λ1,...,λl) (λ1,...,λl) If we restrict to the intersection cohomology sheaf of Grµ ⊂ Gr , then we get a weight subspace

 ∗ (λ1,...,λl) H Grµ ' Vλ1 ⊗ · · · ⊗ Vλl [µ]

33 by the construction of the geometric Satake. Functoriality gives compatibility of these isomorphisms

 ∗ (λ1,...,λl) ∼ H Gr Vλ1 ⊗ · · · ⊗ Vλl

i∗ π (2.9) ∗ (λ1,...,λl) ∼ H Grµ Vλ1 ⊗ · · · ⊗ Vλl [µ] where i∗ is the pullback by the inclusion

λ λ i: Grµ → Gr and π is the natural projection to a direct summand. Now let us say a couple of words on the relation of these constructions to the

λ equivariant cohomology. The spaces Grµ admit a T-invariant cell structure, namely λ the Bia lynicki-Birula decomposition. Then Grµ are T-equivariantly formal, which imply two things

∗ λ ∗ • The equivariant cohomology HT Grµ, C is a free module over HT (pt, C).

• The ordinary cohomology can be obtained from the equivariant cohomology by ”killing” equivariant variables:

∗ (λ ,...,λ ) ∗ (λ ,...,λ ) H Gr 1 l , ' ⊗ ∗ H Gr 1 l , (2.10) µ C C HT(pt,C) T µ C

is an isomorphism of graded algebras.

This implies that

 ∗ (λ1,...,λl) ∗ ∗ (λ1,...,λl) HT Grµ , C ' HT (pt, C) ⊗C H Grµ , C

∗ as HT (pt, C)-modules (not as algebras!). By applying the geometric Satake,

∗ (λ1,...,λl) ∗ HT Grµ , C ' HT (pt, C) ⊗C Vλ1 ⊗ · · · ⊗ Vλl [µ] . 34 We will see later a similar statement on the localized level, but we derive it only from the combinatorics of the T-ﬁxed points, with no use of the geometric Satake.

∗ λ There an analogue of the geometric Satake for the equivariant cohomology HT Grµ , given by V. Ginzburg and S. Riche[GR]. However, we do not use it here.

35 Chapter 3

T-equivariant geometry

To perform computations in T-equivariant cohomology, it’s important to know T-ﬁxed locus, T-invariant curves and T-equivariant line bundles.

In this section we ﬁx dominant cocharacters µ ≤ λ, an split of λ into a sequence of

λ fundamental coweights λ = (λ1, . . . , λl). We write X = Grµ to simplify notation.

3.1 T-ﬁxed points

Proposition 3.1.1. The ﬁxed locus XT is the following disjoint union of ﬁnitely many points

  ∀i µ − µ is a weight of V (λ ) T  ×l i i−1 i  X = ([µ1] ,..., [µl]) ∈ Gr .

 µl = µ, µ0 = 0 

T Proof. Since the T-action is diagonal, we have that (L1,...,Ll) ∈ X iﬀ ∀i Li has form [µi] for some coweight µi. Next, conditions for such points to be in X are

λ λi a A condition to be in Gr . That is, ∀i Li−1 −→ Li which gives that µi − µi−1 is a

weight of V (λi) with convention µ0 = 0 to make L0 = [0] = e · G (O).

−1 b A condition to be in mλ (Grµ). That is, Ll = mλ(L1,...,Ll) = [µ], which is

equivalent to µl = µ. 36 This gives the description of XT in the proposition.

Remark. If λi is minuscule, the constraint of νi to be a weight of V (λi) just means that

µi − µi−1 ∈ W λi.

Remark. If all λi are minuscule, the ﬁxed points are in bijection with a basis of the

∨ weight µ subspace of the G -representation V (λ1) ⊗ · · · ⊗ V (λl).

Notation. For a ﬁxed point p ∈ XT there are two related sequences of coweights. Let

p = ([0] , [µ1] ,..., [µl−1] , [µ]) .

Then for all i, 0 < i < l, we deﬁne

Σi (p) = µi, and for coherent notation we set

Σ0 (p) = 0,

Σl (p) = µ.

This gives a sequence

Σ(p) = (Σ0 (p) ,..., Σl (p)) .

We also deﬁne for all i, 1 ≤ i ≤ l

∆i (p) = Σi (p) − Σi−1 (p) .

One forms the second sequence from these coweights

∆ (p) = (∆1 (p) ,..., ∆l (p)) .

Informally speaking, ∆ (p) is the sequence of increments, so Σ (p) is the sequence of their partial sums. This explains the notation. 37 µ = ω1 + ω2

ω2

ω1

Figure 3.1: A typical path associated to a ﬁxed point.

λ Remark. One useful way to think about the ﬁxed points of Grµ is that they are one- to-one certain polygonal paths in the coweight lattice. Namely, the ones which has l segments and starts at the origin and ends at µ. The ith segment must be a weight of V (λi) (as said earlier, this is equivalent to being a Weyl reﬂection of λi if λi is minuscule). See Figure 3.1 for an example of a typical path in the case G = PSL3 and

(ω1,ω1,ω1,ω1,ω2) Grω1+ω2 .

3.2 Tangent weights at T-ﬁxed points

Weights in the tangent spaces to ﬁxed points are an important data in equivariant computations. In this section we describe this data for the resoltion of slices of the Aﬃne Grassmannian and related spaces.

3.2.1 Tangent spaces of the aﬃne Grassmannian

First let us consider the tangent space at T-ﬁxed point [λ] ∈ Gr and denote it as Tλ.

The following proposition describes Tλ explicitly as a T-representation. Then we can generalize it to the following

Proposition 3.2.1. There is an isomorphism of T-representations

∼ ∼ −1 −1 Tλ = g((t))/ Adtλ (g[[t]]) = Adtλ t g[t ] .

38 Proof. The left action gives a natural map:

pλ : G (K) → Gr, g 7→ g · [λ]

Let us equip G (K) with the T-action, where A acts by conjugation and the loop- rotating × acts naturally (they commute in this case, so we can say that it gives a C~ T-action).

The map pλ is

• A-equivariant because A commutes with tλ and A ⊂ G (O).

• ×-equivariant since for any s ∈ × we have C~ C~

s · tλ = tλλ(s)

and by deﬁnition λ(s) ∈ A ⊂ G (O), so (s · g) · [λ] = s · (g · [λ]).

Thus the map pλ is T-equivariant. The derivative at the identity is a surjection of vector spaces

dpλ : g((t)) Tλ

and since pλ is T-equivariant and the identity of G (K) is a T-ﬁxed point, this is a map of T-representations.

The kernel of pλ is easy to ﬁnd. A vector ξ ∈ g((t)) is in the kernel iﬀ Adt−λ ξ ∈ g[[t]] since being in gOt is equivalent to being killed by G (O) and Adt−λ comes from commuting with tλ. Finally,

ker dpλ = Adtλ (g[[t]]) which gives the isomorphism in the proposition.

We will need the following lemma to explicitly describe the weight subspaces of Tλ.

39 × Lemma 3.2.2. Let V be a A(K) -representation and denote by V ∨ the T-weight o C~ µ ∨ subspace with a T-weight µ . Then the action map atλ induces isomorphisms

atλ : Vµ∨ → Vµ∨+hµ∨,λi~

∨ ∨ Proof. A straightforward computation of the weight of Adtλ Vµ∨ gives µ + hµ , λi~, so we get the required map. Since Adtλ is invertible, this is an isomorphism.

Corollary 3.2.3. For the tangent space Tλ ⊂ g((t)) the following is true

1. Tλ is the direct sum of all T-weight subspaces of g((t)) satisfying condition

k − hω∨, λi < 0

on the T-weight ω∨ + k , where ω∨ is the A-part and k is the ×-part of this ~ ~ C~ weight.

2. The only non-zero weight subspaces of Tλ are

• (rk g)-dimensional subspaces of weight −k~ for k ∈ Z≥0;

∨ ∨ • 1-dimensional subspaces of weight α + hα , λi~ − k~ for k ∈ Z≥0 and each root α∨ of g.

Proof. To prove part1 ﬁrst note that this it true when λ = 0 and then Tλ = T0 = t−1g[t−1].

For a general cocharacter λ use Tλ = Adtλ T0. The image given by the isomorphisms described in Lemma 3.2.2 is exactly the direct sum needed. The part2 is a simple corollary if one takes into account well-known dimensions of T-weight subspaces of g((t)).

3.2.2 Tangent spaces of the G (O)-orbits of the aﬃne Grassmannian

We will also need the following fact about the closed cells Grλ

40 Proposition 3.2.4. Given a dominant cocharacter λ and any element w ∈ W of the Weyl group, the closed cell Grλ is smooth at the ﬁxed point [wλ]. Let us denote the

λ tangent space at this point as Twλ. Then the action by G (O) gives an isomorphism

λ ∼ −1 −1 Twλ = g[[t]] ∩ Adtwλ t g[t ] of T-representations.

Proof. Recall that the smooth locus Grλ ⊂ Grλ is a G-subvariety by Proposition 2.2.2. Since the W -action on A-invariant point [λ] can be made by taking representatives for elements W , we can get that all the points in the Weyl orbit are in the smooth locus. Hence for all w ∈ W the ﬁxed point [wλ] is in the smooth locus.

λ λ The tangent space Twλ by the deﬁnition of Gr is the subspace of Twλ given by the inﬁnitesimal action of G (O), i.e. the intersection

g[[t]] ∩ Twλ

taking into account that Twλ was identiﬁed with a subspace of g(()) by the inﬁnitesimal action of a larger group G (K). Finally, use Proposition 3.2.1 to get

−1 −1 g[[t]] ∩ Twλ = g[[t]] ∩ Adtwλ t g[t ]

3.2.3 Tangent spaces of the resolutions of slices

λ Now let us return to X = Grµ. We assume that µ and λ are such that they satisfy conditions in Theorem 2.3.2, so X is smooth. We want to describe the tangent space of X at a ﬁxed point. In what follows it will be more convenient to consider X as a subscheme of Gr×(l+1),

41 ×l not Gr . I.e. we will think of points in X not as l-tuples (L1,...,Ll), but as an (l+1)- tuple (L0,L1,...,Ll) with condition L0 = e · G (O). Then the incidence condition deﬁning Grλ is expressed in a uniform way

λ1 λ2 λl−1 λl L0 −→ L1 −→ ... −−→ Ll−1 −→ Ll.

Consider a ﬁxed point p ∈ XT. By deﬁnition the T-action on X is induced from the T action on Gr×(l+1), so the inclusion X,→ Gr×(l+1) is T-equivariant. So we have

×(l+1) TpX ⊂ TpGr as a T-representation. Then we have associated sequences of coweights

Σ(p) = (Σ0 (p) , Σ1 (p) ,..., Σl (p)) ,

∆ (p) = (∆1 (p) ,..., ∆l (p)) .

In what follows we do not vary p, so we drop p in the notation:

Σ = (Σ0, Σ1,..., Σl) ,

∆ = (∆1,..., ∆l)

This gives an explicit description

×(l+1) TpGr = TΣ0 ⊕ TΣ1 ⊕ · · · ⊕ TΣl

and we have

TpX ⊂ TΣ0 ⊕ TΣ1 ⊕ · · · ⊕ TΣl

42 as a T-representation. This allows us to write ξ ∈ TpX as

ξ = (ξ0, . . . , ξl) .

By the deﬁnition X is a subscheme of Gr×(l+1) by three types of conditions:

1. Left boundary condition:

L0 = e · G (O) ; (3.1)

2. Right boundary condition:

Ll ∈ Grµ; (3.2)

3. Incidence condition:

λ1 λ2 λl−1 λl L0 −→ L1 −→ ... −−→ Ll−1 −→ Ll. (3.3)

Each gives linear relations on a tangent vector at p. Taking them all into account, we ﬁnd TpX.

Notation. To write the relations in the simplest form we will need the following natural projection in (T-weight) graded spaces. Given a direct sum

M V = Vi i∈I and a graded subspace V 0 where some graded components are missed, i.e. given J ⊂ I set

0 M V = Vi i∈J

0 Then we have a natural projection V V which we denote as restriction v 7→ v|V 0 .

In our case V will be one of the spaces Tλ are weight graded, i.e. I is the T-weight

0 lattice. The subspaces V will be intersections Tλ ∩ Tµ and one can check from the description in Corollary 3.2.3 that these are of the required type. 43 Proposition 3.2.5. The relations on ξ0, . . . , ξl from the conditions (3.1),(3.2) and (3.3) are

1. The left boundary condition (3.1) gives

ξ0 = 0; (3.4)

2. The right boundary condition (3.2) gives

−1 −1 ξl ∈ Tµ ∩ t g[t ]; (3.5)

3. The incidence condition gives

ξi|Ti = ξi+1|Ti for 0 ≤ i < l, (3.6)

where Ti = TΣi ∩ TΣi+1 .

Proof. The left boundary condition is the easiest, L0 is constant ⇒ ξ0 = 0.

In case of the right boundary condition Ll ∈ Grµ implies that Ll changes only by the left multiplication by G1. Recall that we used he left multiplication by G (K) to identify the tangent space with a subspace of g((t)). Under this identiﬁcation the

−1 −1 inﬁnitesimal G1 action is the subspace t g[t ] which gives the condition on ξl.

λi+1 To analyze the incidence condition Li −−→ Li+1 in a neighborhood of Li = [Σi] and

Li+1 = [Σi+1], we look at perturbation of Li, Li+1 by gi, gi+1 ∈ G (K):

Li = gi · [Σi]

Li+1 = gi+1 · [Σi+1]

λi+1 Then the condition gi · [Σi] −−→ gi+1 · [Σi+1] is equivalent to saying that

Σi −1 −1 − Σi −1 (git ) gi+1 [Σi+1] = Adt− Σi gi gi+1 t [Σi+1] = Adt− Σi gi gi+1 [∆i+1] (3.7)

44 ∆i+1 is in the closure of the G (O)-orbit of [∆i+1]. In other words, it is in Gr for which we know the tangent space at [∆i+1], by Proposition 3.2.4 it is

∆i+1 −1 −1 T = g[[t]] ∩ Ad ∆ t g[t ] . ∆i+1 t i+1

Diﬀerentiating the expression (3.7) with respect to gi and gi+1 at the identity, we get that

∆i+1 Ad − Σ (ξ − ξ ) ∈ T + Ad ∆ (g[[t]]) , t i i+1 i ∆i+1 t i+1 where the ﬁrst term is the stabilizer of [∆i+1] from Proposition 3.2.1. Equivalently,

−1 −1 ξi+1 − ξi ∈ ‘ AdtΣi (g[[t]]) ∩ AdtΣi+1 t g[t ] + AdtΣi+1 (g[[t]])

Note that the second term is contains exactly all weight subspaces of AdtΣi which are not in AdtΣi+1 . This means

ξi+1 − ξi ∈ AdtΣi (g[[t]]) + AdtΣi+1 (g[[t]]) .

Saying that this diﬀerence belongs to this space is equivalent to the statement that the

restriction to the complement space Ti = TΣi ∩ TΣi+1 is zero:

(ξi+1 − ξi) |Ti = 0

(here we once again used that any weight subspace of g((t)) either belongs to

AdtΣi (g[[t]]) + AdtΣi+1 (g[[t]]) or intersects trivially with it).

One observation which simpliﬁes the analysis in the following

Proposition 3.2.6. The equations in Proposition 3.2.5 restrict to each T-weight sub- 45 space independently.

Proof. This follows from T-equivariance of conditions (3.1), (3.2) and (3.3) which implies that their inﬁnitesimal forms in Proposition 3.2.5 respect T-weight grading.

This allows us to analyze the tangent space TpX by restricting to T-weight subspaces

×(l+1) of g((t)) (only these weights are present in TpGr and hence in TpX). However it’s more convenient to restrict to A-weight subspaces, i.e. consider T-weights which diﬀer by a multiple of ~ simultaneously. More explicitly, we’re interested in intersections

TpX ∩ h((t)) and TpX ∩ gα∨ ((t)).

Proposition 3.2.7. TpX ∩ h((t)) = 0.

Proof. Since Ad[λ] for any cocharacter λ acts trivially on h((t)), we have that

−1 −1 Tλ ∩ h((t)) = Ad[λ] t g[t ] ∩ h((t))

−1 −1 = Ad[λ] t g[t ] ∩ h((t))

= t−1g[t−1] ∩ h((t)) = t−1h[t−1].

Note that this doesn’t depend on λ. This implies that if we restrict a tangent vector

ξ = (ξ0, . . . , ξl) ∈ TpX to TpX ∩ h((t)) we get

(ξ0|t−1h[t−1], . . . , ξl|t−1h[t−1])

This gives that the incidence condition restricted to TpX ∩ h((t)) is simple:

ξi|t−1h[t−1] = ξi+1|t−1h[t−1].

Combined with the left boundary condition ξ0 = 0 we have

ξi|t−1h[t−1] = 0 for all i. 46 k k

hα∨, µi

hα∨, λi

0 l i

α∨ α∨ α∨ (a) Weight subspaces of Tλ (b) Weight subspaces of TΣ1 ⊕ · · · ⊕ TΣl Figure 3.2: Tangent spaces

This is equivalent to TpX ∩ h((t)) = 0.

α∨ α∨ Let us denote Tλ ∩ gα∨ ((t)) as Tλ and similarly TpX ∩ gα∨ ((t)) as Tp X.

L α∨ Corollary 3.2.8. TpX = Tp X. α∨ root

α∨ Our goal is to describe Tp X. For this purpose we introduce a convenient graphical representation. Let us start

∨ with Tλ. Its T-weight subspaces are 1-dimensional with weights α + k~, k ∈ Z. Then we can represent nonzero weight subspaces of Tλ as dots on k-axis. By Corollary 3.2.3 we have that these are all integer points strictly below hα∨, λi. This is shown on Figure 3.2a. In a similar way we can represent T α∨ ⊕ · · · ⊕ T α∨ . If we plot a nontrivial weight Σ0 Σl subspace of T α∨ with weight α∨ + k as a point with coordinates (i, k) in the plane, Σl ~ α∨ then we have all points with integer coordinates strictly below the graph Γp piecewise

∨ linear connecting points hα , Σii (and 0 ≤ i ≤ l), as shown on Figure 3.2b. From the

Proposition 3.1.1 a sequence Σ corresponding to a ﬁxed point in X is has Σ0 = 0 and

∨ Σl = µ, so the graph starts at (0, 0) and ends at (l, hα , µi). Moreover if all λi are minuscule, then all increments in this path are ±1 or 0 since Σi+1 − Σi = ∆i+1 is a

Weyl reﬂection of λi+1. In this pictorial language one can represent the conditions (3.1)-(3.3).

47 k k k

hα∨, µi

0 0 0

hα∨, µi

∨ (b) Right Condition changes T α if hα∨, µi > 0 (a) Left Condition Σl and doesn’t change if hα∨, µi ≤ 0.

Figure 3.3: Left and Right Conditions

First, both Left Condition and Right Condition say that restrictions to certain T- weight subspaces of T α∨ or T α∨ must be trivial, i.e. we just have to exclude these Σ0 Σl weight subspaces from the picture. We’ll denote these by crosses in our picture. The Figure 3.3a shows how the Left Condition changes the leftmost column, corresponding α∨ −1g −1 to TΣ0 = t [t ]. On the Figure 3.3b we show two possible cases which can appear in the rightmost column, corresponding to T α∨ = T α∨ . Σl µ The Incidence condition says that restrictions to T-weight subspaces of T α∨ and Σi T α∨ are equal if the weight subspace belongs to their intersection (i.e. is non-trivial Σi+1 in both of them). We will show this by connecting corresponding dots with a straight lines. This is shown on Figure 3.4 in all three cases which can occur in the case of minuscule λi’s. If the dots are connected by a line, they give one tangent vector, since these components must be equal. If the left or right end of a chain of lines end with a cross of Left or Right boundary condition, then the restrictions to each dot on this line must be zero. These lines are ”irrelevant”, they don’t give a tangent vector. To make the picture clearer we’ll denote these ones by dashed lines and won’t put dots on them. Typical pictures we get are shown on Figure 3.5. The connected components of red

α∨ ∨ (non-dashed) lines give 1-dimensional subspaces of Tp X of weight α + k~ where k is the height of this line on the plot. 48 k k k

∨ ∨ hα , Σi+1i hα , Σii ∨ ∨ hα , Σii hα , Σi+1i ∨ ∨ hα , Σii hα , Σi+1i

∨ ∨ Figure 3.4: Incidence conditions on T α and T α . Σi Σi+1

0 i

α∨ Figure 3.5: A typical picture of Tp X at a ﬁxed point p.

α∨ Let us mention that this also gives an explicit basis in Tp X. If one ﬁxes a basis vector eα∨ ∈ gα∨ then every segment at height k in the constructed plot gives a basis

k α∨ vector which is e ∨ t for every T if the vertical line corresponding to i intersects with α Σi the segment (possibly at endpoints) and is zero for other T α∨ . Σi Note that the plots for α∨ and −α∨ are reﬂections of each other, we can use it to draw a plot for a pair of roots simultaneously. Under the graph we draw the same picture for α∨ as we used to, above it we draw a reﬂected picture for −α∨. For a visual convenience we draw it in blue. A typical picture is shown on Figure 3.6.

3.2.4 Classical examples

Let us show how this works in classical examples listed before and coincides with well-known weights of ﬁxed points tangent spaces.

(ω,...,ω) (n+1)ω Example 3.2.1. Recall An-singularity resolution is Gr(n−1)ω → Gr(n−1)ω, where G =

49 k

0 i

α∨ −α∨ Figure 3.6: A typical picture of Tp X ⊕ Tp X at a ﬁxed point p.

PSL2 and ω is the fundamental coweight. There are n + 1 ﬁxed points p0, . . . , pn:

∆ (p0) = (−ω,ω,ω,...,ω) ,

∆ (p1) = (ω, −ω, ω, . . . , ω) ,

...

∆ (pn) = (ω, ω, ω, . . . , −ω) .

∨ ∨ The tangent weights at the point pi are α + (i − 1)~ and − (α + i~). An example of such computation is shown on Figure 3.7. This coincides with computation via toric geometry (the singularity is a toric variety and hence admits a toric resolution).

