UNIVERSITY OF CALIFORNIA, SAN DIEGO

The Gelfand-Zeitlin Algebra and Polarizations of Regular Adjoint Orbits for Classical Groups

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy

in

Mathematics

by

Mark Colarusso

Committee in charge:

Professor Nolan Wallach, Chair Professor Thomas Enright Professor George Fuller Professor Russell Impagliazzo Professor Allen Knutson

2007 Copyright Mark Colarusso, 2007 All rights reserved. The dissertation of Mark Colarusso is approved, and it is acceptable in quality and form for publi- cation on microfilm:

Chair

University of California, San Diego

2007

iii To Vince E. Fazari (1982-2004)

iv TABLE OF CONTENTS

Signature Page ...... iii

Dedication ...... iv

Table of Contents ...... v

Acknowledgements ...... vii

Vita and Publications ...... viii

Abstract of the Dissertation ...... ix

1 Background ...... 1 1.1 Introduction ...... 1 1.2 Summary of current research ...... 3 1.2.1 Summary of current research for M(n)...... 3 1.2.2 Gelfand-Zeitlin theory for orthogonal Lie algebras ...... 4 1.3 Poisson algebras and the Poisson structure on U(g)...... 5 1.3.1 The Poisson structure on P (g)...... 9 1.3.2 Hamiltonian vector fields on g ...... 11 1.4 Some basic symplectic geometry ...... 12 1.5 The symplectic structure on the adjoint orbits ...... 13 1.6 A classical analogue of the Gelfand-Zeitlin algebra ...... 16 1.7 Commuting vector fields derived from the classical analogue of the Gelfand-Zeitlin algebra ...... 21

2 The group A and the polarization of generic adjoint orbits in M(n) . . . 24 2.1 Integration of commuting vector fields from Gelfand Zeitlin theory and the group A ...... 24 2.2 Strongly Regular Elements and Polarizations ...... 26 2.3 The action of A on generic matrices ...... 31 2.4 The action of A on degenerate matrices ...... 32 2.5 Summary of the Γn Construction ...... 33

3 Solution Varieties ...... 35 3.1 Extension Problems ...... 35 3.2 The generic solution variety ...... 45 3.3 Ξi in the case of overlap ...... 51 ci,ci+1

v 4 The Γn Construction ...... 54 4.1 Structure of Stabilizer groups Stab(X) for X ∈ Ξi ...... 54 ci,ci+1 4.2 The Γn maps ...... 61 4.3 The analytic structure of ImΓn ...... 66 4.4 The tangent space to ImΓn ...... 72 4.5 The submanifolds ImΓn and A orbits ...... 75 sreg 4.6 The Γn construction and Mc (n)...... 77

5 Regular elements and maximal orbits of the solution variety Ξi . . . 80 ci,ci+1

sreg 6 Counting A orbits in Mc (n)...... 83 6.1 The new set of generic matrices Θn ...... 85 sreg 6.2 The nilfibre M0 (n)...... 86

7 Gelfand-Zeitlin Theory in Types Bl and Dl ...... 93 7.1 Preliminaries ...... 93 7.2 The integration of vector fields derived from the classical analogue of the Gelfand-Zeitlin algebra for Types Bl and Dl ...... 94 7.2.1 The differential of the Pfaffian ...... 99 7.3 The generic matrices for the orthogonal Lie algebras ...... 102 7.4 Generic solution varieties for orthogonal Lie algebras ...... 104 7.4.1 Extending from Type Bl to Type Dl+1 ...... 108 7.4.2 Extending from Type Dl to Type Bl ...... 119 7.5 The Γn construction for generic elements of so(n) ...... 124 7.6 The action of B on generic fibres so(n)c ...... 132

Bibliography ...... 141

vi ACKNOWLEDGEMENTS

I would like to thank Nolan Wallach for being such a wonderful thesis advisor. He has taught me much about the fields of Lie theory, algebraic geometry, and differential geometry in the last few years. I also want to thank him for his patience as I struggled to learn difficult material and for his support when I was discouraged. I am also very grateful to Bertram Kostant with whom I had many fruitful discussions. Our discussions helped me to further understand various aspects of Lie theory and helped me to form some of the results in my thesis. I would also like to thank Hanspeter Kraft for teaching me about the theory of algebraic groups and sheets and decomposition classes. Our discussions helped me to understand the larger context for some of the results in my thesis and gave me ideas for future research. Lastly, I would like to thank Beresford Parlett and Gilbert Strang for bring- ing to light the connections between my thesis research and applied matrix theory.

vii VITA

June, 1981 Born, Hamilton, Ontario, Canada

2003 B. Sc., Queen’s University

2003-2007 Teaching assistant, Department of Mathematics, Uni- versity of California San Diego

2005 M. A., University of California San Diego

2007 Associate Instructor, Department of Mathematics, University of California San Diego

2007 Ph. D., University of California San Diego

viii ABSTRACT OF THE DISSERTATION

The Gelfand-Zeitlin Algebra and Polarizations of Regular Adjoint Orbits for Classical Groups

by

Mark Colarusso Doctor of Philosophy in Mathematics

University of California San Diego, 2007

Professor Nolan Wallach, Chair

The main results of this thesis describe and construct polarizations of reg- ular adjoint orbits for certain classical groups. The thesis generalizes recent work of Bertram Kostant and Nolan Wallach (see [KW]). Kostant and Wallach con- struct polarizations of regular adjoint orbits for n × n complex matrices M(n). (X ∈ M(n) is said to be regular if it has a cyclic vector in Cn.) They accomplish n(n−1) this by defining an 2 dimensional abelian complex Lie group A that acts on M(n) and stabilizes adjoint orbits. We study the A orbit structure on M(n) and generalize the construction to complex orthogonal Lie algebras so(n). Since the n(n−1) A orbits of dimension 2 contained in a given regular adjoint orbit form the leaves of a polarization of an open submanifold of that orbit, we then obtain de- scriptions of these polarizations. For so(n) we construct polarizations of certain regular semi-simple orbits. In the case of M(n), we study the action of A on matrices whose principal i×i submatrices in the top left hand corner (i×i cutoffs) are regular. We determine the A orbit structure of Zariski closed subvarieties of these matrices defined by each cutoff having a fixed characteristic polynomial. We relate the number of A orbits of n(n−1) maximal dimension 2 in such closed sets to the number of eigenvalues shared by adjacent cutoffs. The number of such A orbits always turns out to be a power of 2. We are also able to describe all of the A orbits of maximal dimension as orbits

ix of Zariski connected algebraic groups acting on certain quasi-affine subvarieties of M(n). This work gives an explicit description of the leaves of polarizations of all regular adjoint orbits in M(n). In the case of the orthogonal Lie algebras, we construct a completely inte- grable system of commuting Hamiltonian vector fields on so(n) using a classical analogue of the Gelfand-Zeitlin algebra of the universal enveloping algebra of so(n) in the algebra of polynomials on so(n). Integrating these vector fields gives rise to an action of Cd/2, where d is the dimension of a regular adjoint orbit in an orthogonal Lie algebra. The action of Cd/2 stabilizes adjoint orbits. We then use the orbits of Cd/2 to construct polarizations of certain regular semi-simple adjoint orbits in so(n). We realize the leaves of these polarizations as orbits of a Zariski connected algebraic group.

x 1 Background

1.1 Introduction

In recent work Bertram Kostant and Nolan Wallach have revisited Gelfand- Zeitlin theory from the perspective of classical mechanics see [KW], [KW1]. In doing so they constructed polarizations of all regular adjoint orbits in the n × n n(n−1) n
complex matrices, M(n). They accomplished this by constructing an 2 = 2 dimensional abelian, simply-connected, complex Lie group, denoted by A, that acts n(n−1) on M(n) and stabilizes adjoint orbits. The A orbits of dimension 2 contained in a given regular adjoint orbit then form the leaves of a polarization of an open submanifold of that orbit. The group A is constructed by integrating certain commuting vector fields derived from Gelfand-Zeitlin theory. The associative commutator on the universal enveloping algebra of M(n), U(n), gives GrU(n) the structure of a Poisson algebra. Using the trace form, the Poisson structure on GrU(n) can be carried over to the polynomials on M(n), P (n), making P (n) into a Poisson algebra. If m ≤ n, we can think of M(m) as Lie a subalgebra of M(n) by embedding it in the m × m upper left hand corner of an n×n matrix. Using the fact that the restriction of the trace form to M(m) is non-degenerate, we can think of the polynomials on M(m), P (m), as a Poisson subalgebra of P (n). Given an X ∈ M, we refer to the i × i submatrix in the top left hand corner as the i × i cutoff and denote it by Xi. We have a classical analogue to the Gelfand-Zeitlin algebra of U(n) in P (n), namely

J(M(n)) = P (1)G1 ⊗ · · · ⊗ P (n)Gn where Gi = GL(i). J(M(n)) is a Poisson commutative subalgebra of P (n). If we

1 2

j let fi,j(X) = tr(Xi ), then using the Poisson structure on P (n), we can define a