(ν,ιν) ∼ ∗ − Example 3.2.2. Let λ = ν + ιν for a minuscule ν. Then Gr0 = T G/Pν as

− ∨ T-varieties, where Pν is a maximal parabolic corresponding to −ν. I.e. the roots α

− ∨ ∗ − of Pν are exactly such that hν, α i ≤ 0. Recall that the ﬁxed points of T G/Pν are of

− form wPν and are classiﬁed by left cosets wWP in the Weyl group w ∈ W (here WP in

− the Weyl group of the Levi factor of Pν ). Then the tangent weights to the zero section

− − ∨ ∨ G/Pν at wPν are roots α such that hwν, α i > 0 and the weights of the cotangent

50 k k k

0 i 0 i 0 i

(a) For ∆ (p0) = (−ω, ω, ω, ω, ω) (b) For ∆ (p1) = (ω, −ω, ω, ω, ω) (c) For ∆ (p2) = (ω, ω, −ω, ω, ω) ∨ the tangent weights at p0 are the tangent weights at p1 are α the tangent weights at p2 are ∨ ∨ ∨ ∨ ∨ α − ~ and −α . and − (α + ~). α + ~ and − (α + 2~). k k

0 i 0 i

(d) For ∆ (p3) = (ω, ω, ω, −ω, ω) (e) For ∆ (p4) = (ω, ω, ω, ω, −ω) the tangent weights at p3 are the tangent weights at p4 are ∨ ∨ ∨ ∨ α + 2~ and − (α + 3~). α + 3~ and − (α + 4~).

Figure 3.7: Tangent spaces at the T-ﬁxed points of the resolution of type A4

51 k k

0 i 0 i

(a) At point with Σ1 = wν there’re no (b) At point with Σ1 = wν there’s a pair of weights ∨ ∨ ∨ ∨ ∨ weights of form ±α + ~Z for such roots α with weights α and −(α +~) for each root α such that hwν, α∨i = 0. that hwν, α∨i = 1.

∗ − Figure 3.8: Tangent spaces at the T-ﬁxed points of T G/Pν

fiber over this point are −α∨ − ~ for the same α∨. Under identification of the fixed loci

T ∗ −T ∼ (ν,ιν) T G/Pν −→ Gr0

wP 7→ pwP,

where Σ (pwP) = (0, wν, 0), one gets the same answer from the Figure 3.8. Here we (ν,ιν) used that −wν ∈ W · ιν to get that pwP ∈ Gr0 .

Example 3.2.3. Let G = PSLn and ω1 be the highest weight of the deﬁning representation of SLn (hence a minuscule coweight of PSLn). Then for an n-tuple λ ∼ ∗ λ = (ω1, . . . , ω1), Gr0 = T G/B as T-varieties.

Denote the elements of the Weyl orbit of ω1 as {ei|1 ≤ i ≤ n} with e1 = ω1 and

∗ ei − ei+1 being simple roots. Then T-ﬁxed points of T G/B are wB for w ∈ W . In our case, this is a root system of type A, so W = Sn, w is a permutation. Moreover, w ∈ W acts on {ei} is by applying permutation to the index: ei 7→ ew(i). This allows us to identify the ﬁxed loci by following assignment

T ∗ T ∼ λ (T G/B) −→ Gr0

wB 7→ pw, where −1 −1 −1 ∆ (pw) = ew0w (1), ew0w (2), . . . , ew0w (l)

52 k

−1 −1 0 w0w (i) w0w (j) i

Figure 3.9: Tangent spaces at the T-ﬁxed points of T ∗G/B

and w0 is the longest element in the Weyl group (explicitly w0(i) = n − i in our case). n P We used well-known fact that ei = 0 to get the sum of all components to be zero, i=1 λ λ which means that the point in the image is in Gr0 , not only in Gr . The tangent weights at wB are wα∨ and −wα∨ − ~ for all negative roots α∨. λ The Figure 3.9 shows the tangent weights at a ﬁxed point pw of Gr0 corresponding

∗ ∨ −1 −1 to a ﬁxed point wB, w ∈ W in T G/B. If α = ei − ej and w0w (i) < w0w (j)

−1 ∨ ∨ ∨ (equivalently, w0w α is positive), then the weights of form ±α + ~Z are α and −(α∨ + ~). From this we conﬁrm that the weights are the same as described before, if one takes into account that

1. All roots are of form ei − ej, i 6= j,

2. ei − ej is negative iﬀ i > j,

∨ ∨ 3. α is negative iﬀ w0α is positive.

3.2.5 Combinatorial descripltion of weight multiplicities

λ As previously, let X = Grµ andµ and λ are such that they satisfy conditions in Theorem 2.3.2, so X is smooth. We want to describe weight multiplicities in the tangent space to p ∈ XT with a corresponding sequence of coweights

Σ = (Σ0, Σ1,..., Σl)

where we assume usual conditions Σ0 = 0 and Σl = µ and do not write p explicitly in the notation for Σ.

53 k 0 i

1 n + 2 i k 1 1 (a) n + < 0 and n + < hα∨, µi. 2 2 1 n + 2 i k 1 0 (b) 0 < n + < hα∨, µi. i 0 2 i 1 n + 2 i k 1 (c) hα∨, µi < n + < 0. 2 1 n + 2 i 1 1 (d) n + > 0 and n + > hα∨, µi. 0 2 2 i

1 ∨ Figure 3.10: Typical pictures for four possible cases of k = n + intersecting with Γα . 2 p

∨ α∨ As previously, for each root α we use a graph Γp piecewise linearly connecting

∨ points (i, hα , Σii) in i, k-plane. We are interested in the intersection of this graph with 1 1 the line k = n + , n ∈ . Let N α∨, n + be the number of the such crossings 2 Z − 2 ∨ 1 where Γα goes down as i decreases. Similarly, N α∨, n + is the number of p + 2 α∨ crossings where Γp goes down. The Figure 3.10 shows how this intersections may 1 look like. Blue dots contribute to N α∨, n + and green squares contribute to − 2 1 N α∨, n + . + 2

Proposition 3.2.9. Let α∨ be a root and n ∈ Z. Then the multiplicity of weight ∨ α + n~ in TpX is 1 1 1. N α∨, n + − 1 if 0 < n + < hα∨, µi, + 2 2 1 2. N α∨, n + otherwise. + 2 1 Proof. Every point which is counted by N α∨, n + gives rise to the left end of + 2 a segment corresponding to a non-zero tangent vector with tangent weight if we don’t take into account the Right condition. And this condition vanishes the rightmost vector 54 if and only if 1 0 ≤ n < hα∨, µi ⇐⇒ 0 < n + < hα∨, µi . 2

This gives −1 in this case, as stated in the Proposition. 1 1 If one wants to use N α∨, n + in place of N α∨, n + , there is an easy − 2 + 2 relation between them. 1 1 Proposition 3.2.10. The numbers N α∨, n + and N α∨, n + are related + 2 − 2 in the following way 1 1 1 1. N α∨, n + = N α∨, n + + 1 if 0 < n + < hα∨, µi, + 2 − 2 2 1 1 1 2. N α∨, n + = N α∨, n + − 1 if hα∨, µi < n + < 0, + 2 − 2 2 1 1 3. N α∨, n + = N α∨, n + otherwise. + 2 − 2 Proof. This is a consequence of elementary topology.

These Propositions allow us to get the following similar description in terms of 1 N α∨, n + . − 2

Corollary 3.2.11. Let α∨ be a root and n ∈ Z. Then the multiplicity of weight α∨ +n~ in TpX is 1 1 1. N α∨, n + − 1 if hα∨, µi < n + < 0, − 2 2 1 2. N α∨, n + otherwise. − 2 Proof. Straightforward case-by-case consideration. 1 Remark. We made the shift of n by to make it manifest that the the multiplicities 2 of α∨ + n~ and −α∨ − (n + 1)~ are the same. Indeed,

1 1 −(n + ) = −(n + 1) + 2 2

55 −α∨ α∨ and Γp is a reﬂection of Γp , so in these cases we’re looking at mirror pictures of intersections. The roles of ”going upwards” and ”going downwards” are swapped, so

 1 1 N α∨, n + = N −α∨, −(n + 1) + + 2 − 2 1 1 N α∨, n + = N −α∨, −(n + 1) + − 2 + 2 and the conditions

1 1 0 < n + < hα∨, µi ⇐⇒ h−α∨, µi < −(n + 1) + < 0 2 2 are equivalent.

This is expected because of the presence of the symplectic structure of weight ~, which gives a nondegenerate pairing between spaces of these weights. Moreover, one can keep track which tangent vectors have a non-zero pairing by the symplectic form. It’s non-zero if the end of the segment corresponding to one vector is the beginning of a segment for another vector. It’s not hard to compute the explicit number (depending on normalization of the symplectic form), but we since we won’t need it later we don’t compute it here.

There’s a reformulation of the Proposition 3.2.9 and Corollary 3.2.11 in a way which doesn’t refer to the graph of pairings and might be convenient for a reader who wants to know only about weight multiplicities, not how the tangent vectors behave. It is based on the following two notes

∨ 1 • The intersections of graph Γα with the line k = n + are in bijection with the p 2 1 intersection of the hyperplane h•, α∨i − n + = 0 with the piecewise linear 2 path corresponding to the ﬁxed point p (i.e. connecting the points in the coweight

lattice 0 = Σ0, Σ1,..., Σl−1, Σl = µ );

• To avoid the second case in both Proposition 3.2.9 and Corollary 3.2.11 we can take 1 the number N α∨, n + of intersections with the path going in the direction 0 2 56 µ = 2ω1 + ω2 µ = 2ω1 + ω2

α∨ 2 ω2 ω2

ω1 ω1

∨ 0 0 α1

∨ ∨ ∨ ∨ (a) Weight multiplicities of α2 − ~ and −α2 are (b) Weight multiplicities of α1 and −α2 − ~ are 1. 2.

Figure 3.11: Weight multiplicities in TpX for p represented by the red path.

of k = 0. In the language of lattice paths it’ll be all intersections with hyperplanes where the path goes to the haﬂspace containing the origin (i.e. ”returns” back to the origin halfspace, since it starts at the origin).

λ T As before, X = Grµ, p ∈ X is a smooth ﬁxed point with associated coweight sequence Σ. Let Pp be the piecewise linear path connecting points

0 = Σ0, Σ1,..., Σl−1, Σl = µ in the coweight lattice.

∨ ∨ Theorem 3.2.12. The only weights in TpX are of form α + n~ for a root α and ∨ α∨ n ∈ Z. Moreover, the multiplicity of weight α + n~ is the number of crossings of Γp with the hyperplane 1 h•, α∨i − n + = 0 2 in the direction of the halfspace containing 0.

Proof. This follows from Corollary 3.2.8, Proposition 3.2.9 and Corollary 3.2.11.

An illustration of the statement of Theorem 3.2.12 are shown on Figure 3.11. The dashed lines and green halfphanes correspond to the listed pair of weights, the blue dots show the intersections under considerations. We used the standard identiﬁcation

57 ∨ ∨ of (short) roots α1 and α2 with vectors of length squared 2 via a properly normalized invariant inner product to visualize them.

If we are interested in tangent A-weights, we can just set ~ = 0 in the statements above. Then we have the following statement

Corollary 3.2.13. Let p ∈ XA = XT. Then

∨ ∨ 1. A weight α can have non-zero multiplicity in TpX only if α is a root,

∨ ∨ 2. If α is a positive root, then the multiplicity of α in TpX is the number of such i that

∨ h∆i (p) , α i = −1.

Equivalently, this is N − hµ, α∨i ,

where N is the number of such i that

∨ h∆i (p) , α i = +1.

∨ ∨ 3. If α is a negative root, then the multiplicity of α in TpX is equal to the multi-

∨ plicity of −α in TpX.

3.3 Behavior on walls in T

The goal of this section is to give a description of how the ﬁxed locus jumps on the torus walls for both A and T tori. As a consequence we classify T-invariant closed

λ curves in X = Grµ.

3.3.1 Walls

We ﬁrst need to explain what we mean by ”walls”. Deﬁne

tQ = X∗(T) ⊗Z Q 58 T nT For each point T ∈ tQ one can define the fixed point locus as X = X , where n ∈ Z is any integer such that nT ∈ X∗(T). Then for a generic T the fixed locus is XT. The locus where XT is larger is a union of hyperplanes called walls. Let us remind how positions of walls are related to the tangent weights of the fixed locus. Let us first prove the following auxiliary statement

Lemma 3.3.1. Let T be a torus and π : X → X0 be a T-equivariant proper morphism to an aﬃne variety. Assume, moreover, X0 has the unique T-ﬁxed point x ∈ X0 and

× there is a T-cocharacter λ: C → T contracting X0 to x. Then for any T-invariant closed subvariety Y ⊂ X has a T-ﬁxed point.

Proof. The image π (Y ) is closed T-invariant. By the action of λ on any point, we get that x ∈ π (Y ). So Y intersects non-trivially with π−1 (x). The then Y ∩ π−1 (x) is non-empty and proper, invariant T. Since T is a torus, there is a T-ﬁxed point in Y ∩ π−1 (x) (by a theorem due to Borel[B]), and hence in Y .

Remark. We could not apply the theorem from [B] immediately to Y since X is not proper, it only has a proper map to X0. However, extra conditions on X0 allowed us to reduce our case to the well-known result.

Recall that we have such a proper map

λ λ mλ : Grµ → Grµ

λ to a T-equivariant aﬃne space. By Proposition 2.2.5 Grµ is contracted to the unique ﬁxed point by × ⊂ T. So the statement holds for X = Grλ. C~ µ Then there’s an immediate corollary of Lemma 3.3.1 and computation of tangent weights.

T T Corollary 3.3.2. Given a T ∈ tQ the ﬁxed locus X is greater than X only if there is an aﬃne root χ∨ = α∨ + n~ such that T ∈ ker χ∨.

Proof. Any component of XT is a closed T-invariant smooth subvariety of X Take any 59 connected component Z ⊂ XT which is not a point. Then by Lemma 3.3.1 it has a

T-ﬁxed point p. The tangent space TpZ is not trivial and TpX has only weights of form α∨ + n~. Thus the value of some aﬃne α∨ + n~ is zero on T as stated.

Remark. The statement of this Corollary can be formulated in the way that the only

∨ walls in tQ are of form α + n~ = 0.

Now we want to describe the ﬁxed locus of the generic point on a wall, i.e. subvarieties Xker χ∨ ⊂ X.

3.3.2 Aﬃne root subgroups

ker χ∨ For an explicit description of X we need the SL2-subgroups of G (K) corresponding to the aﬃne roots. One could deﬁne the SL2-subgroups by referring to the fact that G (K) is a quotient of a Kac-Moody group, so it inherits such root subgroups from it. We provide a more explicit description.

For the group SL2 let us denote the standard torus as H, the standard Borel and the opposite Borel subgroups as B+ and B−, the corresponding unipotent radicals as

U+ and U−. ∼ Fix an isomorphism H −→ C×. ∨ First, recall what the SL2-subgroups corresponding to non-aﬃne roots. Let α be a root of G and α: C× → A the corresponding coroot. Then there is a homomorphism of algebraic groups

ια∨ : SL2 → G

such that the diagram

ια∨ SL2 G

∼ α H C× A

60 commutes. This is unique up to a precomposition with an adjoint action of H on

SL2. By abuse of notation we use the same notation for

ια∨ : SL2 → G (K) given by postcomposing with the natural inclusion G ,→ G (K).

One can construct from ια∨ similar maps for aﬃne roots. Take

Adtnω : SL2 → SL2 (K)     a b a btn   7→   c d ct−n d

ω (even though t is not a well-defined element of SL2 (K), conjugation by it is well- defined). Then define

ια∨+n~ = ια∨ ◦ Adtnω .

n This maps sends the positive nilpotents of sl2(2, C) to t gα∨ = gα∨+n~, as desired.

3.3.3 Invariant curves in Gr.

We use the root subgroups to construct T-invarint curves in a way, similar to the case of ordinary ﬂag variety.

Now one can act by the image of SL2 under ια∨+n~ on a T-ﬁxed point [λ]. Denote the orbit as

Cλ,α∨+n~ = ια∨+n~ (SL2) · [λ] .

The Cartan of SL2 is in the stabilizer of [λ] because the point is T-ﬁxed. By the computation of tangent weights of T[λ]Gr, the positive unipotent radical U+ is in

∨ the stabilizer iﬀ n ≥ hλ, α i. Similarly, the negative unipotent radical U− is in the

61 stabilizer iﬀ n ≤ hλ, α∨i. This gives that the orbit is a point if n = hλ, α∨i, and the following isomorphism if n 6= hλ, α∨i

1 Cλ,α∨+n~ ' SL2/B± ' P ,

∨ where B± is either B+ or B−, depending if n is greater or lesser than hλ, α i. I.e.

Cλ,α∨+n~ are rational curves in Gr. It’s an easy fact that they are T-invariant, and, moreover, all T-invariant curves are of this form.

Curves Cλ,α∨+n~ have two T-ﬁxed points. One is, obviously, [λ]. The other is given by the image of (any representative in SL2 of) the nontrivial element of the Weyl group of SL2 (N(H)/H ' Z/2Z). A straightforward computation shows that it is [sα∨+n~λ] the image of λ under the aﬃne Weyl action:

∨ sα∨+n~λ := sα∨ λ + nα = λ + (n − hλ, α i) α.

∨ I.e. sα∨+n~ is the reﬂection preserving h•, α i = n hyperplane in the coweight space. ∨ ∨ The tangent weight of Cλ,α∨+n~ for n < hλ, α i at [λ] is exactly α + n~ and it’s ∨ ∨ −(α + n~) at [sα∨+n~λ]. For n > hλ, α i the weights are exchanged.

3.3.4 Some T-invariant curves in the resolutions of slices

λ Similar T-invariant curves in Grµ can be constructed. Fix i, j such that 0 ≤ i < j ≤ l. Consider the following homomorphism

i,j ×l ι ∨ : SL → G α +n~ 2 ∨ ∨ A 7→ e, . . . , e, ιαn ~A, . . . , ιαn ~A, e, . . . , e , where i is the number of the ﬁrst nontrivial component and j is the last one. This

×l ×l homomorphism and natural component-wise action of G on Gr gives an SL2- action. 62 λ ×l i,j Take a T-ﬁxed point p ∈ Gr ⊂ Gr . In general, the orbit under ι ∨ (SL ) is µ α +n~ 2 λ not contained in Grµ, conditions are no longer satisﬁed. However, if

∨ ∨ hΣi (p) , α i = hΣj (p) , α i = n or

∨ hΣi (p) , α i = n and l = j, then the orbit is in Grλ. This is because the ith component (and jth in the ﬁrst case) is ﬁxed, and moreover

λk λk ∨ ∨ Lk−1 −→ Lk =⇒ ιαn ~(A) · Lk−1 −→ ια +n~(A) · Lk

for any i < k ≤ j and A ∈ SL2(C). The stabilizer might fail to be greater than H, but if we require in addition

∨ hΣk (p) , α i > n for all i < k < j (3.8)

then B− is in the stabilizer. Similarly for

∨ hΣk (p) , α i < n for all i < k < j, (3.9)

we have B+ is in the stabilizer.

λ In these cases of [λ] in Gr is P1 (or a point in a degenerated case j = i+1 which we ij λ ignore). Let us call this curve C ∨ ⊂ Gr . If (3.8) holds, then the tangent vector p,α +n~ ij to C ∨ at p corresponds exactly to a segment in the description of tangent vectors p,α +n~ λ ∨ ∨ in TpGrµ with weight α + n~. Similar for (3.9) and the weight equal to −(α + n~). ij The curve C ∨ has two points. One is p by construction. Another one is q p,α +n~

63 uniquely determined by conditions:

Σk (q) = sα∨+n~ Σk (p) if i ≤ k ≤ j,

Σk (q) = Σk (p) otherwise.

Geometrically, this means that one can get the second ﬁxed point by reﬂecting the part of the path between ith and jth vertex with respect to the hyperplane h•, α∨i = n.

15 See Figure 3.12 for an example. In this ﬁgure G = PSL , the curve C ∨ in X = 3 p,α +~ (ω2,ω2,ω2,ω2,ω2,ω2) T Gr3ω1 connects p, q ∈ X with ∆ (p) = (ω2, ω2, ω2, ω2 −ω1, ω2 −ω1, ω2 −ω1) and ∆ (q) = (ω2, ω2 − ω1, ω2 − ω1, ω2, ω2, ω2 − ω1).

µ = 3ω1 α∨ 2 ω2

ω1

Figure 3.12: An example of two T-ﬁxed points connected by a curve.

λ λ The last thing we have to care about is if this orbit stays in Grµ ⊂ Gr . If j < l the last component Ll is ﬁxed and this is trivially true. If j = l then we have three cases:

∨ ij λ • If n = hΣ (p) , α i then L is ﬁxed and all C ∨ is in Gr . This case is similar j l p,α +n~ µ to the j < l case.

• If n < 0 and (3.8) holds or n > 0 and (3.9) holds, then the remaining unipotent

ij radical acts on L by a subgroup of G . This means that open part of C ∨ is l 1 p,α +n~ λ inside Grµ. Firstly, this part must be T-invariant. Secondly, it must not contain ij q (the fixed point of C ∨ not equal to p) because the affine reflected µ is not p,α +n~ ij λ ij equal to µ. This means that the part of C ∨ inside Gr is C ∨ \{q}, i.e. p,α +n~ µ p,α +n~ P1 \ {∞} ' A1. 64 • In other cases the remaining unipotent radical of SL2 acts on Ll moving it away

ij λ from Gr . Then p is the only point of C ∨ inside Gr . µ p,α +n~ µ

ij Remark. Not all T-invariant curves (even irreducible) are of form C ∨ . However p,α +n~ only these (and chains of them) contribute to the quantum multiplication by certain vanishing arguments.

ij In the ﬁrst two cases we’ll still call the curve C ∨ since it won’t cause confusion. p,α +n~ Now we claim that knowing these actions are enough to construct loci ﬁxed by walls.

ij First remark in that the tangent vectors of T-invariant curves C ∨ are exactly p,α +n~ the ”segments” in the description of tangent weights. An easy check shows that all

ij ”segments” are tangent vectors to a curve of form C ∨ . p,α +n~ Remark. The second case for j = l (giving a A1-curve) is exactly the case when the corresponding tangent weight ”segment” goes all the way to the right border and is not zero by the right boundary condition.

3.3.5 All T-invariant curves in the resolutions of slices

i1,j1 i2,j2 The next crucial remark is that if j ≤ i , then ι ∨ and ι ∨ commute. In 1 2 α +n1~ α +n2~ particular, they commute for each pair of affine weights ± (α∨ + n~) as long as (3.9) or (3.8) are satisfied for each i < j we consider. Simultaneous action on a fixed point p ∈ XT gives an inclusion map

1m 1 λ P × A ,→ Grµ (3.10) if there’s a ”segment” for weights ± (α∨ + n~) going all the way to the right boundary, and

1m λ P ,→ Grµ (3.11) otherwise. The image is a T-invariant, and at a T-ﬁxed point p the tangent space is equal to

∨ ∨ ker α +n~ ker α +n~ TpX . Thus the image is an open subvariety of the component of X 65 λ containing p. Now note that the image is proper over Grµ: for (3.11) the image is

λ 1 already proper, and for (3.10) the projection to Grµ is exactly the projection to A . So

∨ the image is closed. This shows it the whole component of Xker α +n~. By Lemma 3.3.1

∨ any component of Xker α +n~ has this form.

∨ The last touch is to keep track of T action on Xker α +n~. The weights with with T acts on tangent spaces of fixed points of P1 and A1 are ±(α∨ + n~). ∨ 1 1 Let χ be a non-zero character of T. Then we denote by Pχ∨ the projective line P 1 ∨ on which T acts fixing points 0 and ∞ in such a way that the weight of T0P is χ . 1 1 Similarly, we denote by Aχ∨ the affine line A on which T acts fixing 0 and scaling 1 ∨ T0A by weight χ . Then the results on T-walls can be summarized in the following theorem.

Theorem 3.3.3.

∨ 1. Xker χ = XT if χ∨ is not a multiple of α∨ + n~ for some root α∨ and n ∈ Z.

∨ 2. Let χ∨ = α∨ + n~, where α∨ is a root and n ∈ Z. Then the locus Xker χ as a T-variety is isomorphic to a disjoint union of the following T-varieties

•

1 m 1 Pχ∨ × Aχ∨

if 0 ≤ n < hα∨, µi,

•

1 m Pχ∨

otherwise.

Proof. Follows from preceding discussion.

1 Remark. In the statement of Theorem 3.3.3 m = 0 is allowed, giving Aχ∨ or a point.

Remark. The number m can be determined for each connected component of Xker χ∨ by counting multiplicities of weights ± (α∨ + n~) at any ﬁxed point in this component. 66 This information is enough to describe T-invariant closed curves. An irreducible closed T-invariant curve C must be scaled with some T-weight χ∨ by T. Then C ⊂ Xker χ∨ and we know all the invariant curves in these simple spaces.

Remark. Now we see that T-invariant curves do in general come in families, not isolated. This contradicts one’s expectation from well-known examples.

Proposition 3.3.4.

ij 1. The curves of form C ∨ generate H (X, ) as an abelian group. p,α +n~ 2 Z

ij eﬀ 2. The curves of form C ∨ generate the submonoid H (X, ) ⊂ H (X, ) of p,α +n~ 2 Z 2 Z eﬀective curve classes.

Proof. By Bia lynicki-Birula decomposition, H2 (X, Z) is generated by classes of closures of 1-(C)dimensional attractors with respect to some generic character of T. These are 1 m 1 1 m T-invariant curves, so they are in subspaces Pχ∨ × Aχ∨ or Pχ∨ . In these the 2nd homology are generated by classes which are constant at all components but one.

ij These correspond exactly to curves C ∨ by construction. p,α +n~ The statement about the eﬀective cone follows in a similar manner.

3.4 T-equivariant line bundles

Let us ﬁrst remind facts about line bundles on Gr. The reader can ﬁnd these results in [KNR].

Theorem 3.4.1.

1. There is an abelian group isomorphism

Pic Gr ' Z

2. Gr is ind-projective with respect to one of the generators O(1) ∈ Pic Gr, i.e. there

67 is a stratiﬁcation of Gr:

Gr1 ⊂ ... Gri ⊂ · · · ⊂ Gr, i ∈ N

S such that all Gri are projective with respect to O(1) and Gri = Gr. i Proof. See [KNR] for proofs.

One approach to O(1) comes from the theory of Kac-Moody groups. Let us ﬁrst assume that G is simply-connected. Then there is a non-trivial central extension

1 × G G (K) × 1 C b o C~

where Gb is the Kac-Moody group for G. This is a non-trivial C×-principal bundle. The associated line bundle on G (K) trivializes on right G (O)-orbits and gives rise to the line bundle O(1) on Gr.

This C×-bundle trivializes over orbits of right G (O)-action on G (K). This gives a principle C×-bundle on the quotient, Gr. The associated line bundle is O(1). If G is not simply-connected, Gr has several isomorphic connected components, each isomorphich to the affine Grassmannian of the associated simply-connected group[MV1, MV2]. This gives an independent O(1) on each component. When we work with the resolutions of slices each flag does not leave the connected component since the resolutions are connected. Thus we can ignore the fact that on different components of Gr the power of O(1) can be chosen differently.

A representation-theoretic description gives in general only a power of O(1), but it naturally comes with a G (K) ×-equivariant structure. o C~ Let V be a nontrivial ﬁnite dimensional complex representation of G. Then by extending scalars VK = V ⊗C K we get a representation of G (K). Moreover, loop scaling × (i.e. scaling t) extends this to the action of G (K) ×. C~ o C~

68 Consider VO = V ⊗CO. This is a G (O)-invariant subspace of VK. Then for g ∈ G (K) the subspace gV depends only on the image of g in Gr. This gives a G (K) ×- O o C~ equivariant vector bundle V of infinite rank over Gr. Informally, we would like to consider the determinant line bundle of V. This is ill-defined because of the infinite rank. Then one can correct the definition by ”subtracting” an infinite part of V by ”subtracting” a constant bundle, let’s say the constant vector bundle V0 with fibers VO. Unfortunately, V0 is not a subbundle of V. However, one can make sense of the virtual bundle V − V0 by

V − V0 = V/V ∩ V0 − V0/V ∩ V0, since both quotient on RHS are coherent sheaves. Applying the determinant to this object, we get a line bundle

det (V/V ∩ V0) LV = det (V0/V ∩ V0)

To compare LV to O(1), we need the notion of a Dynkin index[D3]. The representation V gives a scalar product on g by TrV (ξη) for ξ, η ∈ g. Restricting to the real

Cartan aR = X∗(A) ⊗Z R ⊂ a (here we identify a cocharacter with its derivative at e) we get a Weyl-invariant positive-deﬁnite scalar product (•, •)V . For a simple g there is only one such scalar product up to a multiple. Thus we can write

(•, •)V = dV K (•, •)

for dV ∈ R, dV > 0. With normalization of K (•, •) we use (i.e. the length squared for the shortest coroot is 2) the number dV is an integer. The number dV is called the Dynkin index of V . For example, if V is the deﬁning representation of SLn, then dV = 1.

Proposition 3.4.2. LV = O(dV ).

69 Proof. The proof can be found in [KNR].

We will see the traces of this fact in the computation of the weights at T-ﬁxed points.

Proposition 3.4.3. For the line bundle LV the T-weight of the ﬁber at a T-ﬁxed point [λ] is (λ, •) + ~ (λ, λ) V 2 V

Proof. For the determinant line bundle the weight of the ﬁber is the sum of weights of T-eigenspaces counted with multiplicity.

∨ Let us denote the set of all weights of V as WV . For a µ ∈ WV we also denote by

∨ mµ∨ the dimention of the subspace in V of weight µ .

Consider the ﬁber of V/V ∩ V0 at [λ]. By Lemma 3.2.2 we have that the weights of

∨ ∨ T-eigenspaces are of form µ + k~ with an integer k and a weight µ ∈ WV such that 0 < k ≤ hµ∨, λi. All multiplicities are 1. The sum is of all weights is

hµ∨,λi X X ∨ X ∨ ∨ ~ ∨ mµ∨ µ + k~ = mµ∨ hµ , λi µ + hµ , λi + 1 . ∨ ∨ 2 µ ∈WV k=1 µ ∈WV hµ∨,λi>0 hµ∨,λi>0

Similarly, the ﬁber of V0/V ∩ V0 has T-eigenspaces o multiplicity 1 and the weights

∨ ∨ ∨ are of form µ + k~ for an integer k and a weight µ ∈ WV such that hµ , λi < k ≤ 0. The sum is

0 X X ∨ X ∨ ∨ ~ ∨ mµ∨ µ + k~ = mµ∨ h−µ , λi µ + hµ , λi + 1 ∨ ∨ ∨ 2 µ k=hµ ,λi+1 µ ∈WV ∨ ∨ hµ ∈WV ,λi<0 hµ ,λi<0

∨ Subtracting these contributions extending summation to include µ ∈ WV with

∨ hµ , λi = 0, we ﬁnd the weight of LV

X ∨ ∨ ~ X ∨ 2 ~ X ∨ mµ∨ hµ , λi µ + mµ∨ hµ , λi + hmµ∨ µ , λi (3.12) ∨ 2 ∨ 2 ∨ µ ∈WV µ ∈WV µ ∈WV 70 First let us write µ in the ﬁrst term of (3.12) as hµ∨, •i

X ∨ ∨ X ∨ ∨ mµ∨ hµ , λi µ = mµ∨ hµ , λi hµ , •i ∨ ∨ µ ∈WV µ ∈WV

Then, note that hµ∨, λi is the eigenvalue of λ acting on the eigenspace of weight µ. This allows us to rewrite (3.12) in terms of traces:

Tr (λ•) + ~ Tr (λλ) + ~ Tr (λ) V 2 V 2 V

The last term vanishes, since g is simple, that is g = [g, g] and every element can be written as a sum of commutators. Rewriting the result using the scalar product, we get (λ, •) + ~ (λ, λ) V 2 V as desired.

By Proposition 3.4.2 we can define the equivariant structure on O(1) as on dV th root of LV . In general, this is an equivariant structure with respect to a connected cover of T. One can make it an honest T-equivariant structure on each connected component of Gr by tensoring with certain weight, but we work with the structure we defined because it has a simple formula for weights at the fixed points.

Corollary 3.4.4. For the line bundle O(1) the T-weight of the ﬁber at a T-ﬁxed point [λ] is K (λ, •) + ~K (λ, λ) . 2

Proof. This follows immediately from O(1) being dV -th root of L as an equivariant line bundle, and the Proposition 3.4.3.

In particular, Corollary 3.4.4 shows that the equivariant structure does not depend on the choice of V : changing the equivariant structure would give a constant summand to all ﬁber weights.

71 λ The line bundle O(1) gives a collection of line bundles on Grµ via coordinate-wise pullbacks:

∗ Li = πi O(1), where

λ πi : Grµ → Gr, 0 ≤ i ≤ l

(L0,L1,...,Ll) 7→ Li.

Then we have the following direct corollary of Corollary 3.4.4.

A Corollary 3.4.5. The weight of a line bundle Li on a ﬁxed point p ∈ X is

K (Σ (p) , •) + ~K (Σ (p) , Σ (p)) i 2 i i

Proof. Follows directly from Corollary 3.4.4.

Corollary 3.4.6. Let C be an irreducible T-invariant curve connecting p, q ∈ XA. Let

∨ A scale TpC with weight α . Then the degree of Li restricted to C is

Z K (Σ (p) − Σ (q) , α) c (L ) = i i . 1 i 2 [C]

Proof. By A-equivariant localization and Corollary 3.4.5 we have

∗ A ∗ A Z ι c (Li) ι c (Li) c (L ) = p 1 + q 1 1 i α∨ −α∨ [C] K (Σ (p) , •) − K (Σ (q) , •) = i i . α∨

By dimension argument and using that the Chern classes are in non-localized coho-

0 mology, we have that this is in HA (pt, Q) = Q. To ﬁnd this number, we can pair both numerator and denominator with α. This gives the statement of the Corollary.

72 One can pose a questions if the collection {Li} has ”enough” line bundles. Given a weight χ∨ we denote the corresponding trivial line bundle as χ∨ (i.e. the line bundle where T scales the ﬁber by χ∨). The check here means that χ∨ a weight; it has nothing to do with taking the dual line bundle.

Proposition 3.4.7.

1. The line bundles L0 and Ll are trivial:

L0 = 0,

L = K (µ, •) + ~K (µ, µ). l 2

2. The Q-linear space Pic (X) ⊗Z Q associated to the Picard group is generated over

Q by Li, 0 < i < l.

Proof.

1. The bundles L0 and Ll are pulled back from a point and Grµ respectively. Both have trivial Picard group, so the bundles are trivial. Now the question is what the equvariant structure is, i.e. how the ﬁber is scaled. This can be done by computing the weight at any ﬁxed point via Corollary 3.4.5.

2 λ λ 2. First we show that the second cohomology H Gr , Q of Gr is generated by

λ1,...λl λ1,...λl−1 c1 (Li), 1 ≤ i ≤ l. The space Gr is a bundle over Gr with a ﬁber

λl λl Gr . The cell Gr is the ﬂag variety for a maximal parabolic (λl is minuscule), so its second cohomology is one-dimensional and is generated by the ﬁrst Chern

class of any ample line bundle. Restriction of Ll is ample (and even can be identiﬁed with one of equivariant bundles), so it generates the second cohomology

of the ﬁber. Both Grλ1,...λl−1 and Grλl are connected, so by the spectral sequence of the ﬁbration the second cohomology is generated by the second cohomology

λ1,...λl−1 ∅ of Gr and c1 (Ll). Applying induction on l and using Gr = pt as the

base case, we ﬁnd that the rational second cohomologies are generated by c1 (i), 73 1 ≤ i ≤ l.

By the compatibility (2.9) we know that the restriction

∗ λ i∗ ∗ λ H Gr , Q −→ H Grµ, Q

2 λ is a surjection. Since H Gr , Q is generated as a linear space by c1 (Li), 0 < i ≤ ∗ λ l and the functoriality of the Chern classes, we get that H Grµ, Q is generated

by the same line bundles Li (or more pedantically, by their restrictions which have

the same name by slightly abusing the notation). Finally, Ll is trivial, so we can drop i = l and take only 0 < i < l to generate the second cohomology.

Remark. The generation of Pic (X) ⊗Z Q by ”tautological” bundles Li is an analogue of Kirwan surjectivity for Nakajima quiver varieties.

The line bundles Li are not independent. Let us give a couple of examples how this dependence might look like.

Example 3.4.1. If λ = µ the space is a point and all the bundles are trivial. As a more interesting generalization, one can say that if µ is dominant, λi is minuscule fundamental coweight and K (λ − µ, λi) = 0, then Li and Li−1 diﬀer by a weight.

Example 3.4.2. Let G be of type Dn. Let ωn−1 and ωn be the highest weights of the half-spin representations, and let αn−1, αn are corresponding coroots. Then if

λ = n1ωn−1 + n2ωn, µ = λ − ωn−1 − ωn we have that the product of the line bundles corresponding to components with ωn is trivial. Similar holds for ωn−1.

Let us introduce line bundles which are more convenient for our purposes.

Ei = Li/Li−1 for 1 ≤ i ≤ l. (3.13)

If Li generate Pic (X) ⊗ Q, then Ei generate it as well.

74 2 We have an isomorphism Pic (X) ' H (X, Z), given by the ﬁrst Chern class. Then we have a surjective map

l 2 Q H (X, Q) X (z1, . . . , zl) 7→ zic1 (Ei) . i

Dualizing, we get

l H2 (X, Q) ,→ Q (3.14)

We call the standard basis in the target e1, . . . , el. The image is determined by the relations on Chern classes. For example, the relation

l X 0 = c1 (Ll) = c1 (Ei) i=1 gives l X ei = 0 i=1 which is a standard relation on weights in Al root system.

Later we identify the homology classes with a linear combination of ei in the image under (3.14). However, one should keep in mind that there might be relations between ei’s.

ij Proposition 3.4.8. The homology class of C ∨ is p,α +n~

j ij K (α, α) X ∨ C ∨ = h∆ (p) , α i e p,α +n~ 2 k k k=i+1

Proof. From Corollary 3.4.6 we get

Z K (α, α) c (L ) = [hΣ (p) , α∨i − n] 1 k k 2 h ij i C ∨ p,α +n~

75 for i < k < j and Z c1 (Lk) = 0

h ij i C ∨ p,α +n~ otherwise.

Then passing from Lk to Ek we get

Z K (α, α) c (E ) = h∆ (p) , α∨i 1 k k 2 h ij i C ∨ p,α +n~ for i < k ≤ j and Z c1 (Ek) = 0

h ij i C ∨ p,α +n~ otherwise. Since pairing with c1 (Ek) gives coordinates in ek by deﬁnition of ek, we get the statement of the Proposition.

Now we can explicitly ﬁnd what the eﬀective cone is in term of ek.

Corollary 3.4.9.

Heﬀ (X, ) = span {e − e |i < j } ∩ H (X, ) . 2 Z Z≥0 i j 2 Z

Proof. By Proposition 3.3.4 we know that the eﬀective cone is generated by classes ij C ∨ . The classes in Proposition 3.4.8 are sums of e − e with positive coeﬃ- p,α +n~ k+1 k cients, because

j X ∨ ∨ ∨ h∆k (p) , α i = hΣj (p) , α i − hΣi (p) , α i = 0 k=i+1 and

∨ ∨ ∨ hΣk (p) , α i > hΣj (p) , α i = hΣi (p) , α i for all i < k < j.

76 The coeﬃcients are in Z because

K (α, α) ∈ 2 Z and

∨ h∆k (p) , α i ∈ Z.

This proves the statement.

Remark. The classes c1 (Li) are closely related to the fundamental weights ωi in the root system Al. Then the classes c1 (Ei) are relatives of ei in the same root system, that is the classes for which the roots are written as ei −ej. By ”related” here we mean that in the generic case the cohomology classes do form a root system of type Al (there is only one relation) and these cohomology classes match with the weights under this identiﬁcation.

3.5 More on Beilinson-Drinfeld Grassmannian

In this section we summarize essential facts about A-ﬁxed points of the Beilinson- Drinfeld deformation

(λ1,...,λl) l Grµ → A and, more importantly, relation of this deformation to the bundles Ei. Let us start with the A-fixed points. The A-action is fiberwise, so we can compare the A-fixed points for different fibers. The family of A-fixed points

A (λ1,...,λl) l Grµ → A (3.15) is a finite flat morphism, so it makes sense to say that A-fixed points in one fiber correspond to A-fixed points in another fiber. Often we use this fact by saying that

77 0 λ an A-fixed point p in some fiber has a limit p in the fiber Grµ over (0,..., 0). The A-weights of the tangent bundle ant these points are the same. Finally, under the isomorphism (2.6) there is a description of the A-fixed points, similar to the case of

λ Grµ.

Now we introduce the deformations of line bundles Ei. Deﬁne the following projections

×l πi : Grf → Gr

(L1,...,Ll) 7→ ρti (Li) for 1 ≤ i ≤ l and set

×l π0 : Grf → Gr

(L1,...,Ll) 7→ e · G (O) for consistency. Then deﬁne

∗ ∗ Eei = πi O(1)/πi−1O(1).

λ ×l We are interested in the restriction of these line bundles to Grf µ ⊂ Grf which we λ denote by the same symbols. It’s easy to check that on the central ﬁber Grµ these line bundles restrict to Ei and come with a similar T-equivariant structure.

λ ×l λ The Beilinson-Drinfeld deformation Grf µ ⊂ Grf is a twistor deformation of Grµ with respect to E1,..., El simultaneously. A shift in the variable ti is a deformation with respect to Ei. Since there are relations between the bundles Ei (or, equivalently, the

ﬁrst Chern classes c1 (Ei)), some directions become trivial. For example, the relation

l X c1 (Ei) = 0 i=1 gives that simultaneous shift of all ti by the same quantity gives an isomorphic ﬁber. 78 2 By the Proposition 3.4.7 we know that c1 (Ei) generate H (X, Q), so the universal λ l base of deformations for Grµ is a quotient of the Beilinson-Drinfeld base A by shifts in ”trivial” directions coming from the relations on c1 (Ei). The same statement is independently proved the incoming work of G. Bellamy, D. Muthiah, A. Weekes and O. Yacobi[BMWY]. Since we know that λ Grf µ

eff is the period map, any d ∈ H2 (X, Z) represented by an algebraic curve in a fiber λ Gr must pair trivially with P t c (E ). By the definition of e over the generic f µ,(t1,...,tl) i i 1 i i point in the wall Hij we can have only classes

d = m (ei − ej) for some m.

3.6 Behavior on walls in A

We can study walls in torus A ⊂ T similarly to the walls in T. From the same study of weights we have an analogue of Corollary 3.3.2.

A A Proposition 3.6.1. Given a A ∈ aQ the ﬁxed locus X is greater than X only if there is a (non-aﬃne) root χ∨ such that A ∈ ker χ∨.

For the rest of this section all objects (like ker χ∨) are associated to the torus A, not T.

ker χ∨ We want to show that components of X are the resolution of slices of type A1.

λ λ Let us fix X = Grµ and X0 = Grµ for the fixed G of any type, and in what follows we write Gr explicitly only meaning the A1-type affine Grassmannian.

So far we always assumed that in the sequence λ = (λ1, . . . , λl) all λi are non-zero.

In this section it’s convenient to drop this assumption for the slices of A1-type. This

79 doesn’t give anything new: the condition

0 Li−1 −→ Li simply means

Li−1 = Li.

So, if one drops all λi = 0, an isomorphic slice is obtained.

For A1-type the connected group of the adjoint type is PSL2. We denote the fundamental coweight as ω and the positive root as α = 2ω. Let p ∈ XA and χ∨ be a root of G. Then we consider

(k1ω,...klω) Z = Grmω , where

m = hµ, χ∨i ,   ∨ 0 if hΣi−1 (p) − Σi (p) , χ i = 0, ki =  1 otherwise.

In Z there is a unique point p0 satisfying

0 ∨ Σi (p ) = hΣi (p) , χ i ω for all i. It’s an easy check that it’s indeed in Z. Then we can deﬁne an inclusion map

i: Z,→ X

80 given by

 0 0 Σ1(p ) Σl(p ) g1t PSL2 (O) , . . . glt PSL2 (O)

Σ1(p) Σ1(p) 7→ ιχ∨ (g1) t G (O) , . . . ιχ∨ (gl) t G (O) ,

where g1, . . . , gl ∈ SL2 (C). We use the same name ιχ∨ for a base change

SL2 (K) → G (K)

of the root subgroup inclusion ιχ∨ : SL2 → G. Here we used the fact that for any

ν coweight ν any point in Gr can be presented in a form gt PSL2 (O) for some g ∈

SL2 (C). Then we get that the image of Z is an A-invariant subvariety containing p. Cochar-

∨ ∨ acters in ker χ act trivially on the SL2-subgroup corresponding to χ , so i(Z) ⊂

∨ Xker χ . Since the multiplicity formulas for weights of form ±χ∨ + n~ are the same for ker χ∨ Z and X, Z is an open subspace of X . Moreover, Z is proper over X0 as one can see from the following diagram

Z i X

P kiω Grmω X0 where the bottom inclusion is deﬁned similarly to i. From this we see that Z is a closed, and hence a union of components of Xker χ∨ . P kiω Z has a proper birational morphism to a normal irreducible scheme Grmω , so Z is connected. Thus Z is a component of Xker α∨ . From Lemma 3.3.1 every component of Xker χ∨ is of this form. Keeping track of the T-action on components of Xker χ∨ we can summarize our consideration in the following theorem.

Theorem 3.6.2. If α∨ is a root, then any connected component Z ⊂ Xker χ∨ is iso-

81 morphic to a resolution of a PSL2-slice:

(ω,...,ω) ' Grmω −→ Z

(ω,...,ω) Moreover, this morphism is T-equivariant if A ⊂ T acts on Grmω via character α∨.

Theorem 3.6.3. Let χ∨ be a nontrivial character of A and p, q ∈ XA. Then p and q are in the same component of Xker χ∨ if and only if the following conditions are satisﬁed

1. χ∨ is a multiple of a root α∨,

2. For all i

Σi (p) − Σi (q) ∈ Z α (3.16)

for the coroot α corresponding to root α∨ and ω∨.