Hamiltonian, algebraic vector field ξfi,j on M(n). ξfi,j is tangent to the adjoint orbits in M(n) and agrees with the usual symplectic Hamiltonian vector field generated by fi,j on the orbit. Kostant and Wallach then show that ξfi,j integrates to a global action of C on M(n) given by a rather simple formula: " #! exp(t j Xj−1) 0 Ad i · X (1.1) 0 Idn−i

for t ∈ C. Since J(M(n)) is Poisson commutative, the vector fields ξfi,j for 1 ≤ i ≤ n − 1, 1 ≤ j ≤ i all commute and thus their flows in (1.1) commute, giving (n) rise to an action of a group A ' C 2 on M(n). We note that the dimension of this group is exactly half of the dimension of a regular adjoint orbit of GL(n) in M(n). Kostant and Wallach show that the set of A orbits of maximal dimension n
2 is a non-empty Zariski open subset of M(n) and that a regular adjoint orbit always contains A orbits of maximal dimension. This fact allows one to construct polarizations of regular adjoint orbits of M(n). In [KW], the authors study the A action on a special set of regular semi- simple matrices, denoted by Ωn.Ωn is defined to be the set of matrices for which each cutoff Xi is regular semi-simple and with the property that two adjacent cutoffs do not share any eigenvalues. The motivation for considering what appears to be a very strange set of matrices comes from the theory of orthogonal polynomials on R. One can define a suitable measure ν on R and consider the Hilbert space L2(R, ν). The measure ν is defined so that any polynomial function φ(t) is integrable with respect to ν

(see [KW, pg 28]). We find an orthonormal sequence of polynomials {φk(t)}k∈N k by applying the Gram-Schmidt process to the functions {t }k∈N. The zeroes of the polynomials {φk(t)}k∈N are an interesting object of study. The amazing fact is that they can be realized as the eigenvalues of cutoffs of certain real symmetric matrices in Ωn (see Theorem 2.20 in [KW, pg 29]). n
Kostant and Wallach show that for X ∈ Ωn, dim(A · X) = 2 and describe n n
× (2) the action of A on Ωn via the action of an 2 dimensional complex torus (C ) .

They show that Zariski closed set of matrices in Ωn consisting of matrices each 3 of whose cutoffs have a prescribed spectrum is exactly one A orbit and that the action of the group A on such matrices can be realized by an a simply transitive, × (n) algebraic action of the torus (C ) 2 (see Theorems 3.23 and 3.28 in [KW, pgs 45, 49] ).

1.2 Summary of current research

Our work took on two directions. The first was to generalize Theorems 3.23 and 3.28 in [KW, pgs 45, 49] about the orbit structure of A to a larger classes of matrices. This work gives explicit descriptions of leaves of polarizations of all regular adjoint orbits in M(n) as orbits of abelian, connected algebraic groups acting regularly on certain quasi-affine subvarieties of M(n). The second direction of our recent work was to generalize the theory of Kostant and Wallach to orthogonal Lie algebras denoted by so(n). This involves using a classical analogue of the Gelfand-Zeitlin algebra of U(so(n)) in the polyno- mials on so(n) to construct d/2 commuting Hamiltonian vector fields, where d is the dimension of a regular adjoint orbit in an orthogonal Lie algebra. We will see that we can integrate such vector fields to an action of a complex analytic group B = Cd/2 on so(n). We study the action of B on the analogous set of regular semi-simple elements Ωn ⊂ so(n) and show that we can polarize the adjoint orbits of such elements.

1.2.1 Summary of current research for M(n)

We study a larger set of matrices Θn ⊃ Ωn. For matrices in Θn, we still insist that adjacent cutoffs Xi and Xi+1 share no eigenvalues and that each cutoff

Xi is regular, however we relax the condition that Xi is semi-simple. In this case, we find analogous results to those found by Kostant and Wallach in the case n
of Ωn. For X ∈ Θn we show that dim(A · X) = 2 . Matrices in Θn each of whose cutoffs have a prescribed characteristic polynomial form one orbit under the action of A. However, the action of A on these Zariski closed subvarieties of n
Θn is no longer realized by an algebraic torus of dimension 2 , but by different 4 commutative, connected complex algebraic groups which often have non-trivial unipotent subgroups. For a slightly more detailed summary see section 2.3 below. For the full statement of the results, please see Chapter 6, section 6.1.

We also consider the A orbit structure of matrices whose cutoffs Xi are regular for 1 ≤ i ≤ n − 1, but we allow for degeneracies in the eigenvalues of adjacent cutoffs. We give an algebraic parameterization of A orbits of arbitrary dimension for such matrices. In these cases, it turns out that the set of matrices each of whose cutoffs have a prescribed spectrum no longer form one A orbit. We are able to relate the number of A orbits of maximal dimension in this set to the number of degeneracies in the spectrum at each of the levels. It turns out that the number of such A orbits is always a power of 2. For a slightly more detailed summary see section 2.4 below. For the full statement of the results see Chapter 6. In particular, we consider the action of A on a special subset of matrices each of whose cutoffs Xi are principal nilpotent elements of M(i). We show that n
the closure of A orbits of maximal dimension 2 in this set are exactly nilradicals of Borel subalgebras that contain the standard Cartan. We find that there are 2n−1 such A orbits and in this way we are choosing 2n−1 special elements of the Weyl group of M(n) (see Chapter 6, section 6.2).

1.2.2 Gelfand-Zeitlin theory for orthogonal Lie algebras

For our purposes, we think of so(n) as n × n complex skew-symmetric matrices. This allows us to embed easily so(i) ,→ so(n) as a Lie subalgebra of so(n), by embedding an i × i skew-symmetric matrix in the top left hand corner of an n × n skew-symmetric matrix. As in the case of M(n), we again have that P (so(n)) is a Poisson algebra and P (so(i)) can be considered as a Poisson subalgebra. The key observation in the orthogonal case is that the Gelfand-Zetilin algebra of U(so(n)) has exactly the right number of generators to give rise to d/2 commuting Hamitlonian vector fields on so(n), which on an open subset of so(n) define a tangent distribution of dimension d/2. As in the case of M(n), these d/2 vector fields are derived from the analogous Poisson commutative subalgebra of 5

P (so(n)) J(so(n)) = P (so(2))SO(2) ⊗ · · · ⊗ P (so(n))SO(n)

We define Hamiltonian vector fields which are tangent to adjoint orbits ξfi,j , SO(i) where fi,j(Xi) ∈ P (so(n)(i)) is a fundamental adjoint invariant. We then show that each of these fields ξfi,j integrate to a global action of C on so(n) (see Chapter 7, section 7.2).

The vector fields ξfi,j for 2 ≤ i ≤ n − 1, 1 ≤ j ≤ rank so(i) commute on account of the Poisson commutativity of the subalgebra J(so(n)), just as in the case of M(n). Thus, we get an action of B = Cd/2 on so(n). However, unlike in the case of M(n), it is not clear that there are elements whose B orbits are of maximal dimension d/2. We show that there are in fact such elements (see Chapter

7, section 7.6). More specifically, we show that if we consider the set Ωn ⊂ so(n) defined by X ∈ Ωn if and only if Xi is regular semi-simple for each 1 ≤ i ≤ n and Xi and Xi+1 share no eigenvalues then elements of Ωn have the property that their B orbits are of maximal dimension. We also show the analogues of Theorems 3.23 and 3.28 in [KW, pgs 45, 48] hold in this case. i.e. We show that the closed subsets of matrices in Ωn for which each cutoff Xi evaluated on the fundamental invariants fi,1, ··· , fi,l (l = rk(so(i))) is constant form one B orbit. We also show that such closed subsets are homogeneous spaces for a free, algebraic action of the torus SO(2)d/2 ' (C×)d/2.

1.3 Poisson algebras and the Poisson structure on U(g)

Poisson algebras play an important role in symplectic geometry and in the theory of Lie algebras and the orbit method. In much of what follows, we will be using Poisson algebras to describe the geometry of regular adjoint orbits in the reductive Lie algebra of n × n matrices M(n) over C and the n × n complex orthogonal Lie algebra so(n). A Poisson algebra can roughly be defined as an a commutative associative algebra that is equipped with a Lie bracket that is compatible with the underlying 6 associative structure.

Definition 1.3.1. A commutative algebra P over C equipped with a bilinear bi- nary operation {·, ·} : P × P → P is said to be a Poisson algebra if the following two conditions are satisfied:

1) {·, ·} makes P into a Lie algebra.

2) Liebniz Rule: {a, bc} = {a, b}c + b{a, c}.

In other words, the bracket operation {·, ·} is not only a derivation of the Lie structure, but also is a derivation of the underlying associative structure. As mentioned above, Poisson algebras naturally arise in geometry, but they can also be constructed in a purely algebraic setting. Our discussion will follow that in section 1.3 [CG, pg 26]. Let B be a filtered, associative, but not necessarily commutative algebra over C. Recall that this means that there is an increasing sequence of C vector spaces C = B0 ⊂ B1 ⊂ · · · ⊂ Bn ⊂ · · · such that Bi · Bj ⊂ S∞ Bi+j and B = i=0 Bi. Now, given any associative, filtered algebra, we can form the associated graded algebra Gr(B) which is given by

∞ ∞ M M i Gr(B) = Bi/Bi−1 = Gr(B) i=0 i=0

i where we set Gr(B) = Bi/Bi−1 and we take B−1 = {0}. It is easy to check that we have a well-defined multiplication:

Gr(B)i × Gr(B)j → Gr(B)i+j

given by

a mod Bi−1 × b mod Bj−1 = ab mod Bi+j−1 7 that makes Gr(B) into a graded algebra over C. Even though we may not have that B is commutative, it can be the case that Gr(B) is commutative. In this situation, we make the following definition.

Definition 1.3.2. The filtered associative algebra B is said to be almost commu- tative if Gr(B) is commutative.

Remark 1.3.1. It is easy to see that B is almost commutative if and only if we have [Bi,Bj] ⊂ Bi+j−1 In this case we have a natural Poisson structure on Gr(B).

Proposition 1.3.1. If B is almost commutative, then Gr(B) is naturally a Pois- son algebra.