Proof. Let p, q be in the same connected component of Xker α∨ . The condition on χ∨ being equal to a root follows from Proposition 3.6.1. Since p, q in a connected component of Xker α∨ , for any equivariant line bundle L the A-weights of ﬁbers over p and q must be equal modulo α∨. By Corollary 3.4.5 we have for all i, 0 < i < l,

∨ K (Σi (p) , •) = K (Σi (q) , •) mod α , which implies

∨ Σi (p) − Σi (q) ∈ α Q.

Now one uses that Σi (p) − Σi (q) is in the coroot lattice to derive

∨ Σi (p) − Σi (q) ∈ α Z.

If χ∨ is a multiple of a root α∨ and p, q ∈ XA satisfy (3.16). Then by construction of the isomorphism of the component Z ⊂ Xker α∨ containing p, one can ﬁnd a point 82 which maps to under i. Thus q is in Z.

This immediately implies the following statement we need later.

∨ ∨ Corollary 3.6.4. Let χ1 and χ2 be Q-linearly independent characters of A. If p, q ∈ A ker χ∨ ker χ∨ X are in the same components in both X 1 and X 2 , then p = q.

83 Chapter 4

Stable Envelopes

In this chapter we introduce the stable envelopes and derive how multiplication by a divisor acts on them.

From now on the base ring for all cohomology is the ﬁeld Q, unless stated otherwise. All the results generalize straightforwardly to the case of any ﬁeld of characteristic 0.

4.1 Preliminaries on Equivariant Cohomology

Computations in equivariant cohomology heavily rely on the well-known localization Theorem[AB].

Theorem 4.1.1. (Atiyah-Bott) The restriction map in the localized equivariant cohomology

∗ ∗ T HT (X)loc → HT X loc is an isomorphism.

∗ ∗ ∗ Here and later we use notation H (X) := H (X) ⊗ ∗ Frac H (pt). T loc T HT(pt) T A If X is proper then there is the fundamental class [X] ∈ H2 dim X (X). By this gives

84 a pushforward to a point homomorphism

∗ ∗ HT (X) → HT (pt) Z γ 7→ γ

X quite often called integration by analogy with de Rham cohomology. Then the localization theorem allows one to compute this using only restrictions to the ﬁxed locus.

Z X Z ι∗ γ γ = Z (4.1) T T e NZ/X X Z⊂X Z where the sum is taken over all components Z ⊂ XT, eT (•) is the equivariant Euler class, NZ/X is the normal bundle to Z in X. If X is not proper the fundamental class lies in Borel-Moore (equivariant) homology

T,BM [X] ∈ H2 dim X (X). This pairs naturally with the equivariant cohomology with compact ∗ support HT,c (X) and gives a pushforward map

∗ ∗ HT,c (X) → HT (pt) .

In general, we have the following natural maps

∗ ∗ HT,c (X) HT (X)

ι∗ ι∗ (4.2)

∗ T ∗ T HT,c X HT X

The vertical arrows are restrictions. XT is closed subvariety, so the inclusion ι: XA → X is proper and the pullback is well-deﬁned in the cohomology with compact support). After the localization the vertical arrows in (4.2) become isomorphisms. Assume now that XT is proper. Then the bottom arrow in (4.2) becomes an isomorphism because the compact support condition becomes vacuous. Then we get an

85 isomorphism

∗ ∼ ∗ HT,c (X) −→ HT (X) which gives the pushforward

∗ ∗ HT (X)loc → HT (pt)loc

We still denote it by R γ. The relation (4.1) still holds. Some authors use it to X define R γ. As a side affect, a priori, the answer one gets by this definition is in X ∗ ∗ HT (pt)loc = Frac HT (pt) because of the division by Euler classes. However, we see ∗ ∗ that the image of non-localized compactly supported classes HT,c (X) is inside HT (pt). ∗ ∗ This defines on HT (X)loc the structure of the Frobenius algebra over HT (pt)loc. In particular, we often use the natural pairing

Z hγ1, γ2iX = γ1 ∪ γ2. X If X is clear we omit it in the notation.

∗ We think of HT (pt) as the ground ring. We usually write the multiplication in ∗ HT (pt) by · to simplify notation. ∗ The multiplication in HT (X) is still denoted by ∪. We also refer to it as classical multiplication as opposed to its deformation, quantum multiplication, which we discuss in the next chapter.

4.2 Stable Envelopes

4.2.1 Assumptions

Here we introduce the stable envelopes and some of their properties following [MO2]. Let us list necessary assumptions for the existence of stable envelopes. Let X be a non-singular algebraic variety and ω ∈ H0 (Ω2X) be a holomorphic symplectic form on X. Let a pair of tori A ⊂ T act on X in such a way, that: 86 1. ω is an eigenvector of T.

2. There is a proper T-equivariant map π : X → X0 to an aﬃne T-variety X0.

3. X is a formal T-variety. [GKM]

We denote the T-weight of ω as ~. Let A ⊂ T be a subtorus preserving ω.

4.2.2 Chambers

In previous chapter we introduced the walls in aQ. For psychological reasons we now consider a larger space

aR = X∗(A) ⊗Z R.

∨ The walls Hα∨ are given by linear equations α = 0 and split this space into disconnected domains called chambers:

[ G aR \ Hα∨ = Ci. α∨

For generic resolutions of slices the conﬁguration is exactly the Weyl chambers from representation theory. Even thought in certain examples we might miss walls corresponding to some roots, we keep calling them Weyl chambers.

4.2.3 Attracting manifolds

Let λ be an A-cocharacter. Then the existence and value of the limit

lim λ (t) · x ∈ XA t→0 does not change if we vary λ without crossing the walls. I.e. it depends only on a choice of chamber C and we denote it by limC x. For any subvariety Z ⊂ X let us denote

AttrC (Z) = {x ∈ X |limC x ∈ Z } . 87 We call this set the attractor of Z. More generally, the attracting manifolds can be deﬁned for a face of smaller dimension in the stratiﬁcation of aR by walls. We will use it later.

4.2.4 Partial order

The attractors deﬁne a partial order on components of XA. We take the relation

0 0 AttrC (Z) ∩ Z 6= ∅ =⇒ Z≥CZ and take its transitive closure. Now we can deﬁne the full attractor as

f G AttrC (Z) = AttrC (Z) . 0 Z ≤CZ This set is closed.

4.2.5 Support and degree in A

∗ We say that a class γ ∈ HT (X) is supported on a closed T-invariant subset Z if the pullback vanishes i∗γ = 0 for the inclusion i : X \ Z,→ X. Notation is the following

supp γ ⊂ Y.

Since A acts on XA trivially,

∗ A ∗ A ∗ H X = H X ⊗ ∗ H (pt) . T T/A HT/A(pt) T

88 ∗ ∗ We have an increasing filtration of HT/A (pt)-modules on HT (pt) induced by the ∗ ∗ A degree filtration on HA (pt). This induces a filtration on HT X . We call this degree degA, the degree in A.

4.2.6 Polarizations

Deﬁnition of stable envelopes depends on a certain sign choice provided by polarizations.

Let C be a face in the stratiﬁcation of aR by walls (possibly not of the highest dimension, so it’s not necessary a chamber). Given Z ⊂ XA a component of the ﬁxed locus, the restriction of the tangent bundle TX of X to Z splits into attracting, constant and repelling parts:

C −C TX |Z = NZ/X ⊕ TZ ⊕ NZ/X

(repelling part is exactly the attraction part with respect to −C, the opposite face). If we consider the normal bundle to Z in X, we get a splitting

C −C NZ/X Z = NZ/X ⊕ NZ/X .

C −C The symplectic form pairs NZ/X and NZ/X , which gives an isomorphism of T- equivariant bundles

C ∨ −C NZ/X ' ~ ⊗ NZ/X

Since the restriction of ~ to A is trivial,

2 A A C A −C codim Z/2 h A −C i e NZ/X = e NZ/X e NZ/X = (−1) e NZ/X .

∗ A polarization on Z is a choice of ∈ HA (Z), such that

2 codim Z/2 A = (−1) e NZ/X .

89 A −C One can easily see that = e NZ/X is a polarization. Moreover, any polarization A −C is ±e NZ/X , so a choice of polarization is just a choice of sign. This sign we denote later by

C, q σZ/X = ∈ {±1} . A −C e NZ/X

The polarization on Z, dual to is deﬁned as

 = (−1)codim Z/2.

A A A polarization on X is a choice of polarization Z for each component Z ⊂ X .

4.2.7 Deﬁnition of stable envelopes

Now we are ready to give the deﬁnition of stable envelopes. Fix a chamber C and a polarization on XA.

∗ Theorem 4.2.1. (Maulik-Okounkov) There exists a unique map of HT (pt)-modules

 ∗ A ∗ StabC :HT X → HT (X)

A ∗ such that for any component Z ⊂ X and γ ∈ HT/A (Z), the stable envelope Γ = StabC (γ) satisﬁes

f 1. supp Γ ⊂ AttrC (Z),

∗ A −C ∗ 2. ιZ Γ = ±e Np ∪γ, where the sign is ﬁxed by the polarization: ιZ Γ|A = Z ∪ γ|A,

∗ 1 0 3. degA ιZ0 Γ < 2 dim X for any Z

A ∗ A In the spaces of our interest X is ﬁnite discrete, so HT X has a basis 1p of classes those only non-zero restriction is 1 at p ∈ XA. To simplify notation we write

 StabC (p)

90 instead of

 StabC (1p) .

The classes StabC (p) may be thought of as a refined version of (the Poincaré-dual h i class to) the fundamental cycle ± AttrC(p) . This explains the first two conditions in the Theorem 4.2.1. The last condition is required to ensure these classes behave well under deformations of the symplectic variety.

4.2.8 First properties

Let us recall some results about stable envelopes.

∗ From the deﬁnition of stable envelopes the restriction matrix ιqStabC (p) is triangular with respect to order

Theorem 4.2.2. Let X be a symplectic resolution. Then for any components Z,Z0 ⊂

A 0 ∗ X , Z

∗ ∗ 0 ιZ0 StabC (γ) ∈ ~ HT (Z ) .

Then if one restricts StabC (p) to the torus A (sets ~ = 0), it has a non-zero coeﬃ- ∗ cient only at p. Since the ﬁxed point is a basis in the localized cohomology HT (X)loc, stable envelopes also form a basis in the localized cohomology.

 Another nice property is that the basis, dual to {StabC (p)}, is given by taking the stable envelopes for the opposite chamber and polarization, −C and .

Theorem 4.2.3. For p, q ∈ XA, we have

 Stab−C (q) , StabC (p) = δq,p,

91 where δp,q is the Kronecker delta

  1, if p = q, δp,q =  0, otherwise.

∗ Remark. Note, in particular, that the pairing lies in non-localized cohomology HT (pt) since the class

Stab−C (q) ∪ StabC (p) has compact support because of the support conditions on the stable envelopes and the choice of opposite chambers. That is, the support is π−1 (x) ⊂ X, a proper subset since π is proper. Similar arguments are crucial later for our computation.

4.2.9 Steinberg correspondences

Let us ﬁrst introduce the notion of Lagrangian correspondence.

Recall that a subvariety L ⊂ M of an algebraic symplectic variety (M, ωM ) is

∗ called Lagrangian if the restriction ιZ ωM of the symplectic form vanishes and dim Z = 1 2 dim M.

Now let (X, ωX ) and Y be two algebraic symplectic varieties. We equip X × Y with

∗ ∗ a symplectic form pX ωX − pY ωY (here pX and pY are natural projections of X × Y to X and Y ). Then we say that L ⊂ X × Y is a Lagrangian correspondence if it’s Lagrangian with respect to the chosen symplectic structure. If both X and Y are T-varieties and L is T-invariant and proper over X, this deﬁnes a map ∗ ∗ pY ∗ pX,∗ ∗ ΘL :HT (Y ) −→ HT (L) −−→ HT (X) .

92 We say that a Lagrangian correspondence L ⊂ X × Y is Steinberg if there exist proper T-equivariant maps Y

X V to a T-equivariant aﬃne variety V such that

L ⊂ X ×V Y.

Let (M, ωM ) be an algebraic symplectic variety with an A-action preserving ωM , and let L ⊂ M be an A-invariant Lagrangian subvariety. Choose a polarization on

∗ A M. Then we can deﬁne the Lagrangian residue ResZ L ∈ HT M to be the unique class such that

∗ ιM A [L] = ∪ ResZ L + ...

∗ where dots stand for terms of smaller A-degree and [L] is the pushforward of 1 ∈ HT (L) along the inclusion L,→M (i.e. the Poincar´edual class to the fundamental cycle of L). Now let M = X × Y with algebraic symplectic varieties X and Y . Then M A =

A A A A X × Y . Fix polarizations X and Y on X and Y respectively. Then we deﬁne

A A the polarization on X × Y to be X Y .

Let Θ be the map given by a Steinberg correspondence L, and ΘA the map by

ResXA×Y A L (this depends on choice X and Y as above). Then we have the following theorem.

Theorem 4.2.4. The diagram

∗ A StabC ∗ HA Y HA (Y )

ΘA Θ

∗ A StabC ∗ HA X HA (X)

commutes. The polarization above is Y , below it is X .

93 4.2.10 Torus restriction

Let C be a chamber and let C0 be a face of C of some dimension. Let A0 ⊂ A be a connected subtorus, such that X (A0) ⊗ = a0 = span C0 ⊂ a . Then there’s a ∗ Z R R R projection of the cone C to a /a0 which we denote by C00. R R Moreover, ﬁx polarizations and 0 on XA and XA0 respectively. This induces a polarization 00 of XA as a subspace of XA0 by the following. Let p ⊂ XA and Z ⊂ XA0 , such that p ⊂ Z. If we present p as a product of A-weight

Y ∨ p = α , then we can write

∗ 0 Y ∨ ι = ± α . p Z α∨| 6=0 A0 A0

Then we set

00 Y ∨ = ± α p α∨| =0 A0 A0/A

∗ 0 with the same sign as for ιpZ . One can rewrite this in terms of signs:

C, C0,0 C00,00 σp/X = σZ/X σp/Z

A/A0 −C00 A −C using the fact that the polarization e Np/Z is induced in this sense by e Np/X A0 −C0 and e NZ/X . With these choices we have the following statement.

Theorem 4.2.5. The diagram

Stab ∗ A C ∗ HT X HT (X)

00 0 Stab StabC00 C0 ∗ A0 HT X commutes.

94 2 4.3 Stable Envelopes restrictions modulo ~ : A1 type

In this case we have enough recursive relations similar to the ones which apper in [S2]. They arise from action by certain Steinberg correspondences. These correspondences appeared in the context of the convolution aﬃne Grassmannian of type A1 at least in in [CK]. Even before that these correspondences were studied by H. Nakajima[N3], since the slices in type A appear to have representation as Nakajima quiver varieties[MV3].

4.3.1 Steinberg correspondences in type A1

In type A1 the connected group of the adjoint type is G = PSL2, the fundamental coweight ω is minuscule, the simple root is α = 2ω. Let us denote the slice as X0 =

lω Grkω. One can think ≥ 0 to have kω dominant, but it won’t aﬀect the computations.

We do assume l > 1 and |k| < l, otherwise X0 is just a point or an empty space. Then λ X = Grkω is the resolution of X0, where λ is an l-tuple

λ = (ω, . . . , ω) . | {z } l

First let us introduce the following subvarieties of X. Recall that by deﬁnition a point in X is an (l + 1)-tuple (L0,L1,...,Ll) of points in Gr (here we use the trick

L0 = e · G (O) as before). Then for every i, 1 ≤ i < l deﬁne

i X = {(L0,L1,...,Ll) ∈ X|Li−1 = Li+1} .

I.e. Xi is the equalizer of the projections to (i − 1)th and (i + 1)th factor.

(l−2)ω ν We will also need a slice Y0 = Grkω . In case l > 2, its resolution is Y = Grkω, where ν is an (l − 2)-tuple ν = (ω, . . . , ω) . | {z } l−2

In case l = 2, we have Y0 = Y = {e · G (O)} is the point if k = 0, and Y0, Y are

95 empty spaces if k 6= 0. Then there are the following natural T-equivariant ”forgetful” projections:

i πi : X → Y

(L0,...,Ll) 7→ (L0,...,Li−2,Li−1 = Li+1,Li+2,...,Ll).

In English, we forget Li and identify Li−1 and Li+1. It’s an easy check that for l > 2 the resulting (l − 2)-tuple satisﬁes the deﬁning relation of Y (or is the only point e · G (O) on the case l = 2).

Proposition 4.3.1. Xi → Y is a P1-bundle.

Proof. Since Xi adds one more element of Gr to a sequence from Y , it is natural to consider Xi → Y as a subbundle of the trivial ﬁbration

Xi Y × Gr

Y deﬁned by the equations ω Li−1 −→ Li

where L0,...,Li−2,Li−1 = Li+1,Li+2,...,Ll are projections of the point in Y , and Li is an element in Gr (the one we want to reconstruct). The ﬁber is P1 and this ﬁbration is locally trivial.

Recall that X and Y have symplectic forms ωX and ωY as described in Chapter2.

Proposition 4.3.2. The natural pullbacks of the symplectic forms from X and Y to Xi are equal:

∗ ∗ ιXi ωX = πi ωY

i i 1 1 i i Then we get that X ×Y X is a P × P -bundle over Y . In particular, X ×Y X is 96 smooth. We have the following commutative diagram

Y πi Xi X

Y0 X0

The vertical arrows are natural maps from the deﬁnition of resolutions, the inclusion

(l−2)ω lω Y0 ,→ X0 is induced by the inclusion of closed cells Gr ,→ Gr .

i i i i This gives that, ﬁrst of all, there’s a natural inclusion X ×Y X ⊂ X ×X0 X , where the second ﬁber product is with respect to the dashed line on the diagram. This is because the dashed line factors through πi. Secondly, we have a natural inclusion

i i i X ×X0 X ⊂ X ×X0 X because the inclusion X ,→ X commutes with the maps to X0 as the diagram shows. Finally, because all morphisms are T-equivariant, the inclusions are also T-equivariant.

i i i i Let us denote L = X ×Y X . The statements above allows us to write L ⊂

X ×X0 X.

Example 4.3.1. In the case of the resolution of An-singularity, i.e. when l = n + 1,

(l−2)ω i k = l − 1, we have Y0 = Gr(l−2)ω = pt, so Y = pt. The subvariety X ⊂ X is an i 1 i i irreducible component Ei of the exceptional divisor E, X ' P . Then X ×Y X is 1 1 i i P × P , and the inclusion X ×Y X ⊂ X × X is just Ei × Ei ⊂ X × X. Since E maps i i to one point in X0, we even have X ×Y X ⊂ X ×X0 X.

i Proposition 4.3.3.L ⊂ X ×X0 X is a Steinberg correspondence.

i Proof. First, we have L ⊂ X ×X0 X for a proper morphism X → X0 as shown before. So we only need to check if Li is Lagrangian.

i Let p1, p2 : L → X be the projections to the ﬁrst and second factor in X ×X. Then we need to check

∗ ∗ p1ωX − p2ωX = 0.

We also have the natural inclusion Li ⊂ Xi × Xi and corresponding projections 97 i i p˜1, p˜2 : L → X . These are related to pk’s by

pk = ιXi p˜k.

i i i By deﬁnition L = X ×Y X , so

i i π p˜1 = π p˜2.

Then from the Proposition 4.3.2 we can write

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ p1ωX − p2ωX = (˜p1 − p˜2)(ιXi ωX ) = (˜p1 − p˜2)(πi ωY ) = 0, which ﬁnishes the proof.

Let us discuss what are the elements (Xi)A and what are the ﬁbers of (Xi)A → Y A.

Proposition 4.3.4.

1. A point p ∈ XA is in (Xi)A if and only if

Σi−1 (p) = Σi+1 (p) .

A i A 2. Every point q ∈ Y has exactly two preimages q+, q− ∈ (X ) uniquely determined by the following conditions:

Σj (q±) = Σj (q) for all j < i,

Σj (q±) = Σj−2 (q) for all j > i,

Σi (q±) = Σi−1 (q) ± ω.

Proof. The both parts follow directly from the deﬁnitions of Xi, Xi → Y and the description XA.

The following statements are useful in localization computations.

98 i A A Proposition 4.3.5. Let q+, q− ∈ (X ) and q ∈ Y be as in Proposition 4.3.4. Then we have the following formulas

∗ T ∨ ∨ i ιq± e NX /X = ∓ α + α , Σq± (i) ~

2. T e N i q±/X = ± (α∨ + hα∨, Σ (i − 1)i ) T q ~ e Nq/Y

Proof.

1. Let us check what eﬀect on the tangent space the condition Li−1 = Li+1 has. It gives an extra relation that the restrictions to (i − 1)th and (i + 1)th components must be equal. However, due to adjacency conditions with ith component, all but one weight vectors must be already equal in X. The only new condition is on the part of weight

∨ ∨ ∓ α + α , Σq± (i) ~ .

Instead of two independent components, we now have one (note that it can’t be that components from both sides must be 0). This gives the relation of the Euler classes.

2. By comparison of tangent spaces of Xi and Y (or even at the weight multiplicities) we see that Xi has one more tangent vector of weight

∨ ∨ ± (α + hα , Σq (i − 1)i ~) .

This proves the statement.

i A i A If p ∈ (X ) we denote by sip the other point in (X ) which has the same image

99 under Xi → Y . In notation of Proposition 4.3.4

siq+ = q−,

siq− = q+.

If p ∈ (X\Xi)A, then we deﬁne

sip = p.

A One can easily see that si are involutions on X .

The motivation for this notation is that this operation swaps ∆i (p) and ∆i+1 (p) as an adjacent transposition. This leads to the following statement.

A Proposition 4.3.6. Any two points in X can be related by a sequence of si’s.

Proof. Since for any p ∈ XA we have

X ∆i (p) = Σl (p) = kω

∆i (p) = ±ω, then any two points in XA diﬀer only by permuting positions of +ω and −ω in ∆ (p). Any permutation can be written as a product of adjacent transpositions, which gives a relation by a sequence of si’s.

The correspondence Li is T-invariant, so it deﬁnes a map which we denote by

i ∗ ∗ Θ :HT (X) → HT (X). A polarization on XA induces a polarization on XA × XA as in Theorem 4.2.4.

i i A Proposition 4.3.7. The Lagrangian residue ResXA×XA L is supported on (L ) . The A-equivariant restrictions to (p, q) ∈ (Li)A is the following

 −1, if p = q, ∗ i  ι(p,q)ResXA×XA L = p  , if p 6= q. q 100 Proof. By dimension count

∗ i 0 ι(p,q)ResXA×XA L ∈ HA (pt) = Q

Restricting the deﬁnition of of the Lagrangian residues to the subtorus A, we get

∗ i ∗ i ι L = ι Res A A L (p,q) A p q (p,q) X ×X

So A ∗ i e N(p,q)/Li ι(p,q)ResXA×XA L = pq

One can compute

 (−1)dim Y/2(α∨)dim X , if p = q, A  e N(p,q)/Li =  dim Y/2 ∨ dim X −(−1) (α ) , if p 6= q. and

p dim X/2 ∨ dim X p pq = qq = (−1) (α ) q q

This gives the statement of the Proposition.

i ∗ A ∗ A We denote by Res Θ :HT X → HT X the map given by the Lagrangian

i residues ResXA×XA L .