Proof: The idea is to note that in any associative algebra A, the associative com- mutator makes A into a Lie algebra and is also a derivation of the underlying associative structure. In other words, the associative commutator on A satisfies both conditions 1) and 2) of definition 1.3.1. However, A is itself not commutative, so it is not a Poisson algebra. If A is filtered, conditions 1) and 2) in Definition 1.3.1 are not be affected when we move to GrA. In our case Gr(B) is commuta- tive, so to use the associative commutator to get a non-trivial Poisson structure, we have to take its second order symbol. In other words, we define

{·, ·} : Gr(B)i × Gr(B)j → Gr(B)i+j−1

given by

{a1, a2} = a1 a2 − a2 a1 mod Bi+j−2

One can easily check that this operation is well-defined and can be bilinearly extended to define a skew-symmetric bilinear operation on Gr(B) × Gr(B) → Gr(B) which satisfies conditions 1) and 2) in Definition 1.3.1. We now consider one of the main examples of this type of Poisson structure. 8

Example 1.3.1. Let g be a finite-dimensional Lie algebra over C. Then one knows that the universal enveloping algebra U(g) is a filtered algebra and that Gr(U(g)) is commutative, so that U(g) is almost commutative. Thus, by Proposition 1.3.1, Gr(U(g)) is a Poisson algebra. We recall that using the Poincar´e-Birkhoff-Witt (PBW) theorem, we have a canonical isomorphism as graded algebras

Ψ: S(g) ' Gr U(g) (1.2) with S(g) being the symmetric algebra on g. Recall that this map can be described in terms of a basis for U(g) and S(g) as follows. Let g have basis {X1, ··· ,Xn}. Then by PBW, the monomials Xj1 ··· Xjk mod U (g) with P j = n and with i1 ik n−1 l n i1 < ··· < ik form a basis for Gr(U(g)) . The above isomorphism can be described in terms of the basis by noting that it sends a coset Xj1 ··· Xjk mod U (g) to the i1 ik n−1 monomial Xj1 ··· Xjk which forms part of a basis for Sn(g) (see [Knp, pg 222]). i1 ik Using this isomorphism, we can carry over the Poisson structure on Gr(U(g)) to S(g).

Remark 1.3.2. It is important to note that if x, y ∈ S1(g), then {x, y} = [x, y] recovers the Lie bracket of x, y. We can see this as follows. We note U1(g) = C⊕g, so that Gr (U(g))1 ' g as vector spaces. Thus for x, y ∈ Gr (U(g))1 ' g, we have that {x, y} = x · y − y · x = [x, y] by the defining relations for U(g).

Now, we also note that S(g) ' P (g∗) where P (g∗) denotes the polynomials on g∗. Recall that the homogenous polynomials of degree k on g∗ are the linear span of the functions k ∗ Y ∗ v → hv , xii i=1 with xi ∈ g (see [GW, pg 622]). Thus, we have a canonical Poisson structure on P (g∗). In our work, we will be concerned solely with finite-dimensional reductive Lie algebras (namely M(n) and so(n)). Such a Lie algebra g comes equipped with an invariant, non-degenerate, symmetric bilinear form. This means a bilin- ear form β which has the following invariance property: β([x, y], z) = β(x, [y, z]) 9 for all x, z ∈ g. (In the case of M(n) and so(n), we will use the trace form β(x, y) = tr(xy). In the case of a general semi-simple Lie algebra, the Killing form can be used.) Using this form one can canonically identify g ' g∗. φ ∈ g∗ is identified with unique gφ in g such that φ(x) = β(x, gφ) for all x ∈ g. Using this isomorphism, one may then identify P (g∗) with P (g), the polynomial functions ∗ on g itself. Specifically, if u ∈ P (g ) and gφ is as above, then u is thought of as polynomial on g by the rule: u(gφ) = u(φ). The upshot is that P (g) becomes a Poisson algebra. We now describe the Poisson structure on P (g) in more detail.

1.3.1 The Poisson structure on P (g)

In this section g is a reductive Lie algebra over C with non-degenerate, symmetric, invariant bilinear form β(·, ·). It will be useful to have a description of the Poisson bracket of two polynomials on f, g ∈ P (g). To do this we have to make a few preliminary definitions. Let H(g) denote the algebra of holomorphic functions on g. Then if we are ∗ ∗ given ψ ∈ H(g) and x ∈ g, then dψx ∈ Tx(g) ' g . Thus, using the form β, we can naturally identify dψx with an element of g, denoted by dψ(x) defined via the rule d β(dψ(x), z) = | ψ(x + tz) = (∂zf) (1.3) dt t=0 x z for all z ∈ g.((∂ f)x denotes the directional derivative of f in the direction of z evaluated at x). We claim that we have the following simple description of the Poisson bracket of two polynomials.

Proposition 1.3.2. If f, g ∈ P (g) and if {f, g} denotes their Poisson bracket as described in the previous section, then we have:

{f, g}(x) = β(x, [df(x), dg(x)]) (1.4) with df(x), dg(x) ∈ g are as defined in equation (1.3).

Proof: This proof is adapted from [CG, pg 36]. We observe that since both sides of 10 equation (1.4) satisfy the Liebniz rule that it suffices to prove (1.4) for f, g ∈ g∗, since g∗ generates P (g) as an algebra. We also observe that both sides of (1.4) are ∗ bilinear, so that is suffices to prove if for a basis {α1, ··· , αn} of g . To that effect, let {e1, ··· , en} be the corresponding basis of g. This means that for any y ∈ g, k αi(y) = β(y, ei). We have structure constants cij ∈ C defined via

n X k [ei, ej] = cijek k=1

∗ Now, we consider first the LHS of (1.4) {αi, αj}. Now, let γ : P (g ) → P (g) be the isomorphism described in the last section. From the definition of the Poisson structure on P (g) as coming from the Poisson structure on S(g) = P (g∗), and by Remark 1.3.2 we have that

n ! n X k X k {αi, αj} = γ({ei, ej}) = γ cijek = cijαk k=1 k=1 Thus, we obtain n X k {αi, αj}(x) = cijβ(ek, x) (1.5) k=1 for any x ∈ g. Now, we consider the RHS of (1.4). We claim that dαi(x) ∈ g = ei for any x ∈ g. To see this, consider d d | α (x + ty) = | β(e , x + ty) = β(e , y) dt t=0 i dt t=0 i i Thus, the RHS of (1.4) becomes

n X k β(x, [dαi(x), dαj(x)]) = β(x, [ei, ej]) = cijβ(x, ek) (1.6) i=1 This is equal to (1.5), which completes the proof. Q.E.D.

Remark 1.3.3. Although we will not need it in what follows, it is worthwhile to note that the RHS (1.4) can be used to define the Poisson bracket of any pair of holomorphic functions f, g ∈ H(g). This is actually a workable definition, since the RHS of (1.4) can be computed in terms of polynomial functions. We just need 11

to observe that if we have f ∈ H(g) and {y1, ··· , yn} a basis of g and {x1, ··· , xn} the dual basis in g∗, then

n X yi dfx = (∂ f)x (dxi)x i=1

yi d where (∂ f)x = dt |t=0f(x + tyi) is the directional derivative in the direction of yi. Thus, by the bilinearity of the formula in (1.4), we need only know the Poisson brackets {xi, xj} to compute {f, g} for f, g ∈ H(g). As we showed above, we can compute {xi, xj} in terms of structural constant of the Lie algebra. More precisely, the Poisson bracket of two holomorphic functions f, g ∈ H(g) is given by

X yi yj X yi yj X k {f, g}(x) = (∂ f)x (∂ g)x {xi, xj} = (∂ f)x (∂ g)x cijxk i,j i,j k

1.3.2 Hamiltonian vector fields on g

Recall from definition 1.3.1 that the Poisson bracket on P (g) is a deriva- tion of the underlying associative structure. This allows us to define algebraic

Hamiltonian vector fields ξf for f ∈ P (g) by

ξf (g) = {f, g} (1.7)

(By Remark 1.3.3, we could also make the same definition for f, g ∈ H(g)). We will justify the term Hamiltonian in section 1.5.

Using Proposition 1.3.2, we can see that the vector fields ξf have a partic- ularly simple expression in coordinates.

ξf (g)(x) = {f, g}(x) = β(x, [df(x), dg(x)]) (1.8) d = β([x, df(x)], dg(x)) = dt |t=0 g(x + t([x, df(x)]))

Thus, the tangent vector (ξf )x is just the directional derivative in the direc- tion of [x, df(x)] ∈ g evaluated at x and thus, in coordinates, it has the expression

[x,df(x)] (ξf )x = (∂ )x (1.9) 12

1.4 Some basic symplectic geometry

We now have a notion of a Poisson structure on P (g) which is purely al- gebraic in the sense that it comes from the Poisson structure on U(g) described in section 1.3. One also knows that the adjoint orbits in g have a natural sym- plectic structure, and therefore holomorphic functions on the adjoint orbits form a Poisson algebra. We want to see that this is the same Poisson structure as the one comming from U(g). Before, we do that let us review a few of the basic facts from symplectic geometry. For an introduction to symplectic manifolds [see Lee, pgs 480-490] and for a much more detailed exposition see [CdS]. Recall that a complex (real differentiable) manifold (M, ω) is symplectic if ω is a holomorphic (smooth) two form on M, which is non-degenerate and closed (i.e. dω = 0). In particular, this means that at every point x ∈ M, the tangent space Tx(M) comes equipped with a non-degenerate, anti-symmetric bilinear form

ωx. This fact allows us to define a bundle isomorphism:

ω˜ : T (M) → T ∗(M)

Using this bundle isomorphism, for any holomorphic (smooth) function f on M we can define a holomorphic (smooth) vector field Xf , referred to as the Hamiltonian vector field of f as follows,

ωx((Xf )x, v) = −dfx(v) (1.10) for v ∈ Tx(M). Using (1.10), we can define a Poisson structure on the algebra of holomorphic (smooth) functions on M. If f, g ∈ H(M) then we define

{f, g}(x) = ωx((Xf )x, (Xg)x) (1.11)

See [Lee, pg 489] or [CG, pg 21] to see this definition actually makes H(M) into a Poisson algebra. One of the most important facts about this Poisson structure is contained in the following proposition.