4.3.2 Recursion

The main ingredient for the recursion is interaction of Θi with the stable envelopes and the ﬁxed points basis.

i Proposition 4.3.8. The correspondence Θ has the following (left) action on StabC (p), p ∈ XA:

i p Θ ◦ StabC (p) = − StabC (p) + StabC (sip) . sip

101 Moreover, p ∈ {±1} . sip Proof. By the Theorem 4.2.4 we get

i i Θ ◦ StabC (p) = StabC Res Θ 1p

i A i If p ∈ (X\X ) , both Res Θ 1p = 0 and

 p − StabC (p) + StabC (sip) = 0, sip

i A since sip = 0 by deﬁnition. This proves the Proposition for all p ∈ (X\X ) . Now assume p ∈ (Xi)A. FromProposition 4.3.7 and Proposition 4.3.4, we have

 i p StabC Res Θ 1p = StabC −1p + 1sip sip p = − StabC (p) + StabC (sip) . sip

This proves the formula for p ∈ (Xi)A.

p ∨ dim X/2 The remark about ∈ {±1} follows from the fact that q = ± (α ) in type sip A A1 for any q ∈ X .

∗ There’s a similar fact for the restrictions ιq.

i ∗ Proposition 4.3.9. The correspondence Θ has the following (right) action on ιq, q ∈ XA: α∨ + hα∨, Σ (q)i ι∗ ◦ Θi = i ~ −ι∗ + ι∗ q ∨ ∨ q siq α + hα , Σi−1 (q)i ~ Proof. This is a straightforward localization computation.

In the recursive relations it’s more convenient to work with reduced stable envelopes

102 Stab, i.e. stable envelopes with a diﬀerent normalization:

1 StabC (p) = StabC (p) . p

These classes no longer depend on the polarization. The only restriction which is not divisible by ~ is the diagonal element

∗ ιpStabC (p) = 1 mod ~.

As a side eﬀect, these classes lie only in localized cohomology, but we won’t use this classes when we need non-localized statements. One can rewrite all recursions in terms of original Stab, just inserting appropriate polarizations.

The recursion relation appear from application of both Proposition 4.3.8 and Propo- sition 4.3.9 to the following expression

∗ i ιq ◦ Θ ◦ StabC (p) .

This gives the following relations similar to the ones in [S2].

Proposition 4.3.10. Let p, q ∈ XA, C be a Weyl chamber. Then

hα∨, ∆ (q)i α∨ + hα∨, Σ (q)i ι∗Stab (s p) = − i ~ ι∗Stab (p) + i ~ ι∗ Stab (p) q C i ∨ ∨ q C ∨ ∨ siq C α + hα , Σi−1 (q)i ~ α + hα , Σi−1 (q)i ~ and

hα∨, ∆ (q)i α∨ + hα∨, Σ (q)i ι∗ Stab (p) = i ~ ι∗Stab (p) + i−1 ~ι∗Stab (s p) siq C ∨ ∨ q C ∨ ∨ q C i α + hα , Σi (q)i ~ α + hα , Σi (q)i ~

∗ Proof. From Proposition 4.3.9 and Proposition 4.3.8 we have two ways to express ιq ◦

103 i Θ ◦ StabC (p):

α∨ + hα∨, Σ (q)i ι∗ ◦ Θi ◦ Stab (p) = i ~ −ι∗Stab (p) + ι∗ Stab (p) q C ∨ ∨ q C siq C α + hα , Σi−1 (q)i ~ ∗ i ∗ p ∗ ιq ◦ Θ ◦ StabC (p) = −ιqStabC (p) + ιqStabC (sip). sip

By algebraic manipulations with these expressions we get the relations in the Propo- sition.

Remark. One can use this relation to recursively compute the stable envelopes, starting either from p for which all restrictions are known (i.e. the minimal p with respect to C), or starting from q restriction to which are all known (i.e. the maximal q with respect to C).

Remark. These relations modulo ~ just give

ι∗ Stab (p) = ι∗Stab (s p) mod . siq C q C i ~

So if we know that only the diagonal restriction is 1 mod ~ and all other are zero (say, from StabC (p) for the minimal p with respect to C), then all other stable envelopes have this property. We know this from the deﬁnition of the stable envelopes combined with the property of oﬀ-diagonal terms of a symplectic resolution. Here we derive it just from commutativity with Steinberg correspondences and knowing the stable envelope of the minimal point.

4.3.3 Formulas for stable envelopes restrictions: type A1

Since for our main application we are interested in the stable envelopes modulo ~2, let us present the following version of 4.3.11.

Proposition 4.3.11. Let p, q ∈ XA, C be a Weyl chamber, 1 ≤ i < l. Then

∗ ∗ ∨ ~ 2 ι StabC (s p) = ι StabC (p) + hα , ∆ (q)i [δ − δ ] mod . q i siq i p,siq p,q α∨ ~ 104 Proof. This is a straightforward computation using that

∗ ιqStabC (p) = δp,q mod ~. as discussed above.

Let us also introduce the following numerical quantity for p ∈ XA

1 X |p| = hα∨, Σ (p)i , C 2 i i

∨ where αC is the positive root with respect to C. Since all σpi are coweights, the pairings are integers, so |p|C ∈ Z. It is easy to check that

A This implies that the minimal |p|C is for the minimal point of X with respect to C, C C pmin. I.e. pmin is the unique point of the following form:

C ∆ pmin = (−ωC,..., −ωC, ωC, . . . ωC) ,

where ωC is the minimal strictly dominant coweight with respect to C.

A l−k l+k Remark. One can identify points p ∈ X with partitions in a square 2 × 2 . Then

|p|C is the size of the partition for one chamber C = C+, and the size of the complement for the opposite chamber, C = C− (both up to adding a constant). Moreover, the order

Now we are ready to prove the main result in type A1.

A Theorem 4.3.12. Let p, q ∈ X be two ﬁxed points. Let αC be the positive coroot with

∨ respect to C and αC the corresponding root. Then

105 1. If p = q, then

 ∗ ~ 2 ιpStabC (p) = p 1 + (|p|C + C) ∨ mod ~ , αC

where the constant C ∈ Z does not depend on p.

2. If there are i, j, 0 ≤ i < j ≤ l such that

∆i (q) = ∆i (p) − αC,

∆j (q) = ∆j (p) + αC, (4.3)

∆k (q) = ∆k (p) for all k, k 6= i, k 6= j,

then

∗ p 2 ιqStabC (p) = ~ ∨ mod ~ . αC

3. Otherwise

∗ 2 ιqStabC (p) = 0 mod ~ ,

Proof. We work with Stab, the statements for Stab are recovered by a simple multiplication on polarization .

The proof is by induction on |p|C. Base case

C C The minimal value of |p|C is pmin C as mentioned above. Moreover, pmin is the unique point with this weight.

Since pmin is the minimal point with respect to C, for all q 6= pmin

∗ C ιqStabC pmin = 0.

This agrees with the statement of the theorem, because there is no q satisfying (4.3)

C for i < j: one can only subtract αC from ∆i pmin = ωC which are in the second half C C of the sequence ∆ pmin , and αC can be added to ∆j pmin = −ωC which are in the

106 C ﬁrst half of ∆ pmin . −C The check of the diagonal is trivial: the weights of Np/X are of form

 ∨ ∨ ~ −(αC + n~) = −αC 1 + n ∨ , n ∈ Z αC

So T −C ∨ dim X/2 ~ 2 e Np/X = (−αC ) 1 + N ∨ mod ~ ,N ∈ Z. αC This implies

∗ C ~ ι C StabC pmin = 1 + N . pmin ∨ αC

After a shift of N by an integer constant |pmin|C we get the statement of the Theorem. Thus the base case is proved.

Inductive step

ﬁnd a ωC followed by −ωC:

∆m (p) = ωC,

∆m+1 (p) = −ωC

for some m. Then smp

|smp|C = |p|C − 1 = N, because

Σ(smp) = (Σ0 (p) ,..., Σm−1 (p) , Σm (p) − αC, Σm+1 (p) ,..., Σl (p)) .

We have 3 cases of q:

107 1. q = p.

By Proposition 4.3.11 we have

ι∗Stab (p) = ι∗ Stab (s p) + hα∨, ∆ (p)i ~ mod 2 p C smp C m m α∨ ~ = ι∗ Stab (s p) + hα∨, ∆ (p)i ~ mod 2 smp C m C m ∨ ~ αC = ι∗ Stab (s p) + ~ mod 2 smp C m ∨ ~ αC

We know

ι∗ Stab (s p) = 1 + (|s p| + C) ~ mod 2 smp C m m C ∨ ~ αC ~ 2 = 1 + (|p|C − 1 + C) ∨ mod ~ αC

by the induction assumption.

This proves this case.

2. q = smp.

In this case the Proposition 4.3.11 says

ι∗ Stab (p) = ι∗Stab (s p) − hα∨, ∆ (s p)i ~ mod 2. smp C p C m m m α∨ ~

∗ We have ιpStabC (smp) = 0 since smp

∆m (smp) = ∆m+1 (p) = −ωC,

∨ hα , ∆m (smp)i = −1.

Finally,

108 ι∗ Stab (p) = ~ mod 2. smp C α∨ ~

Note that such a pair of q, p satisfy the property (4.3) for i = m and j = m + 1. After restoring from the normalization, we get the statement of the Theorem.

3. Both q 6= p and q 6= smp.

The Proposition 4.3.4 gives in this case

∗ ∗ 2 ιqStabC (p) = ιsmqStabC (smp) mod ~ .

Note that if smp and smq are satisfy relations (4.3), then p and q are also satisfy

these, possibly for diﬀerent i, j. The reason is that sm only permutes components

of ∆ (smp) and ∆ (smq) in the same way. The only think we have to check is that

the order of ∆i (smp) and ∆j (smp) is not reversed. Since sm permutes ∆m (smp)

and ∆m+1 (smp), this happens only if i = m, j = m + 1. But then q = smp, which is prohibited in this case.

Applying the induction assumption

∗ ~ 2 ιqStabC (p) = ∨ mod ~ αC

if p and q are related by (4.3), and

∗ 2 ιqStabC (p) = 0 mod ~

otherwise.

Combining the results from these three cases, we prove the inductive step.

Remark. The constant C in the formula for the diagonal can be found at any point p comparing with the result on weight multiplicities obtained in the previous chapter. 109 For example, at the minimal or the maximal point. In what follows we’re not interested in the diagonal restrictions, so we don’t write the computation of C.

4.4 Stable Envelopes restrictions modulo ~2: general case

4.4.1 Wall-crossing for stable envelopes

The ﬁrst fact we need to reduce the general case to type A1 is the wall-crossing phenom- ena for restrictions of stable envelopes modulo ~2. It is a straightforward consequence of the torus restriction Theorem 4.2.5 and a close relative of R-matrices in [MO2].

Theorem 4.4.1. Let X, A, be as in the deﬁnition of the stable envelopes and XA

0 discrete. Pick two Weyl chambers C+ and C− sharing a face C (of any codimension). This gives associated subtorus A0 ⊂ A, i.e. such connected subgroup that a0 = X (A0) ⊗ = span C0 ⊂ a . For any p, q ∈ XA, p 6= q we have the following R ∗ Z R R

1. If p and q are in diﬀerent components of XA0 then

∗ ∗ 2 ιqStabC+ (p) = ιqStabC− (p) mod ~

2. If p and q are in the same component of Z ⊂ XA0 then

∗ ∗ 0 h ∗ 00 ∗ 00 i 2 ι Stab (p)−ι Stab (p) = ι Stab 00 (p) − ι Stab 00 (p) mod (4.4) q C+ q C− q q/Z C+ q/Z C− ~

∗ 00 ∗ 00 A0 where ι Stab 00 (p) and ι Stab 00 (p) are the stable envelopes on X as in the q/Z C− q/Z C+ torus restriction Theorem 4.2.5. The polarization 00 on XA as a subspace of XA0 is induced by 0 on XA0 and on XA as in Theorem 4.2.5. On the RHS we use

∗ notation ιq/Z to emphasize that we restrict to q from Z, not from X.

Proof. The torus restriction Theorem 4.2.5 gives us that the following diagram is com-

110 mutative Stab ∗ A C+ ∗ HT X HT (X) . 00 0 Stab Stab C00 C0 + ∗ A0 HT X

0 00 00 The induced polarizations , and the projected chamber C+ are the same as in Theorem 4.2.5. Then we have

∗ ∗ ∗ 0 00 ι Stab (p) = ι ι Stab 0 ◦ Stab 00 (p). q C+ q/Z Z C C+

Similar holds if we replace C+ by C−:

∗ ∗ ∗ 0 00 ι Stab (p) = ι ι Stab 0 ◦ Stab 00 (p). q C− q/Z Z C C−

This gives

∗ ∗ ∗ ∗ 0 h 00 00 i ι Stab (p) − ι Stab (p) = ι ι Stab 0 Stab 00 (p) − Stab 00 (p) . q C+ q C− q/Z Z C C+ C−

A0 00 If the fixed point if p is in the component Zp ⊂ X , then the class Stab 00 (p) − C+ 00 Stab 00 (p) is supported on Zp. One gets this from the definition of the stable envelopes C− 0 and that zp contains refined attractor to p via any evolution by A/A .

A0 Denote the component of X containing q as Zq.

Let us ﬁrst consider the case when p and q are in diﬀerent components of XA0 , i.e.

Zp 6= Zq. Then

• The class

00 00 Stab 00 (p) − Stab 00 (p) C+ C−

is divisible by ~ since the diagonal contributions cancel and the oﬀ-diagonal contributions are divisible by ~ because X is a symplectic resolution.

111 ∗ A0 • For any class γ ∈ HT X supported on Zp the restriction

∗ ιZq StabC (γ)

is divisible by ~ because X is a symplectic resolution.

∗ Then by HT (pt)-linearity of the stable envelopes and restrictions we get that

∗ ∗ ιqStabC+ (p) − ιqStabC− (p) is divisible by ~2 which gives the ﬁrst statement of the theorem.

Now let p and q be in the same component Z = Zp = Zq. Then by one of the deﬁning properties of the stable envelopes we get

 C0,0 T −C0 h 00 00 i Stab (p) − Stab (p) = σ e N Stab 00 (p) − Stab 00 (p) . C+ C− q/Z Z/X C+ C−

Restricting to q we get

∗ ∗ C0,0 T −C0 h ∗ 00 ∗ 00 i ι Stab (p) − ι Stab (p) = σ e N ι Stab 00 (p) − ι Stab 00 (p) q C+ q C− q/Z q/X q/Z C+ q/Z C−

0 h ∗ 00 ∗ 00 i 2 = ι Stab 00 (p) − ι Stab 00 (p) mod . q q/Z C+ q/Z C− ~

0 00 00 Remark. In the case when C is codimension one (a ”wall”) we have C+ = −C− and there’s only one non-zero contribution on the RHS of (4.4).

Remark. Informally, the Theorem 4.4.1 says that if one ﬁxes p, q ∈ XA (p 6= q) and a

∗ 2 polarization variation of the chamber C doesn’t change ιqStabC (p) modulo ~ , unless one crosses a wall C0 where p and q are in the same component of Xspan C0 .

112 4.4.2 Formulas for stable envelopes restrictions: general case

λ Let us return back to the case X = Grµ. Now we can prove the following generalization of Theorem 4.3.12.

Theorem 4.4.2. Given ﬁxed points p, q ∈ XA, p 6= q, a Weyl chamber C and a polarization we have the following restriction formula

1. If there are i, j, 0 ≤ i < j ≤ l and a coroot α positive with respect to C, such that

∆i (q) = ∆i (p) − α,

∆j (q) = ∆j (p) + α, (4.5)

∆k (q) = ∆k (p) for all k, k 6= i, k 6= j,

then ι∗Stab (p) = ω ~ mod 2, q C p,q α∨ p ~

0 ker α∨ where the coeﬃcient ωp,q can be found from any polarization on X :

0 q ωp,q = 0 . p

0 This coeﬃcient does not depend on the choice of . In general, ωp,q is not in Q.

2. Otherwise

∗ 2 ιqStabC (p) = 0 mod ~ .

∗ Proof. Let q

∗ ∗ Then it implies that ιqStab−C (p) = 0. We’ll use this to ﬁnd ιqStabC (p) by wall- crossing. One can connect −C to C by a sequence of adjacent chambers

−C = C0, C1,..., Ck−1, Ck = C. 113 Moreover, if H1,...,Hk are the hyperplanes which split the space into chambers, then one can choose the sequence to be such that each Hi is crossed exactly once (for example, make a choice by following a straight line connecting a generic point in −C with a generic point in C).

∨ ∨ In our case all the hyperplanes have form Hα = ker α for a root α by Proposi- tion 3.6.1.

By using the Theorem 4.4.1 for the chain of chambers −C = C0, C1,..., Ck−1, Ck = C

2 ∨ we see, that the only change module ~ appears when we cross the wall Hα = ker α such that p and q are in the same component of Xker α∨ . By Corollary 3.6.4 we know that there at most one such wall. To have exactly one wall, p and q most satisfy conditions from Theorem 3.6.3. Since otherwise

∗ ∗ 2 ιqStabC (p) = ιqStab−C (p) mod ~

2 = 0 mod ~ we assume that conditions in Theorem 3.6.3 are satisﬁed for one root α∨. Then by Theorem 4.4.1

∗ ∗ ∗ ιqStabC (p) = ιqStabC (p) − ιqStab−C (p)

0 h ∗ 00 ∗ 00 i 2 = q ιq/Z StabC00 (p) − ιq/Z Stab−C00 (p) mod ~

0 ∗ 00 2 = qιq/Z StabC00 (p) mod ~

00 ∨ ker α∨ where C is the projection of C to aR/ ker α , Z ⊂ X is the component containing both p and q, 0 is any polarization on Z, 00 is induced by and 0.

By identiﬁcation of Z with the resolution of type A1 slice in Theorem 3.6.2 and the result for A1-type in Theorem 4.3.12 we get that

00 ∗ 00 p 2 ι Stab 00 (p) = mod q/Z C ~α∨ ~

114 if p and q satisfy conditions (4.5) and is 0 modulo ~2 otherwise. This gives 00 ι∗Stab (p) = 0 p mod 2. q C ~ q α∨ ~

Recall that by deﬁnition of the induced polarization we have

0 00 p = pp.

Then 0 ∗ q ~ 2 ιqStabC (p) = 0 ∨ p mod ~ . p α The factor 0 q 0 p does not depend on a choice of polarization 0 since all polarizations diﬀer only by ±1 on Z and the extra sign would cancel. This proves the statement.

Remark. To see that ωp,q is not in Q it’s enough to consider the ﬁrst type A2 case, T ∗P2, which is one of the examples with the Picard rank one.

4.5 Classical Multiplication

4.5.1 Stable envelopes and classical multiplication

Let us ﬁrst discuss in general relation between classical multiplication of a stable envelope by a divisor and restrictions of stable envelopes.

Proposition 4.5.1. Let X, T, A, ~, , C be as in the deﬁnition of the stable envelopes. Assume moreover that X is a symplectic resolution with XA ﬁnite.

115 A L 1. For any L ∈ PicT (X) and p, q ∈ X , q

T ∗ T X L c1 (L) ∪ StabC (p) = ιpc1 (L) · StabC (p) + ~ cp,q StabC (q) . (4.6) q

L 2. The number cp,q ∈ Q is uniquely reconstructed from the restriction of the stable ∗ 2 envelope ιqStabC (p) modulo ~ :

∗ ∗ A ∗ A L ιqStabC (p) ιqc1 (L) − ιpc1 (L) cp,q = . ~ A q

Proof.

n o n o 1. Since Stab−C (q) is the dual basis to StabC (q) , we have

T X T c1 (L) ∪ StabC (p) = Stab−C (q) , c1 (L) ∪ StabC (p) StabC (q) q∈XA

By localization we can compute the coeﬃcients

X ι∗Stab (q) · ι∗cT (L) · ι∗Stab (p) Stab (q) , cT (L) ∪ Stab (p) = x −C x 1 x C −C 1 C T e Nx/X x∈XA (4.7)

First of all, by the properties of the stable envelopes the only nonzero terms are

the ones where q≤Cx≤Cp. In particular, this coeﬃcient is zero unless q≤Cp.

If q = p there’s only one term in (4.7):

ι∗Stab (p) · ι∗cT (L) · ι∗Stab (p) Stab (p) , cT (L) ∪ Stab (p) = p −C p 1 p C = ι∗cT (L) −C 1 C T p 1 e Np/X

If q 6= p, we know that for every term in (4.7) at least one of the stable envelope

restrictions is divisible by ~ since X is a symplectic resolution, see Theorem 4.2.2.

On the other hand, by dimension count the coeﬃcients we have

 T 2 Stab−C (q) , c1 (L) ∪ StabC (p) ∈ HT (pt) 116 ∗ The pairing lies in non-localized cohomology HT (pt) since the class

 T Stab−C (q) ∪ c1 (L) ∪ StabC (p)

has compact support. This follows from the support conditions on the stable

−1 λ envelopes and the choice of opposite chambers: the support is mλ ([µ]) ⊂ Grµ =

X, a proper subset since mλ is proper.

Thus

 T 0 Stab−C (q) , c1 (L) ∪ StabC (p) ∈ ~ HT (pt) = ~Q.

Denoting

L 1 T cp,q = Stab−C (q) , c1 (L) ∪ StabC (p) ~ we get the formula (4.6).

2. Let us restrict the formula (4.6) to a point q.

∗ T ∗ T ∗ X L ∗ 0 ιq c1 (L) ∪ StabC (p) = ιpc1 (L) · ιqStabC (p) + ~ cp,p0 ιqStabC (p ). 0 p

The LHS equals

∗ T ∗ ιqc1 (L) · ιqStabC (p).

So we have

∗ T ∗ T ∗ X L ∗ 0 ιqc1 (L) − ιpc1 (L) · ιqStabC (p) = ~ cp,p0 ιqStabC (p ). 0 p

∗ 0 0 2 The terms ιqStabC (p ) are divisible by ~ unless p = q. Then we have modulo ~

∗ T ∗ T ∗ L ∗ 2 ιqc1 (L) − ιpc1 (L) · ιqStabC (p) = ~cp,qιqStabC (q) mod ~ .