Proposition 1.4.1. The assignment f → Xf intertwines the Poisson bracket with commutator of vector fields. i.e. we have that

[Xf ,Xg] = X{f, g} (1.12) 13 for any f, g ∈ H(M).

Two holomorphic functions f, g ∈ H(M) are said to Poisson commute, if we have

{f, g} = ω(Xf ,Xg) = −Xgf = Xf g = 0

Remark 1.4.1. Note, that f, g Poisson commute if and only if f is constant along the Hamiltonian flow of g. (since f, g Poisson commute if and only if Xg f = 0).

1.5 The symplectic structure on the adjoint orbits

Let G be a finite dimensional complex Lie group and let g be its Lie algebra of left invariant vector fields. Then G acts on g via the adjoint action, which in the case of the classical groups is just conjugation. i.e.

Ad(g) · X = g X g−1 g ∈ G and X ∈ g. The contragradient action to the adjoint action is referred to as the coadjoint representation on g∗ and is given by

Ad∗(g)(λ)(X) = λ(Ad(g−1)(X))

∗ ∗ for g ∈ G, λ ∈ g , and X ∈ g. If λ ∈ g , let Oλ be its coadjoint orbit. The beautiful fact is that Oλ is naturally a symplectic manifold.

∗ Theorem 1.5.1. Let λ ∈ g and let Oλ be its coadjoint orbit. Then Oλ is a symplectic manifold with the symplectic form given by

ωλ(x, y) = λ([x, y]) (1.13) for x, y ∈ g.

For a proof see ([CG], pg 23). This symplectic structure is often referred to in the literature as the Kirillov-Kostant-Souriau structure. Some explanation ∗ is required here. Note that the tangent space Tλ(O) ⊂ g is spanned by vectors 14 of the form λ ◦ (−ad(X)) for X ∈ g and isomorphic as a vector space to g/gλ where gλ is the Lie algebra of the isotropy group of λ and is given by gλ = {X ∈ g| λ ◦ (−ad(X)) = 0}. Thus to define a bilinear form on Tλ(Oλ) × Tλ(Oλ) one can define it on g × g as we did in equation (1.13) and then show that it pushes down to the quotient g/gλ, which is clear from its definition. The same argument will also show that the form is non-degenerate. What takes work is to show that it is closed (see [CG]). In the cases we will be dealing with G will be a complex reductive Lie group with reductive Lie algebra g. We can use the non-degenerate, invariant, bilinear form β to transfer the symplectic structure on the coadjoint orbits to the adjoint orbits, making the the adjoint orbits into symplectic manifolds. More precisely if ∗ λ ∈ g , then Oλ 'Oxλ where xλ ∈ g is given by λ(y) = β(xλ, y) for all y ∈ g. (Where the isomorphism is given by the form β as described in section 1.3.) Now, using this isomorphism, we can compute the tangent space to Ox at x ∈ g to be:

Tx(Ox) = span{[y, x]|y ∈ g} ⊂ g (1.14)

The symplectic form on Ox is then given by

αx([y, x], [z, x]) = β(x, [y, z]) for x, y, z ∈ g. The fact that α is a symplectic form on OX follows from that fact that it is the pullback of the form ω on the coadjoint orbit Oλ given in equation (1.13) under the isomorphism induced by β. However, it is an easy computation to see that α is well-defined and non-degenerate. We first show αx is well-defined. Suppose that [y, x] = [z, x]. Then we have:

αx([y, x], [w, x]) = β(x, [y, w]) = β([x, y], w)

= β([x, z], w) = β(x, [z, w]) = αx([z, x], [w, x])

To see that the form is non degenerate suppose that we have αx([y, x], [z, x]) = 0 for all z ∈ g. Then, we compute

αx([y, x], [z, x]) = 0 ⇒ β(x, [y, z]) = 0 ⇒

β([x, y], z) = 0 ⇒ [x, y] = 0 15 by the non degeneracy of β. The fact α is closed follows from the fact that the form ω on the coadjoint orbits is closed. For the rest of our discussion, we always transfer the natural structure on the coadjoint orbits to the adjoint orbits. Now, recall equation (1.9) in section 1.3.2

[x,df(x)] (ξf )x = (∂ )x

Using this fact, we have the following proposition.