117 This implies

ι∗cT (L) − ι∗cT (L) · ι∗Stab (p) cL = q 1 p 1 q C p,q ∗ ιqStabC (q) ~ A

ι∗cA (L) − ι∗cA (L) · ι∗Stab (p) = q 1 p 1 q C q ~ A

as desired.

4.5.2 Main result on classical multiplication

To formulate the main result on classical multiplication it’s convenient to introduce auxiliary operators. First we deﬁne two families of diagonal operators

i ∗ A ∗ A H :HA X → HA X (4.8) 1 7→ K (∆ (p) , •) + ~K (∆ (p) , µ) 1 p i 2 i p

ij ∗ A ∗ A Ω0 :HA X → HA X (4.9)

1p 7→ K (∆i (p) , ∆j (p)) 1p

Then we introduce a family of strictly triangular operators. Let i, j be such that 0 ≤ i, j ≤ l, α∨ be a root and be a polarization. We deﬁne

ij ∗ A ∗ A Ω−α, :HA X → HA X (4.10) by the following property: for any p ∈ XA

K (α, α) Ωij (1 ) = σ 1 −α, p p,q 2 q

118 if there exists q ∈ XA satisfying conditions (4.5) for these i, j and α. The signs

 σp,q ∈ {±1} are the following product of signs

eA N −Ce C˜, C˜, p q/X σp,q = σq/X σp/X = , (4.11) A −Ce q e Np/X and where Ce is any Weyl chamber adjacent to the wall α∨ = 0. The sign does not depend on this choice which we will show later. If there’s no such q, we set

ij Ω−α, (1p) = 0.

Remark. By the conditions on ∆ (p)’s the existence for such q ∈ XA is equivalent to the following two conditions on p

∨ h∆i (p) , α i = 1, (4.12)

∨ h∆j (p) , α i = −1. (4.13)

Now for any Weyl chamber we deﬁne the following operator

1 X Ωij = Ωij + Ωij . (4.14) C, 2 0 α, α>C0

ij Remark. The notation ΩC, suggests that these operators are related to Casimir operators in representation theory. We’ll make this statement more precise later. The minus

ij sign for −α in the deﬁnition of Ω−α, is introduced to make this comparison easier.

A −C ij Remark. If one chooses polarization p = e Np/X , then the signs in Ωα, with a simple α (with respect to C) are all +1. We will need it later to choose the right basis in representation-theoretic language.

Similar to stable envelopes, for a simpler notation we write

i ij H (p) and ΩC, (p)

119 instead of

i ij H (1p) and ΩC, (1p) .

Now we are ready to state the main result on classical multiplication in the stable basis.

Theorem 4.5.2. The classical multiplication is given by the following formula

" # T X i X ij c1 (Lk) ∪ StabC (p) = StabC H (p) − ~ Ω−C, (p) . i≤k i≤k

Proof. From the ﬁrst part of Proposition 4.5.1 we get that

T ∗ T X k c1 (Lk) ∪ StabC (p) = ιpc1 (Lk) · StabC (p) + ~ cp,q StabC (q) . q

Let us first rewrite the first term. By Corollary 3.4.5 we know the restriction of the first Chern classes

ι∗cT (L ) = K (Σ (p) , •) + ~K (Σ (p) , Σ (p)) p 1 k k 2 k k = K (Σ (p) , •) + ~K (Σ (p) , µ) − ~K (Σ (p) , µ − Σ (p)) k 2 k 2 k k X X = K (∆ (p) , •) + ~K (∆ (p) , µ) − ~ K (∆ (p) , ∆ (p)) , i 2 i 2 i j i≤k i≤k

P i ij where we used ∆i (p) = µ. Using deﬁnitions of H and Ω0 , we can write " # X X 1 ι∗cT (L ) · Stab (p) = Stab Hi (p) − Ωij (p) (4.15) p 1 k C C ~ 2 0 i≤k i≤k

k Now let us compute the oﬀ-diagonal terms. The coeﬃcients cp,q can be found from the second part of Proposition 4.5.1.

∗ ∗ A ∗ A k ιqStabC (p) ιqc1 (L) − ιpc1 (L) cp,q = . ~ A q

By Theorem 4.4.2 these are zero unless p and q are related by (4.5) for some i < j and

120 a coroot α positive with respect to C. For such p, q

k p K (Σk (q) , •) − K (Σk (p) , •) cp,q = ωp,q ∨ α q

ker α∨ where one can compute the multiplier ωp,q by choosing the polarization on X to

A −C0 ∨ be e N ∨ for some Weyl chamber in ker α (the multiplier does not depend Xker α /X on this choice). This gives A −C0 e Nq/X ωp,q = . A −C0 e Np/X

0 A −C Let Ce be any Weyl chamber adjacent to the ”wall” C . Then e Nq/X diﬀers from A −C0 ∨ e Nq/X by a factor which is a power of ±α with the sign chosen to make the weight negative with respect to Ce. We see that this aﬀect numerator and denominator by the same factor, so the multiplier can be computed by

A −Ce e Nq/X ωp,q = . (4.16) A −Ce e Np/X

0 Since the ωp,q did not depend on the choice of C , it does not depend on the choice of

Ce as long as it is adjacent to the hyperplane α∨ = 0. From the relation (4.5) we know that

  Σk (q) + α, if i ≤ k < j, Σk (p) =  Σk (q) , otherwise.

This gives

 K (α, α) − α∨, if i ≤ k < j,  2 K (Σk (q) , •) − K (Σk (p) , •) =  0, otherwise.

k Combining this together, we get that cp,q is not zero if p and q are related by (4.5)

121 for i < j and i ≤ k < j. Then

K (α, α) ck = −σ p,q p,q 2 with the signs eA N −Ce p q/X σp,q = q A −Ce e Np/X

ij equal to the ones introduced in the definition of Ω−α,. ij Comparing with the definition of Ωα,, we find that the off-diagonal contribution is

" # X X ij StabC −~ Ω−α, (p) . (4.17) i≤kC0

Adding (4.15) and (4.17) ﬁnishes the proof of the Theorem.

The connection to the Knizhnik-Zamolodchikov becomes more clear if instead of line bundles Li we use Ei from (3.13). There’s a straightforward reformulation of Theorem 4.5.2.

Theorem 4.5.3. The classical multiplication is given by the following formula

" # T i X ji X ij c1 (Ei) ∪ StabC (p) = StabC H (p) + ~ Ω−C, (p) − ~ Ω−C, (p) . j

122 Chapter 5

Quantum Cohomology

In this chapter we compute the multiplication by a divisor in the quantum cohomology. This gives rise to the quantum differential equation, which we match in the next paper with certain representation-theoretic object, called trigonometric Knizhnik- Zamolodchikov connection. Our computation is based on the idea of the reduction to Picard rank 1, appearing in [BMO] for the case of the cotangent bundle of the full flag variety. We follow this paper quite closely for the definition of the reduced class and the application of a deformation family to pass to the wall cases. The first partial result on the purely quantum cohomology was done in M. Viscardi’s thesis [V]. Let us point out some issues with this computation

1. The computation was done only for the simplest case of λ = (λ1, λ2), it was only mentioned that the general case can be done by reduction to Picard rank 1, but no actual computation,

2. The choice of basis was the ﬁxed point basis, which allowed only to compute the leading part of the purely quantum multiplication. It was completely unprotected

from corrections in ~,

3. The classical part was not computed. Moreover, even if it’s done (for example, by Corollary 3.4.5) in the ﬁxed point basis it must be diagonal, which makes it hard 123 to identify with asymmectic trigonometric Knizhnik-Zamolodchikov connection. As we see from the asymmetry of the ﬁnal answer this means that the corrections to the purely quantum part must be assymmetric (in particular, the corrections do appear!),

4. The diagonal part was explicitly omitted, but never restored, which lead to appearance of ”truncated” Casimir operators, missing exactly the Cartan (diagonal!) part.

We don’t have these problems in our computation.

5.1 Preliminaries on quantum cohomology

Quantum cohomology is a deformation of ordinary equivariant cohomology by three- point Gromov-Witten invariants (by a formal parameter q). We deﬁne it by the following formula

X X d hγ1, γ2 ∗ γ3iX = hγ1, γ2 ∪ γ3iX + hγ1, γ2, γ3i0,3,d q eﬀ d∈H2 (X,Z)

∗ X for any γ1, γ2, γ3 ∈ HT (X). Here h•, •, •i0,3,d are genus zero three-point degree d equivariant Gromov-Witten invariants. The three-point Gromov-Witten invariants are the integrals over a virtual fundamental cycle Z X ∗ ∗ ∗ hγ1, γ2, γ3i0,3,d = ev1γ1 ∪ ev2γ2 ∪ ev3γ3

vir [M0,3(X,d)] of the Deligne-Mumford stack of stable genus 0 degree d maps to X with 3 marked points. The integration in this formula in the pushforward map

∗ ∗ HT M0,3 (X, d) → HT (pt)

124 vir T which is given by the virtual fundamental cycle M0,3 (X, d) ∈ H∗ M0,3 (X, d) .

For the spaces of our interest M0,3 (X, d) are not proper in general and the vir- vir tual fundamental class is only in Borel-Moore (eqivariant) homology M0,3 (X, d) ∈ T,BM H∗ M0,3 (X, d) . However, let us assume that there is proper T-equivariant map to an aﬃne T-variety X0

π : X → X0

× such that X0 is contracted to a point x ∈ X0 by some cocharacter λ: C → T. We λ know that for the spaces X = Grµ it holds. T Under this assumption the T-ﬁxed locus M0,3 (X, d) is proper since all T-invariant mas have the image in π−1 (x) which is proper. Then the integration (or pushforward to a point)

∗ ∗ HT M0,3 (X, d) loc → HT (pt)loc can be deﬁned as mentioned in the previous chapter and computed via the localization formula.

Let us denote by Q[[qd]] the algebra of formal power series in qd, where d ∈ eﬀ H2 (X, Z) are the eﬀective curve classes. The quantum multiplication gives the quantum equivariant cohomology

∗ ∗ d QHT (X) = HT (X) ⊗Q Q[[q ]]

∗ a structure of a supercommutative associative algebra. This is a deformation of HT (X). We will split the quantum multiplication into classical part and purely quantum part

γ1 ∗ γ2 = γ1 ∪ γ2 + γ1 ∗q γ2. (5.1)

2 We are interested in the case when γ3 is a divisor, that is γ3 = D ∈ HA (X). Then we

125 can apply the divisor equation to reduce three-point invariants to two-point invariants

X X hγ1, γ2,Di0,3,d = hD, di hγ1, γ2i0,2,d , (5.2) where hD, di is the pairing of a homology and cohomology class under the natural map

∗ ∗ HT (X) → H (X).

5.2 Reduction to Picard rank 1

5.2.1 Two approaches to the reduced virtual class

One can consider non-equivaraint counts instead of T-equivariant ones. There are two related reasons why these counts must vanish for a symplectic resolution X and d 6= 0. Let us briefly outline them. The first one comes from a surjective cosection in the obstruction bundle. See [KL] for a more general notion of cosection localization. The cosection appears as follows. We work with the tangent-obstruction theory relative to the prestable curves, so let us show how the construction work on a fixed prestable curve C. Let MC (X, d) be the moduli space of stable maps from C to X with the degree class d 6= 0. The standard relative tangent-obstruction theory for MC (X, d) is defined by the natural morphism

∗ ∨ Rπ∗ (ev TX) → LMC (X, d), where

ev: C × MC (X, d) → X,

π : C × MC (X, d) → MC (X, d)

are the natural maps and LMC (X, d) is the cotangent complex of MC (X, d).

Let ωπ be the relative dualizing sheaf. Then the pairing with the symplectic form 126 ω and the pullback of forms induces a map

∗ ∗ ev (TX) → ωπ ⊗ (Cω) .

This gives a map of complexes

∨ ∗ ∨ Rπ∗ (ωπ) ⊗ Cω → Rπ∗ ev (TX) .

Then we truncate

∨ ∗ ∨ τ≤−1Rπ∗ (ωπ) ⊗ Cω → Rπ∗ ev (TX) .

This truncation is a trivial bundle. It carries non-trivial equivariant structure as long as the torus acts non-trivially on ω. This map gives a surjective cosection in the obstructions. The second reason to vanish is the deformations of X. On one hand, the virtual fundamental class is deforamtion invariant. On the other hand, we know that X has a large family of universal deformations and, moreover, a generic deformation has

0 vir no projective curves. This means that M0,3 (X , d) = 0 at a generic symplectic deformation X0 of X for d 6= 0. Combining these two facts, we get that for d 6= 0 the vir virtual fundamental class M0,3 (X, d) = 0 for X itself.

Remark. Same vanishings happen in A-eqivariant count instead of T-equivariant one. The symplectic form has the zero A-weight, so we would still have a non-vanishing cosection of weight zero in the obstruction bundle. Similarly, in the universal deformation the A-action on X extends to the ﬁberwise A-action in the universal deformation family.

Remark. Now we see how the T-equivariant counts destroy the vanishing constructions.

In the ﬁrst approach the cosection gets a non-zero weight −~. The second approach doesn’t work because T scales the base of the deformation with weight ~ and the zero

127 ﬁber becomes distinct.

Each of these reasons suggest how one may modify the counting problem to avoid such vanishing. More precisely, how to track the part which vanishes when one passes to the non-equivariant case, and deﬁne the remaining part, which is called the reduced fundamental class. First, let us show how one modiﬁes the tangent-obstruction bundle. Above we showed that there is a map

∨ ∗ ∨ ι: τ≤−1Rπ∗ (ωπ) ⊗ Cω → Rπ∗ ev (TX) . which is responsible for the appearance of a surjective cosection in the obstruction bundle. Now we can ”kill” it by considering the mapping cone of ι, C(ι). Then the induced map

C(ι) → MC (X, d) give a perfect obstruction theory. This is the reduced obstruction theory, and the associated fundamental class is called the reduced virtual fundamental class.

The second approach comes from one-parametric families of deformations. Let Xe → B be a symplectic deformation of (X, ω) over a smooth curve B. Let us denote the point in the base over which (X, ω) lies as b ∈ B. Pick d ∈ H2 (X, Z) ⊂ H2 X,e Z . We assume that the deformation Xe and are in general position, that is

1. The only ﬁber of Xe → B where there are curves of degree d is (X, ω) over b.

2. The following composition is an isomorphism

2 ∼ 2 ∗ d∩ TbB → H (X,TX) −→ H (X,T X) −→ C

where the ﬁrst map is the Kodaira-Spencer map, the second map is induced by the symplectic structure, the last map is the pairing with the homology class d.

Remark. If X is a symplectic resolution and we have the universal family of the de- 128 formations, then the ﬁrst condition says that the image of B intersects exactly once the hyperplane (or intersection of hyperplanes) associated to d. The second condition means that this intersection is transversal.

The following result is the Theorem 1 of [MP].

Proposition 5.2.1. In the setting above, the embedding

 M0,n (X, d) → M0,n X,e d is an isomorphism of stacks. Under this isomorphism the virtual classes

red h ivir M0,n (X, d) = M0,n X,e d where the right-hand side is the usual virtual fundamental class associated to Xe.

From both points we see that the virtual dimension of the reduced class

red M0,n (X, d) is the virtual dimension of vir M0,n (X, d) plus 1.

5.2.2 Choice of basis

Let (X, ω) be a symplectic variety with a torus T acting on it in such a way that the symplectic form ω is scaled by T-weight ~. Then from the explicit description of the reduced tangent-obstruction theory we see that it misses one trivial line subbundle in the obstructions with weight −~. This gives

vir red M0,n (X, d) = −~ M0,n (X, d)

129 T,BM in H∗ M0,n (X, d) , d 6= 0. This allows us to write for d 6= 0

X,vir X,red hγ1, γ2i0,2,d = −~ hγ1, γ2i0,2,d where the tags vir and red denote the fundamental class we use. In order to recover the usual Gromov-Witten invariant we have to use the non-reduced class in (5.2). However the reduced class allows us to apply the trick from [BMO]. In our case X is a symplectic variety, so it has trivial canonical class. Then the virtual dimension of M0,2 (X, d) is

− hKX , di + dim X + n − 3 = dim X − 1.

red red Thus the degree of M0,n (X, d) is 2 dim X − 2 and the degree of M0,n (X, d) is 2 dim X.

∗ Let γ1, γ2 ∈ HT (X) be non-localized classes. If γ1 ∪ γ2 have compact support, then

∗ ∗ the support of ev1γ1 ∪ ev2γ2 is also proper and the 2-point reduced Gromov-Witten invariant

X,red ∗ hγ1, γ2i0,2,d ∈ HT (pt) lies in non-localized cohomology of a point. We get that for such classes the invariants

X are hγ1, γ2i0,2,d are non-zero by dimension argument only if we have

deg γ1 + deg γ2 ≥ 2 dim X.

∗ Possibility to get invariants in non-localized HT (pt) shows why it is important to work with non-localized classes. We have finitely many terms to compute, it makes the problem more doable. At the same time it shows disadvantages in using the fixed point basis in equivariant cohomology. By definition, it lies in the localized cohomology. If one replaces it via

130 pushforwards ιp,∗1 of the units from the ﬁxed points it is better, because these classes are non-localized. In this case we get that

X,red 2 dim X hιq,∗1, ιp,∗1i0,2,d ∈ HT (pt) which means that we have to compute a polynomial of degree dim X. There is a better choice. Assume that XT is ﬁnite and the conditions for the existence of the stable envelopes are satisﬁed. Then we can take

 γ1 = Stab−C (q) ,

 γ2 = StabC (p) .

As we said in the previous chapter, these classes are dual, so computing these 2-point invariants exactly gives us the matrix of the purely quantum multiplication in the stable basis.

 Recall that the degree of StabC (p) is dim X. Then

X,red 0 Stab−C (q) , StabC (p) 0,2,d ∈ HT (pt) ' Q, so the invariants are just numbers. In particular, if XT = XA these number can be computed via A-equivariant localization instead of T-equivariant one (as usual, we set A ⊂ ker ~ to be a subtorus acting by symplectomorphisms).

 For a symplectic resolution X this is convenient, since the stable envelope StabC (p) restricted to A vanish everywhere except the point p. Then we get

Z X,red Stab−C (q) , StabC (p) 0,2,d = qp 1,

h A ired M0,(p,q)(X,d)

131 A where the integral is deﬁned via A-equivariant localization and M0,(p,q) (X, d) is a subspace of the A-ﬁxed locus

A G A M0,2 (X, d) = M0,(p,q) (X, d) p,q∈XA which maps the ﬁrst marked point to p and the second marked point to q.

5.2.3 Reduction to walls

We reduced the computation of the quantum multiplication to the computation of A-localized integrals Z 1.

h A ired M0,(p,q)(X,d)

The space A-ﬁxed maps to X has components of high dimension, sice we saw that even T-invariant curves are not isolated in X. In principle, it is possible to compute the integral via deformation to the case which has only isolated A-invariant projective curves. This appoach was used in [BMO] for one of the most canonical example of the sypmlectic resolution, the Springer resolution.

Let X be a symplectic resolution and let Xe be its suﬃciently generic twistor deformation over A2 by a pair of line bundles:

X Xe .

0 A2

By ”suﬃciently generic” we mean that

1. The generic fiber of Xe → A2 has no curves. By the discussion of the wall structure in twistor deformations, this means the fibers with curves lie over a union of finitely

many lines H ⊂ A2 passing through the origin 0 ∈ A2.

2. If H ⊂ A2 is such a line passing through 0, that there are curves over H \ 0, then 132 curves there is d ∈ H2 (X, Q) such that the degrees of all such curves are multiples

of d. We consider d up to scaling. Then we denote such a line by Hd and refer to

them as ”walls” in A2.

The main example of such a deformation is the family over a plane A2 ' P ⊂ Bu in the base of the universal symplectic deformation of X

X Xe Xeu

0 P Bu as long as P intersects transversely all the walls. The walls in P are exactly the intersections with the walls in Bu.

2 2 Let L0 ⊂ A be any line passing through the origin 0 ∈ A which is not a wall. We get a family Xe0 over L0

X Xe0 Xe

0 L A2 Then by the Proposition 5.2.1 the integral over the reduced virtual class can be computed via Xe0 Z Z 1 = 1,

h A ired h A ivir M (X,d) M X0,d 0,(p,q) 0,(pe0,qe0)( e )

A where pe0 and qe0 are the components of Xe0 containing p and q respectively. They have exactly one point in the ﬁber over every point A1, they show how the A-ﬁxed points A p, q vary in the family Xe0 . Now we can use the deformation invariance of the virtual fundamental class

h A ivir M X0, d . 0,(pe0,qe0) e

133 Parallel shifts along L0 give us a quotient

2 1 A A

whose ﬁber over 0 is L0 and over another point t 6= 0 is a parallel line Lt. Let Xet be the family Xe restricted to Lt. Then the composition

2 1 Xe → A A

gives a family whose ﬁber over 0 is Xe0

Xe0 Xe Xet

0 A1 t

A In other words, it’s a deformation of Xe0. The ﬁxed components pe0 deform to pet ⊂ Xet . By the deformation invariance of the virtual class

Z Z 1 = 1.

h A ivir h A ivir M X0,d M Xt,d 0,(pe0,qe0)( e ) 0,(pet,qet)( e )

First let us point that some eﬀective curve classes d are repserented only by curves over the origin 0 ∈ A2. Then the moduli space on the RHS is empty and the integral is zero. From now on we assume that eﬀective d has representatives not only over the origin.

By construction, this means that there is a wall Hd which has curves of degree d in the ﬁbers over it. By construction Lt intersects with the hyperplane Hd at exactly one

d point and this intersection is transverse. Call the ﬁber of Xet over this point X and

0 0 dA denote by p , q ∈ X the ﬁbers of pet and qet. Then by Proposition 5.2.1 we have

134 Z Z 1 = 1.

h ivir red A h A d i M Xt,d M 0 0 X ,d 0,(pet,qet)( e ) 0,(p ,q )( ) Combining all the equalities of integrals, we get that the integral of interest

Z Z 1 = 1 (5.3)

red red h A i h A d i M0,(p,q)(X,d) M0,(p0,q0)(X ,d)

if there is a wall Hd, and zero otherwise. This formula relates a count of curves of degree d with a count of curves in a generic

d ﬁber X over the corresponding wall Hd. The invariant curves in Xd are much simpler as we will see in examples. For the

A d spaces of our interest the moduli space M0,(p0,q0) X , d has only isolated points. We summarize the reduction in the following statement

Proposition 5.2.2. Let X be a symplectic resolution satisfying the assumptions in the definition of the stable envelopes and let XA be finite. If for an effective curve class d there is a wall Hd in a sufficiently generic 2-parametric deformation

Z X,vir Stab−C (q) , StabC (p) 0,2,d = −~qp 1 red h A d i M0,(p0,q0)(X ,d) for any p, q ∈ XA and the corresponding A-ﬁxed points p0, q0 in a generic ﬁber Xd over the wall Hd.

If there is no wall of form Hd, then

 X,red Stab−C (q) , StabC (p) 0,2,d = 0.

Remark. Note that it is important for us that we are doing A-localized computation, not T-equivariant one. The torus T would scale the base and make the origin a distinguished point, the deformation invariance would be destroyed.

135 5.3 Computation of wall contributions

5.3.1 Wall deformations

λ Now we restrict to the case X = Grµ. Then the 2-parameter deformation can be made by restricting the Beilinson-Drinfeld Grassmannian to a generic plane P ' A2 containing the origin in the base Al

λ λ Grµ Xe Grf µ

0 P Al

l The walls in P are the intesections of P with the walls in A , that is Hij = {ti = tj}. As mentioned before, all ﬁbers in the complement of these walls have no projective curves. Now let us compute the contribution from a generic ﬁber over each wall.

ij l Let (t1, . . . , tl) ∈ H ⊂ A be a generic point in the wall Hij, that is ti = tj (i < j) and all other tk are pairwise distinct. Then we know that the ﬁber of the global convolution Grassmannian splits over this point in the following way

λ Y Gr ' Grλi,λj × Grλk . f (t1,...,tl) k6=i k6=j

The restriction of the global line bundles Eei, Eej are the pull-backs of E1 and E2

λi,λj respectively from Gr . For other k, Eek restricts to a pullback of E1 ' O(1) from

Grλk . These are isomorphisms as A-equivariant line bundles. λ λ We are interested in the ﬁber of the global subspace Gr ⊂ Gr . We f µ,(t1,...,tl) f (t1,...,tl) denote this ﬁber as Xij: Y Xij ⊂ Grλi,λj × Grλk . k6=i k6=j

From the period map we ﬁnd that any algebraic curve C in Xij must have funda-

136 mental class of form c(ei−ej). This gives that the degree with respect to the restrictions

λ of all line bundles Eek is zero unless k = i or k = j. Since O(1) is ample on Gr and generate the Picard group, this implies that the projection of C to each factor of Grλk is constant. Moreover, the degrees with respect to Eei and Eej must be opposite. That

λi,λj means that the degree with respect to L2 on Gr is zero (as the restricition of EeiEej), which similarly forces the projection to the last coordinate of Grλi,λj to be constant. Now we apply A-equivariant localization to compute the integrals over the stable

ij maps M0,2 (X , d).

λ The A-ﬁxed locus has a description, similar to the A-ﬁxed locus of Grµ in Propo- sition 3.1.1:

Proposition 5.3.1. The ﬁxed locus (Xij)A is the following disjoint union of ﬁnitely many points ([νi] , [νj]) , [ν1] ,..., [cνi],..., [cνj],..., [νl]

A A λi,λj λk where the hats denote omitted terms, ([νi] , [νj]) ∈ Gr and [νk] ∈ Gr for all k, k 6= i, k 6= j, and, ﬁnally, X νk = µ. k6=i Proof. Omitted.

The identification of the line bundles also allow us to compare the fixed points of Xij and the central fiber X, that is, to say which A-fixed points go to which fixed points of X as we approach the central fiber along the hyperplane.

A For any T-equivariant line bundle Le on Xe the A-weights on a point p0 ∈ (Xij) and its limit p ∈ XA must be the same. This is because A-action commutes with × C~ and the limit is taken by the ×. The A-weights of E uniquely determine ∆ of the C~ i i ﬁxed point. Then the limit of

0 p = ([νi] , [νj]) , [ν1] ,..., [cνi],..., [cνj],..., [νl] 137 is p with

∆k (p) = νk if k 6= j (5.4)

∆j (p) = νj − νi. (5.5)

Similarly to the case of the slices one ﬁnds that A-invariant curves in Grλi,λj are of

12 form Cp,α∨ , where A λi,λj p = ([νi] , [νj]) ∈ Gr and

∨ hνi, α i > 0, (5.6)

∨ hνj, α i = 0. (5.7)

12 Recall that Cp,α∨ has two ﬁxed points: p and

q = ([νi − α] , [νj])

12 ∨ ∨ and the tangent weight of Cp,α∨ at p and q are α and −α respectively. These curves are isolated.

5.3.2 Unbroken maps

Now we need to discuss which stable A-invariant maps give zero contributions. This needs the notion of broken maps by Okounkov and Pandharipande [OP]. The classiﬁ- cation of the unbroken maps is identical to the one in the case of the cotangent bundles to a ﬂag variety, see [S2]. Recall that Xij is symplectic because it comes from a twistor deformation. We

ij reduced the tangent-obstuction theory on the moduli stack M0,2 (X , d) to make sure that the obstruction bundle doesn’t have a non-vanishing cosection with zero A-weight. However, there are two cases than an extra such cosection appears and has no way to 138 cancel with deformation.

Let Z be an algebraic symplectic variety. Consider an A-ﬁxed stable map

A [f : C → Z] ∈ M0,n (Z, d)

• We say that f has a breaking subcurve Ce ⊂ C if Ce is contracted by f and the disconnected curve C \ Ce has at least two connected components which have positive degree under f.

• We say that f has a breaking node s ∈ C, if two non-contracted components

C1,C2 ⊂ C meet at s and have tangent weights a1, a2 with a1 + a2 6= 0.

We call a stable A-invariant map f : C → Z broken, if there’s at least one broken subcurve or broken node. If a map f is not broken, we call it unbroken. By combi-

A + natorial description of the components of M0,n (Z, d) (see [HKK ]) we get that in A any component of M0,n (Z, d) either all maps are broken or all maps are unbroken. Both a broken subcurve and a broken node give rise to a non-vanishing cosection of zero A-weight in the obstruction bundle, which implies vanishing of the reduced fundamental cycle on components of broken maps.

Let us return back to the case Z = Xij, n = 2. Then there’s a classiﬁcation of all unbroken maps

ij Proposition 5.3.2. Let [f : C → X ] be an unbroken map with two ﬁxed points c+, c− ∈ C. Then there’s exactly two cases

1 1. C ' P , f is a cover over an A-invariant curve Cp0q0 , branched over A-ﬁxed points 0 0 0 0 p , q of Cp0q0 ; the two marked points are sent to p , q .

2. C has two rational components C0, C1 meeting at a node. C0 is contracted by f

0 to an A-ﬁxed point p and has both ﬁxed points. C1 is mapped by f as C in case 1.

139 Proof. First let us note that two non-contracted components can’t meet at a node since by (5.6) the weights at this node can’t sum to 0, so the node would be broken. If there’s no contracted component, then it must be the case1. Then if there is a contracted curve, it can’t have two nodes because then it would be a breaking subcurve. Then by stability it must contain both marked points and one node. The non-contracted component meeting at this node must connect two points and can’t meet another component at the other point. We get the case2.

The integral we want to compute is

Z 1

red h A d i M0,(p0,q0)(X ,d)

for d a multiple of ei − ej, and we see that we have two cases:

1. If p0 6= q0, then there are only contributions from the case1 of Proposition 5.3.2.

2. If p0 = q0, then there are only contributions from the case2 of Proposition 5.3.2.

Remark. The contribution of curves of type2 in Proposition 5.3.2 is diagonal, and can be uniquely reconstructed from the equation

D ∗ 1 = D.

5.3.3 Oﬀ-diagonal case

In this subsection we assume p0 6= q0.

0 0 In this case there is a curve Cp0q0 connecting p and q if and only if

• p0 and q0 project to the same point in

Y Grλk . k6=i k6=j

140 • Let p00 and q00 be the projections to Grλi,λj of p0 and q0 respectively. Then there must be a coroot α such that

00 00 Σ1 (p ) = Σ1 (q ) − α.

Remark. The ﬁrst condition also implies

00 00 Σ2 (p ) = Σ2 (q ) .

Remark. For the corresponding limit points p, q ∈ XA we get the relations

∆i (q) = ∆i (p) − α,

∆j (q) = ∆j (p) + α, (5.8)

∆k (q) = ∆k (p) for all k, k 6= i, k 6= j, which is almost the same as the condition to have a non-zero off-diagonal contribution in the classical case (4.5). The only difference is that now we don’t require α∨ to be positive with respect to some chamber. If p satisfies this condition, the corresponding q also satisfies it. In the classical case the relation was asymmetric, and in the quantum case it becomes symmetric.

Remark. Given p0 and α such q0 is unique if it exists. It exists if and only if conditions (5.6) and (5.7) are satisﬁed, or, equivalently, for the corresponding limit point p ∈ X we have using (4.12) and (4.13)

∨ h∆i (p) , α i = 1, (5.9)

∨ h∆j (p) , α i = −1. (5.10)

These conditions is the same as (4.12) and (4.13) in the classical case, again, modulo the requirement on positivity of α∨.

0 0 12 λi,λj The curve Cp0q0 connecting such p and q then has projection Cp00,α∨ to Gr and 141 projects to a point in Grλk . The degree by Proposition 3.4.8 is

K (α, α) (e − e ) . 2 i j

∼ ij Let f : C −→ Cp0q0 ⊂ X be the m-fold cover of Cp0q0 mapping points c+, c− ∈ C to p0 and q0 respectively. The degree of f by 3.4.8 is

K (α, α) d = m (e − e ) . f 2 i j

A ij Then [f] ∈ M0,(p0,q0) (X , df ) is an isolated point and the unique unbroken map of

0 0 degree df connecting p and q . Let us ﬁgure out the A-equivariant reduced fundamental cycle. It is cf times the fundamental class of [f]. By A-equivariant localization.

1 1 cf = , m A red e N ij [f]/M0,(p0,q0)(X ,df ) where the factor m appears from the automorphisms of f and the N red is the virtual normal bundle in the reduced tangent-obstruction theory. Namely,

red vir ∨ N ij = N ij + (CωXij ) [f]/M0,(p0,q0)(X ,df ) [f]/M0,(p0,q0)(X ,df )

∨ as virtual A-representations. Torus A preserves ωXij , so (CωXij ) is a trivial A- representation. It is introduced to cancel a trivial cosection in the obstruction bundle, which enters with negative sign to N vir. We want to show that the reduced virtual normal bundle has no trivial weights. For this we keep track where they appear in the usual virtual normal bundle N vir. Recall that usual N vir is the following[HKK+]

vir N ij = − Aut (C, c+, c−, f) + Def (C, c+, c−, f) − Obs (C, c+, c−, f) [f]/M0,(p0,q0)(X ,df )

= − Aut (C, c+, c−) + Def (C, c+, c−) + Def (f) − Obs (f)

142 1 Since C ' P , the inﬁnitesimal automorphisms Aut (C, c+, c−) are 1-dimensional, 1 generated by a vector ﬁeld on P vanishing at both c+, c−. This has A-weight 0. The deformations vanish: Def (C, c+, c−) = 0, there’s only one rational curve with two marked points. Then we look at the deformations and obstructions of f:

Def (f) − Obs (f) = H0 C, f ∗TXij − H1 C, f ∗TXij .

The map f induces an action on C with respect to a cover of A. With respect to this action the vector bundle f ∗TXij is equivariant. We will need the following lemma[MO2].

Lemma 5.3.3. Let T be an A-equivariant vector bundle over C without zero weights

at the ﬁbers Tc+ , Tc− over the ﬁxed points c+, c− ∈ C. Then

A 0 ∨ 1 ∨ deg T +rk T +z A A e H (C,T ⊕ T ) − H (C,T ⊕ T ) = (−1) e Tc+ e Tc− , where z is the number of zero weights in H1 (C,T ⊕ T ∨). In particular, H0 (C,T ⊕ T ∨)− H1 (C,T ⊕ T ∨) has no zero weights.

Proof. By the splitting principle, it’s enough to prove the statement for the case when T is a line bundle: T = L. By symmetry of the expression in terms of exchanging L ↔ L∨, we assume that L has non-negative degree. If L is trivial with weight χ∨, then deg L = 0, H1 (C,L ⊗ L∨) = 0, so z = 0. Moreover, H0 (C,L ⊗ L∨) is two-dimensional with weights χ∨ and −χ∨. Then both sides are − (χ∨)2. Now let us assume the degree of L is positive.

∨ ∨ Let the ﬁbers of TC at the ﬁxed points c+ and c− have weights ω and −ω respectively. It is well-known that then for any equivariant line bundle L on C of positive degree

143 the weights of H0 (C,L) are equidistant

∨ ∨ Lc+ ,Lc+ − ω ,...,Lc− + ω ,Lc− ,

where Lc+ and Lc− are weights of L at c+ and c−. All weights have multiplicity 1. For L∨ we have H0 (C,L∨) = 0. Using Serre duality, we ﬁnd that for the line bundle L∨ of degree < −1 the weights of

1 ∨ 0 ∗ H (C,L ) ' H (C,L ⊗ KC ) are also equidistant and of multiplicity 1:

∨ ∨ ∨ ∨ −Lc− − ω , −Lc− − 2ω , · · · − Lc+ + 2ω , −Lc+ + ω .

∨ 1 Since L ⊗ KC has negative degree, H (C,L) = 0. If a trivial weight is present in H0 (C,L) it’s also present in H1 (C,L∨) by the as-

sumption that Lc+ and Lc− are not zero. Then the trivial weights cancel. All other weights in H1 (C,L∨) have a corresponding weight in H0 (C,L) with the opposite sign. So we cancel

deg L ⊗ KC + 1 − z = deg L − 1 − z terms, each gives −1 sign. This gives a sign factor

(−1)deg L−1−z = (−1)deg L+rk L+z.

0 A A The remaining terms in H (C,L) are Lc+ , Lc− which give e Tc+ e Tc− . This proves the lemma.

Since TXij has a symplectic structure, we can split it into TXij ' T ⊕ T ∨ for some

ij ij equivariant bundle T . The tangent weights Tp0 X and Tq0 X are all non-zero because

144 they are roots, then we are it he setting of Lemma 5.3.3. The lemma gives that the subspaces of weight 0 in Def (f) − Obs (f) cancel.

∨ The remaining 1-dimensional Aut (C, c+, c−) of zero weight cancel with (CωXij ) red introduced in the deﬁnition of N ij exactly for this cancellation. [f]/M0,(p0,q0)(X ,df )

The Lemma 5.3.3 gives us even an expression for cf if we choose a splitting of f ∗TXij. Let a0 be the real space associated to ker α∨ ⊂ a as in previous chapter. We ﬁx a R Weyl chamber C0 ⊂ a0 . R Then we have a splitting

ij C0 α∨ −C0 TX = NC 0 0 /Xij ⊕ NC 0 0 /Xij ⊕ N ij , Cp0q0 p q p q Cp0q0 /X

α∨ ∨ where N ij has all weights which are zero on ker α . Cp0q0 /X By previous consideration we know that

α∨ ∗ N ij = TCp0q0 ⊕ T Cp0q0 ⊕ V, Cp0q0 /X where V is a trivial vector bundle pulled back from Q Grλk . Let us choose a Weyl

0 ∨ ∨ chamber Ce ⊂ aR adjacent to C (that is, we choose whether α or −α is positive). Then we can split V with respect to the chamber Ce

V = V Ce ⊕ V −Ce.

This gives a splitting of TXij| Cp0q0

ij ∨ TX = T ⊕ T Cp0q0 where

−C0 −Ce T = N ij ⊕ TCp0q0 ⊕ V . Cp0q0 /X

145 Then by Lemma 5.3.3 we can compute the Euler class

A 0 ∗ ij 1 ∗ ij deg f ∗T +rk T +z A ∗ A ∗ e H C, f TX − H C, f TX = (−1) e f Tc+ e f Tc−

deg f ∗T +rk T +z+1 A −Ce A −Ce = (−1) e Np0/Xij e Nq0/Xij , where we used that at one of points p0 and q0 the ﬁber of T is the same as N −Ce and at another point is diﬀers by a sign of the weight coming from TCp0q0 .

First we find z. Let us first find all pairs L, L∨ of line subbundles in a larger bundle, λ the tangent bundle of Gr f (t1,...,tl) The pullback of the tangent bundle from Grλk are trivial with no trivial weights.

So such pairs of line bundles must be in (the pullback of) T Grλi,λj . Then the weights at the ﬁxed points Lp00 and Lq00 are roots. Moreover, since the degree

Z L 00 − L 00 deg L = cA (L) = p q 1 α∨ Cp00,q00

∨ is an integer, then the weights of Lp0 and Lq0 must diﬀer by a multiple of α . A zero

1 ∗ ∗ ∨ weight if H (C, f L ⊗ f L ) can appear only if 0 is on the line connecting Lp0 and Lq0 . Then this means that one of this weight must be α, the other is −α∨. The multiplicities of ±α∨ are both 1 by Corollary 3.2.13. Despite the computations

λi,λj λi,λj ∨ of the multiplicities are done for Grµ , not for Gr , the multiplicities for ±α are transferable literally: the only difference is if the Right Boundary Condition 3.2 is A λi,λj 00 ∨ satisfied or not. For the points of Gr with hΣ2 (p ) , α i = 0, as in our case, the Right Boundary Condition 3.2 holds as long as the Left Boundary Condition 3.1 is satisfied. Thus the multiplicity of both ±α∨ at the tangent space at both p0 and q0 is 1. Then there’s at most one pair of subbundles L, L∨ which can contribute to z.

One can easily identify this pair of subbundles. One is the tangent space TCp0q0 ⊂ TXij, it has degree 2. The symplectic form gives the dual subbundle of degree −2. In particular, this pair is a subbundle in TXij, not in the larger tangent space we considered. Finally, it gives exactly one zero in H1 (C, f ∗L ⊗ f ∗L∨), so z = 1. 146 Let us now ﬁnd the degree of T . By A-equivariant localization it is

∗ A ∗ A Z ι 0 c (T ) − ι 0 c (T ) deg T = cA (T ) = p 1 q 1 1 α∨ Cp0q0

We know these Chern classes because we know the multiplicities of tangent A-weigths at the limit points p, q ∈ X from Corollary 3.2.13. The weights χ∨ coming from

∨ counting h∆k (p) , χ i = −1 for k 6= i, k 6= j cancel with similar contributions for q, because ∆k (p) = ∆k (q) for such k. So we only have to take contributions from ∆i (p),

∆j (p) and similar for q. Denote the sums of these A-weights ci,j,p and ci,j,q and get

c − c deg T = i,j,p i,j,q . α∨

The are coweights ∆i (p), ∆i (q), ∆j (p) and ∆j (q) are in the Weyl orbit of a miniscule coweights, so the conditions (5.8) imply

∆i (q) = sα ∆i (p)

∆j (q) = sα ∆j (p) .

Since the weight multiplicities are determined by pairing with these coweights, we get

∨ ci,j,q = sαci,j,p = ci,j,p − hci,j,p, αi α and

deg T = hci,j,p, αi .

Let us consider contributions to this sum of tangent weights which are in a pair of opposite roots ±β∨.

• Assume ﬁrst that

∨ ∨ hβ , ∆i (p)i = hβ , ∆j (p)i = 0.

Then both ±β do not contribute to ci,j,p.

147 • Now let the root β∨ which satisfy

∨ ∨ hβ , ∆i (p)i = hβ , ∆j (p)i= 6 0.

∨ ∨ Without loss of generality we can assume hβ , ∆i (p)i = −1. Then β contributes

∨ ∨ 2 hβ , αi ∈ 2Z to deg T . The root −β does not contribute to ci,j,p.

• Finally, the root β∨ which satisfy

∨ ∨ hβ , ∆i (p)i = − hβ , ∆j (p)i= 6 0

∨ Without loss of generality we can assume hβ , ∆i (p)i = −1. The contributes of

∨ ∨ β to deg T is hβ , αi via coweight ∆i (p) and there is a zero contribution via δpj.

∨ ∨ The second root −β has no contribution via ∆i (p) and contributes h−β , αi via

∨ ∆j (p). Then the total contribution of the pair ±β vanishes.

This proves that the contribution of all roots is even. Then the degree deg T is even, so deg f ∗T = m deg T is also even.

1 ij 1 The rank rk T is 2 dim X = 2 dim X. Knowing z, deg T and rk T we ﬁnd the Euler class

A 0 ∗ ij 1 ∗ ij dim X/2 A −Ce A −Ce e H C, f TX − H C, f TX = (−1) e Np0/Xij e Nq0/Xij .

0 0 Finally, we get that for p 6= q and eﬀective curve class d = m (ei − ej), such that the limits p and q are satisfying (5.8) for some αs, not necessary positive (or, equivalently, p0 and q0 are connected by a curve in the deformation to the generic ﬁber Xij on the wall associated to ei − ej) the contribution of the reduced Gromov-Witten invariants

148 is

Z X,red Stab (q) , Stab (p) = qp 1 −C C 0,2,m(ei−ej )

h A ired M0,(p,q)(X,m(ei−ej )) Z = qp 1

red h A ij i M0,(p0,q0)(X ,m(ei−ej )) dim X/2 (−1) qp = m A −Ce A −Ce e Np0/Xij e Nq0/Xij

1 qp = m A −Ce A −Ce e Np0/Xij e Nq0/Xij

1 qp = m A −Ce A −Ce e Np/Xij e Nq/Xij ˜ ˜ 1 = σC, σC, q/X p/X m 1 = σ , p,q m

 where σp,q is exactly the same sign as appeared in (4.11). If p 6= q and they don’t satisfy (5.8) for these i < j and any α, then the contribution is zero.

We summarize the computation of oﬀ-diagonal terms. Let p 6= q be related by 5.8 for a coroot α and i < j. Then the maps connecting p and q give the following oﬀ-

∞ X K (α, α) 1 D ∗ Stab (p) = − qm·d(i,j,α) D, m (e − e ) σ Stab (q) + ... q C ~ 2 i j p,q m C m=1 qd(i,j,α) K (α, α) = − hD, e − e i σ Stab (q) + ... ~ i j 1 − qd(i,j,α) 2 p,q C qd(i,j,α) = − hD, e − e i Stab Ωij p + ... ~ i j 1 − qd(i,j,α) C α,

K(α,α) ij where d (i, j, α) = 2 (ei − ej), dots stay for terms with StabC (r) for r 6= q and Ωα, is the operator deﬁned in (4.10).

149 5.3.4 Diagonal case

Now let us consider the case p0 = q0. The computation is quite similar to the oﬀ- diagonal case, so we will skip some details. We know by Proposition 5.3.2 that the A-ﬁxed unbroken maps with both marked ponts mapped to p0 = q0 are f : C → Xij where C has two irreducible rational components

C = C0 ∪ C1,

0 0 C0 and C1 meet at a node s, C0 is contracted by f to p = q , both marked points

0 ij A c+, c− are in C0, C1 m-foldly covers Cp0r0 under f, where r ∈ (X ) . The tangent

∨ 0 0 weight α of Cp0r0 at p uniquely determines r . The degree of f by 3.4.8 is

K (α, α) d = m (e − e ) . f 2 i j

For each pair p0, r0 ∈ (Xij)A connected by an A invariant curve, and every positive

A ij integer m there is unique such map f. Thus [f] is an isolated point in M0,(p0,p0) (X , df ).

We are interested in ﬁnding the reduced fundamental class at this point. It’s cf times the class of the point [f]. As previously,

1 1 cf = , m A red e N ij [f]/M0,(p0,p0)(X ,df ) where N red is the reduced virtual normal bundle, which is the same correction of the virtual normal bundle as before.

150 The usual virtual normal bundle to [f] is

vir N ij = − Aut (C, c+, c−) + Def (C, c+, c−) + Def (f) − Obs (f) [f]/M0,(p0,p0)(X ,df )

The inﬁnitesimal automorphism bundle Aut (C, c+, c−) is now 2-dimensional and comes from automorphisms of C1 preserving the node s. One weight subspace is generated by rotations preserving s and f −1(r0). Similarly to the case p0 6= q0, this one

1 has weight 0. The new weight subspace is generated by parallel shifts in A ' C1 \{s}. ∗ −1 0 ∨ This one has the same weight as the weight of f TCp0r0 at f (r ), that is −α /m.

The deformations Def (C, c+, c−) come from the smoothing of the node s. The weight of this deformation is the sum of adjacent tangent weigths

0 + α∨/m

Then we get

∨ A ∨ α /m e − Aut (C, c , c ) + Def (C, c , c ) + ( ω ij ) = = −1. + − + − C X −α∨/m

In particular, all zero A-weights cancel. For the deformations and obstructions we have

Def (f) − Obs (f) = H0 C, f ∗TXij − H1 C, f ∗TXij .

To compute these sheaf cohomology, let us write the following exact sequence of sheaves

OC OC0 ⊕ OC1 Os.

Tensoring it with a vector bundle T f ∗TXij, we get

f ∗TXij f ∗TXij| ⊕ f ∗TXij| f ∗TXij| . C0 C1 s

151 The associated long exact sequence of sheaf cohomology gives

0 ∗ ij 1 ∗ ij Def (f) − Obs (f) = H C0, f TX − H C0, f TX

0 ∗ ij 1 ∗ ij + H C1, f TX − H C1, f TX

− H0 s, f ∗TXij where we used that only these cohomologies don’t vanish by the dimension of the support. Moreover, f ∗TXij| is a constant vector bundle with the same weight as f ∗TXij| , C0 s 1 ∗ ij 0 ∗ ij 0 ∗ ij so H (C0, f TX ) = 0 and H (C0, f TX ) cancels with H (s, f TX ). We get

0 ∗ ij 1 ∗ ij Def (f) − Obs (f) = H C1, f TX − H C1, f TX

The map f restricted to C1 is exactly the same is the oﬀ-diagonal case. By applying the same arguments, we get

A 0 ∗ ij 1 ∗ ij dim X/2 A −Ce A −Ce e H C1, f TX − H C1, f TX = (−1) e Np0/Xij e Nr0/Xij

In particular, Def (f) − Obs (f) has no zero weights. Then the Euler class of the reduced virtual normal bundle is

A red dim X/2+1 A −Ce A −Ce e N ij = (−1) e N 0 ij e N 0 ij [f]/M0,(p0,p0)(X ,df ) p /X r /X

152 X,red Finally, the contribution from f to Stab (p) , Stab (p) is −C C 0,2,df

dim X/2+1 (−1) pp cf pp = m A −Ce A −Ce e Np0/Xij e Nr0/Xij dim X/2+1 (−1) pp = m A −Ce A −Ce e Np/Xij e Nr/Xij dim X/2+1 A (−1) e Np/Xij = m A −Ce A −Ce e Np/Xij e Nr/Xij A Ce (−1)dim X/2+1 e Np/Xij = m A −Ce e Nr/Xij

A −Ce 1 e Np/Xij = − . m A −Ce e Nr/Xij where p, r ∈ XA are limits of p0, r0 ∈ (Xij)A in the central ﬁber X. In what follows we denote

A −Ce e N ij p/X ∗ ωr,p = ∈ HA (pt)loc A −Ce e Nr/Xij the coeﬃcients appearing here. They are the same factors as in (4.16) and by the same reasoning do not depend on a choice of Ce as long as it is adjacent to the wall α∨ = 0.

Remark. Let us show how one can derive this result form

D ∗ 1 = D ⇐⇒ D ∗q 1 = 0.

First, we have X StabC (p) 1 = mod ~ p p∈XA

∗ in HT (X)loc due to the oﬀ-diagonal vanishing and the deﬁnition of the stable envelopes.

153 Then we have X D ∗q StabC (p) D ∗q 1 = mod ~ p p∈XA Then we see that contributions of a cover of an irreducible curve connecting p0 and q0 exactly cancels with a diagonal contribution of the same cover with additional rational component contracting to q0.

5.3.5 Quantum multiplication

Let us deﬁne operators similar to (4.9) and (4.14) used for the purely classical multiplication. Unfortunately, contributions from long and short roots give diﬀerent contribu- K (α, α) tions to the degree because of the factor . This is why we split the operators 2 into two groups

ij ∗ A ∗ A Ωe 0,st :HA X → HA X loc " # X 1p 7→ ωq,p 1p q where the summation runs over all q such that p and q satisfy the relation 5.8 for a short coroot α and i < j. Similarly, we deﬁne

ij ∗ A ∗ A Ωe 0,lg :HA X → HA X loc " # X 1p 7→ 2 ωq,p 1p q where the summation runs over all q such that p and q satisfy the relation 5.8 for a long coroot α and i < j.

154 ij ij X ij Ωe st, = Ωe 0,st + Ωα,, α short

ij ij X ij Ωe lg, = Ωe 0,lg + Ωα,, α long where by our convention the shortest coroot is normalized as K (α, α) = 2. Since we only consider root systems with minuscule weights, long roots appear in types B and

ij C with norm squared equal to 4. The coeﬃcient 2 in the deﬁnition of Ω0,lg is exactly K (α, α) for the long coroots. 2 Now we can sum the computation of this chapter in one statement.

2 λ Theorem 5.3.4. The purely quantum multiplication by a divisor D ∈ HT Grµ is given by the following formula

 ei−ej 2(ei−ej ) X q ij q ij D ∗q StabC (p) = −~ hD, ei − eji StabC Ωe st, (p) + Ωe lg, (p) 1 − qei−ej 1 − q2(ei−ej ) i

Proof. This follows from the Divisor Equation (5.2) and the reduction in Proposi-

ij ij tion 5.2.2. Then the identiﬁcation of terms in operators Ωsh,, Ωlg, and the curve contributions found is straightforward.

T Corollary 5.3.5. The purely quantum multiplication by the ﬁrst Chern classes c1 (Ei) ∗ λ in HT Grµ is the following

 ei−ej 2(ei−ej ) T X q ij q ij c1 (Ei) ∗q StabC (p) = − ~ StabC Ωe st, (p) + Ωe lg, (p) 1 − qei−ej 1 − q2(ei−ej ) i

 ej −ei 2(ej −ei) X q ji q ji + ~ StabC Ωe st, (p) + Ωe lg, (p) 1 − qej −ei 1 − q2(ej −ei) j

Proof. This follows from the deﬁnition of classes ei.

A direct computation shows that

155 Lemma 5.3.6.

ij ij ij Ω0 = Ωe 0,st + Ωe 0,lg + K (λi, λj)

ij For simply-laced types ADE all roots are short, so Ωlg, = 0 and

ij ij ij Ωe := Ωe st, = Ω −K (λi, λj) , where

ij ij X ij ij ij Ω := Ω0,st + Ωα, = ΩC, + Ω−C, α for any Weyl chamber C.

Remark. This explains why we used tilde in the notation for the diagonal contribution ij operator Ωe 0 in the quantum case: it will diﬀer from the Cartan part of the Casimir operator by a constant shift K (λi, λj).

In what follows we use notation

ij K (p) = K (λi, λj) 1p.

Then we can write the following formula for the quantum multiplication

T Theorem 5.3.7. The quantum multiplication by the ﬁrst Chern classes c1 (Ei) in ∗ λ HT Grµ for a simply-laced G is the following

" ei ij ej ij # T i X q ΩC, (p) + q Ω−C, (p) c1 (Ei) ∗ StabC (p) = StabC H (p) + ~ qei − qej j6=i " # ej ij ei ij X q K (p) X q K (p) − ~ StabC + , qei − qej qei − qej ji where the rational function is assumed to be expanded into a power series in the region

|qn| |qn| iﬀ n < m.

156 Proof. Let us expand the fraction

ei ij ej ij q ΩC, (p) + q Ω−C, (p) ~ qei − qej for i < j.

ei ij ej ij ei ij ij ej ei ij q ΩC, (p) + q Ω−C, (p) q ΩC, (p) + Ω−C, (p) + [q − q ] Ω−C, (p) ~ = ~ qei − qej qei − qej ei ij q ij = ~ − Ω−C, (p) + Ω (p) qei − qej

ei−ej ij q ij = −~ Ω−C, (p) − ~ Ω (p) 1 − qei−ej ei−ej ij q ij = −~ Ω−C, (p) − ~ Ωe (p) 1 − qei−ej ei−ej q ij − ~ K (p) . (5.11) 1 − qei−ej

Similarly for the case j < i:

ei ij ej ij ei ej ij ej ij ij q ΩC, (p) + q Ω−C, (p) [q − q ] ΩC, (p) + q ΩC, (p) + Ω−C, (p) ~ = ~ qei − qej qei − qej ej ij q ij = ~ ΩC, (p) + Ω (p) qei − qej

ej −ei ji q ij = ~ Ω−C, (p) + ~ Ω (p) 1 − qej −ei ej −ei ji q ij = ~ Ω−C, (p) + ~ Ωe (p) 1 − qej −ei ej −ei q ij + ~ K (p) , (5.12) 1 − qej −ei

ij ji where we used ΩC, = Ω−C,. Then we see that the ﬁrst term in (5.11) and (5.12) is the classical term in The- orem 4.5.3. The second term in (5.11) and (5.12) is the purely quantum part in Corollary 5.3.5, expanded in the correct region. The last term in both expressions is exactly compensated by terms with Kij.

157 Chapter 6

Representation-Theoretic

Reformulation

6.1 Trigonometric Knizhnik-Zamolodchikov connection

First let us recall some constructions from representation theory. Let G∨ be a semisimple complex connected Lie group and g∨ be its Lie algebra. We denote the Cartan

∨ ∨ ∨ ∨ subalgebra as h ⊂ g and the root subspaces with respect to h as gα. Since g∨ is simple, it admits a non-degenerate symmetric g∨-invariant pairing

∨ ∨ K (•, •): g ⊗ g → C which is unique up to scaling. Taking the dual map and identifying (g∨)∗ ' g∨ via the same symmetric form, we get a g∨-invariant element

Ω ∈ g∨ ⊗ g∨.

The image of this element in the universal enveloping algebra U (g∨) is the quadratic Casimir element. We call Ω the Casimir operator. We normalize K (•, •) by the condition that on the longest coroot α∨ we have K (α∨, α∨) = 2. 158 ∨ ∨ ∨ ∨ ∨ Since K (•, •) is h -invariant, it pairs h with h and gα with g−α. This gives a splitting X Ω = Ω0 + Ωα, α where

∨ ∨ Ω0 ∈ h ⊗ h ,

∨ ∨ Ωα ∈ gα ⊗ g−α.

Later it is important for us that the deﬁnition of Ω implies the following normalization conditions

K (Ωα) = 1, (6.1)

∨ K (Ω0) = rk g . (6.2)

Quite often the terms operators Ωα and Ω0 are written in the following form

X i Ω0 = hi ⊗ h i

Ωα = eα ⊗ e−α

i ∨ ∨ for dual bases hi, h in h and a non-trivial eα ∈ gα. However, by (6.1) this assumes that

K (eα, e−α) = 1.

We use a different normalization later for eα, so we prefer not to write it in this form. Now we fix a Weyl chamber C. Then we can define a ”half-Casimir” operator

1 X Ω = Ω + Ω C 2 0 α α>C0

159 It is easy to see that

Ω = ΩC + Ω−C.

Often this decomposition is made with respect to the dominant chamber C = +, then −C = − is the antidomimant chamber, and

Ω = Ω+ + Ω−.

For our reasons it’s better to keep C arbitrary.

∨ Let V1,...,Vl be ﬁnite-dimensional complex representations of G . Then for any

X ∈ g∨ we deﬁne the operator Xi acting on the tensor product

V1 ⊗ · · · ⊗ Vl

via action of X on Vi:

i X · v1 ⊗ · · · ⊗ vi ⊗ · · · ⊗ vl = v1 ⊗ · · · ⊗ X · vi ⊗ · · · ⊗ vl

Similarly, for any Y ∈ g∨ ⊗ g∨ we can deﬁne the operator Y ij acting on the same tensor product in such a way, that the ﬁrst component of Y acts on Vi and the second component acts on Vj:

ij X (1) (2) Y · v1 ⊗ · · · ⊗ vi ⊗ · · · ⊗ vj ⊗ · · · ⊗ vl = v1 ⊗ · · · ⊗ Yk vi ⊗ · · · ⊗ Yk vj ⊗ · · · ⊗ vl, k where

X (1) (2) Y = Yk ⊗ Yk . k

160 Let us introduce formal variables z1, . . . , zl. The trigonometric Knizhnik-Zamolodchikov connection is a collection of diﬀerential operators ij ij KZ,C ∂ i X ziΩC + zjΩ−C ∇i = zi − H − ~ (6.3) ∂zi zi − zj j6=i on the space

C[~](z1, . . . , zl) ⊗ V1 ⊗ · · · ⊗ Vl.

Here H ∈ h∨ is any element in the Cartan subalgebra. These operators commute. We think geometrially about this object as a connection vector bundle with ﬁber

l V1 ⊗ · · · ⊗ Vl over the base which is an open part of A with coordinates z1, . . . , zl. Commutativity of the operators means the flatness of the connection. To combine the connections for various H in one object, let us extend the base to an open subset of Al ⊗ A (g∨) ' Al+r, where r is the rank of g∨. If we fix a basis a ∨ l ∨ 1 r basis hk in h , then A ⊗ A (g ) has coordinates z1, . . . , zl and h , . . . , h , the vectors hk ∈ h form the dual basis and can be identified with vectors in h∨ via K (•, •). The connection in z-direction will be of form (6.3) with with

i X k i H = h hk. k

For the a-direction we use the operators

X i Hk = hk = hk. i

These are not diﬀerential operators, but we need them for comparison.

∨ ∨ The h -invariance of ΩC and commutativity of h implies that these operators commute.

The commutativity with the Cartan (which are the operators Hk) allows us to restrict these operators from the whole tensor product V1 ⊗ · · · ⊗ Vl in the ﬁber to its

161 weight subspace

V1 ⊗ · · · ⊗ Vl [µ] .

KZ,C For further comparison it is convenient to add to the operators ∇i an extra term

KZ,C KZ,C ~ X i ∇b = ∇ − h H . i i 2 k k k

Equivalently, one replaces Hi in (6.3) by

i i ~ X i Hb = H + h H . 2 k k k

After restriction to the subspace of weight µ this gives operators

 i i ~ X i X k ~ k i Hb = H + h hh , µi = h + K µ, h h . 2 k k 2 k k k

KZ,C i i ∨ The operators ∇b i still commute because Hb and H diﬀer by a shift in A (h ).

To see explicitly the terms of the quantum multiplication in trigonometric KZ connection it is useful to expand the operator in a power series in the region

|z1| |z2| · · · |zl|, i.e. write base change this operator as an operator on

∗ ∨ C((z1/z2 . . . zl−1/zl)) ⊗ Sym (h ⊕ C~) ⊗ V1 ⊗ · · · ⊗ Vl[µ]

Then the connection becomes

162 " # KZ,C ∂ i X ji X ij ∇b = zi − Hb − Ω − Ω i ∂z ~ −C −C i ji " # X (zj/zi) X (zi/zj) − Ωij − Ωij ~ 1 − (z /z ) 1 − (z /z ) ji i j

x ∞ where one treats fractions as power series P xi. We see that it has almost the 1 − x i=1 same terms as the quantum multiplication. In particular, the ﬁrst term in brackets in the expansion corresponds to the classical part in the multiplication by a divisor, and the second term in brackets is the quantum correction (modulo some gauge transform).

6.2 Quantum connection

2 Let us consider a trivial vector fiber bundle over the affine associated to HT (X) with ∗ the fiber QHT (X). The quantum connection is the following collection of differential operators

Q ∇D = ∂D − D∗, (6.4)

2 d d where D ∈ HT (X) and the operators ∂D act as if formally q = e

d d ∂Dq = hD, di q

d ∗ ∗ on q with d ∈ H2 (X, Z), and trivially on q-constant classes HT (X) ⊂ QHT (X). The pairing is well-deﬁned if since it vanishes for all constant equivariant classes D,

∗ ∗ D ∈ HT (pt) ⊂ HT (X). ∗ Because of the vanishing of ∂D when D ∈ HT (pt), we have that for these D the Q operator ∇D just multiplies by D and these directions are less interesting. This allows one to reduce base to

2 2 2 HT (X) / HT (pt) = H (X) ,

163 2 2 by taking any subspace of HT (X) complementary to HT (pt).

A 2 2 From (3.4.7) we know that the classes c1 (Ei) and a basis in HT (pt) generate HT (X), possibly with some relations.

A Taking Di = c1 (Ei) in (6.4) we have diﬀerential operators

∂ Q ei ∇i = q − Di ∗ . ∂qei

 By Theorem 5.3.7 their action on StabC (p) is the following

Q ∇i StabC (p) = −Di ∗ StabC (p)

" ei ij ej ij # i X q ΩC, (p) + q Ω−C, (p) = StabC − H (p) − ~ qei − qej j6=i " # ej ij ei ij X q K (p) X q K (p) + ~ StabC + . qei − qej qei − qej ji

Now we perform a gauge transform to get rid of terms with Kij. Take

 ei X q d ψ = −~ K (λi, λj) ln 1 − ∈ Q[[q ]]. qej i

Then if we do a gauge transform

∂ψ Q −ψ Q ψ Q ei ∇b i = e ∇i e = ∇i + q ∂qei and compute

∂ψ qej qei ei X X q = −~ K (λi, λj) − ~ K (λi, λj) , ∂qei qei − qej qei − qej ji we get

" ei ij ej ij # Q i X q ΩC, (p) + q Ω−C, (p) ∇b i StabC (p) = StabC − H (p) − ~ . qei − qej j6=i

Now the quantum connection has the form almost identical to the one of the trigono- 164 metric Knizhnik-Zamolodchikov connection.

6.3 Identiﬁcation

λ Finally, we show that the quantum connection of Grµ can be identiﬁed with the trigonometric Knizhnik-Zamolodchikov connection. Our assumptions are that G is simply-laced and all λi in λ = (λ1, . . . , λl) We consider the Knizhnik-Zamolodchikov connection on

λ Vµ = Vλ1 ⊗ · · · ⊗ Vλl [µ] ,

where Vλi are irreducible ﬁnite dimensional representations with highest weight λi. The trigonometric Knizhnik-Zamolodchikov operators act on

λ ∗ ∨ λ Vbµ = C((z1/z2 . . . zl−1/zl)) ⊗ Sym (h ⊕ C~) ⊗ Vµ (6.5)

Let us start with writing explicitly the standard generators of Ug∨ . Fix a Weyl

chamber C, let αi, 1 ≤ i ≤ rk g be the simple roots with respect to C. Then on eαi ,

∨ e−αi , αi we have the standard relations

∨ ∨ αi , αj = 0

∨ [eαi , e−αi ] = αi

∨ ∨ αj , e±αi = αj , ±αi e±αi .

∨ This deﬁnes e±αi uniquely up to scaling via the exponent of the adjoint action of h ,

∨ in particular, eαi ⊗ e−αi is deﬁned uniquely. Similarly, if we take eα ∈ gα such that

∨ [eα, e−α] = α ,

∨ ∨ then this uniquely determines eα ⊗ e−α. The space gα ⊗ g−α has dimension one, so

165 eα ⊗ e−α is a multiple of Ωα. To ﬁgure out the coeﬃcient, look at

K (α, α) = K ([eα, e−α] , α)

= K (eα, [e−α, α])

= K (eα, 2e−α) , and using the normalization (6.1), we get

K (α, α) e ⊗ e = Ω . α −α 2 α

In particular, in simply-laced case

eα ⊗ e−α = Ωα.

Then we pick a basis in Vλi . Since λi is minuscule, the weights of this representation are W λi.

Let vν0 be the highest weight vector with respect to C. Then for any ν ∈ W λi we have a sequence

ν0, ν1, . . . , νi = ν,

such that the diﬀerence νj−1 − νj is a simple root with respect to C. Then we deﬁne

vν = eνi−νi−1 . . . eν1−ν0 vν0 .

If there are diﬀerent such sequences, then the this vector doesn’t depend on such a choice.

A straightforwad check shows that for simple αi and vν1 ⊗ vν2 ∈ Vλ1 ⊗ Vλ2   vν1+αi ⊗ vν2−αi if ν1 + αi ∈ W λ1 and ν2 − αi ∈ W λ2 eαi ⊗ eαi · vν1 ⊗ vν2 =  0 otherwise.

166 On the side of quantum connection we ﬁx the polarization

A −C p = e Np/X and we omit it in the notation.

Now we are ready to identify the sides. There is an isomorphism

λ ∼ λT Vµ −→ H∗ Grµ , C

λT given by sending vν1 ⊗ · · · ⊗ vνl to 1p, where p ∈ Grµ is the following:

∆i (p) = νi for all i. Then we can get a isomorphism

T λ ∼ T λT StabC T λ H∗ (pt)loc ⊗ Vµ −→ H∗ Grµ , C −−−→ H∗ Grµ, C , loc ∼ loc

which we denote by SC. Then the following diagrams

ij T λ Ωα T λ H∗ (pt)loc ⊗ Vµ H∗ (pt)loc ⊗ Vµ

SC SC ij HT Grλ, Ωα HT Grλ, ∗ µ C loc ∗ µ C loc

Ωij T λ 0 T λ H∗ (pt)loc ⊗ Vµ H∗ (pt)loc ⊗ Vµ

SC SC Ωij HT Grλ, 0 HT Grλ, ∗ µ C loc ∗ µ C loc commute for a simply-laced G. The operators Hb i and Hi coincide as well.

167 ei Sending zi to q provides a morphism

λ ∗ λ SC ∗ λ Vbµ ,→ ((z1/z2 . . . zl−1/zl)) ⊗ HT (pt) ⊗ Vµ −→ QHT Grµ, C loc ∼ C loc

λ where Vbµ is from (6.5) and the last arrow becomes an isomophism if there is only one relation on ei. We call this composition SC as well because they are obviously related. Then we summarize the computations in the main theorem.

λ Theorem 6.3.1. For a simply-laced G the quantum connection for Grµ matches with the trigonometric Knizhnik-Zamolodchikov connection. That is, the following diagram

∇KZ,C λ b i λ Vbµ Vbµ

SC SC Q ∇b H∗ Grλ, i H∗ Grλ, T µ C loc T µ C loc commutes.

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