Proposition 1.5.1. Let x ∈ g and let Ox be its adjoint orbit. For any p ∈ P (g)

(or more generally for any p ∈ H(g)), we have that the vector field ξp defined in section 1.3.2 is tangent to Ox. i.e. we have

(ξp)x ∈ Tx(Ox)

Thus, we now potentially have two different notions of Hamiltonian vector

field and of Poisson bracket on an adjoint orbit Ox ⊂ g of x ∈ g, one coming from the “global ” Poisson structure of H(g) and the other from the symplectic structure on O. The next result says that these two different structures agree, so that we were justified in calling the vector fields ξf Hamiltonian in section 1.3.2.

Proposition 1.5.2. For φ ∈ H(g), x ∈ g we have that (ξφ)x ∈ Tx(Ox) and if

V ⊂ Ox is open and q = φ|Ox , then

ξφ|V = Xq|V

(where Xq is the Hamiltonian vector field derived from the symplectic structure on O). Thus we have

{φ, ψ}|Ox = {φ, ψ}Ox (1.15) (where the Poisson bracket on the LHS is the restriction of the Poisson bracket on H(g) to the orbit Ox, and the RHS is the Poisson bracket coming from the symplectic structure of Ox).

For a proof of this proposition in the case of g = M(n), see Proposition 1.3 in [KW, pg 12]. The methods used there carry over to a general reductive Lie algebra. For a proof in a slightly more general setting, see [K1, pg 5]. 16

Now, we have an immediate corollary, which will play a very important role in what follows.

Corollary 1.5.1. For f, g ∈ H(g) we have

ξ{f,g} = [ξf , ξg] (1.16)

Proof: The corollary follows immediately from Proposition 1.5.2 and Proposition 1.4.1.

1.6 A classical analogue of the Gelfand-Zeitlin algebra

We expand upon some of the facts about the universal enveloping algebra of a finite dimensional Lie algebra over C in order to set the stage for the definition of the classical analogue of the Gelfand-Zeitlin algebra of U(g) in S(g). For any n ≥ 1, we can define a natural symmetric n-multilinear map

σn : g × · · · × g → Un(g)

given by σ (X , ··· ,X ) = 1 P X X ··· X n 1 n n! τ∈Sn τ(1) τ(2) τ(n) Since this map is symmetric, it descends to a linear map

n σn : S (g) → Un(g)

Taking the direct sum of the maps σn we get a linear map

Sym : S(g) → U(g) which we denote by Sym and refer to as symmetrization. It is easy to see that the map Sym is a G-module map with respect to the G action induced on S(g) and U(g) from the adjoint action of G on g. We will now recall some facts about this map (see [Knp, pgs 225-226]). 17

n Proposition 1.6.1. The map σn : S (g) → Un(g) maps injectively onto a vector space compliment of Un−1(g) in Un(g). i.e., we have that

n Un(g) = σn(S (g)) ⊕ Un−1(g) (1.17)

Moreover, the map Sym is a filtered G-module isomorphism

Sym : S(g) ' U(g) whose associated graded map

Gr Sym : S(g) → Gr U(g) is map Ψ in equation (1.2).

Now, let U(g)G be the G invariants in U(g). U(g) is naturally filtered by G G Un(g) = U(g) ∩ Un(g). Now, we claim that we have

G n G G Un(g) = σn(S (g) ) ⊕ Un−1(g) (1.18)

G We note that inclusion (⊃) is clear. To see the other inclusion, consider x ∈ Un(g) . n By equation (1.17), we note that we have x = y + z with y ∈ σn(S (g)), z ∈

Un−1(g). We let g ∈ G act on x to get g · x = g · y + g · z. We note that Un−1(g) is n G-stable and σn is a G map, so that we have g · y ∈ σn(S (g)) and g · z ∈ Un−1(g). Thus, we have that g · x = g · y + g · z = y + z = x Since the sum in (1.17) is direct, we must have that g · y = y and g · z = z. Using the fact the σn is a G-map and is 1-1, we get the inclusion ⊂ and thus equation (1.18). Equation (1.18) tells us that Gr commutes with taking invariants. In other words, we have that Gr(U(g)G) ' Gr(U(g))G Now, the upshot of this discussion is that Gr Sym gives a map

Gr Sym : S(g)G ' Gr(U(g)G) (1.19) which is the restriction of the map Ψ in (1.2) to the invariants in S(g). Now, we recall from section 1.3 that we use the isomorphism Ψ to define the Poisson structure on S(g). We want to see that (1.19) gives us that S(g)G is a commutative Poisson algebra. This follows immediately from the following lemma. 18

Lemma 1.6.1. Let Z(g) be the centre of U(g), then we have:

U(g)G = Z(g)

Thus, we have that (1.19) becomes

Gr Sym : S(g)G ' Gr(Z(g)) (1.20) and Gr(Z(g)) is clearly a Poisson commutative subalgebra of Gr(U(g)), since Z(g) is abelian. Now, for the remainder of this section g = M(n) or g = so(n). For g =

M(n) we note the we have a natural embedding of Lie subalgebras gi = M(i) ,→ M(n), where we think of M(i) as embedded via " # X 0 X,→ 0 0n−i

If we realize so(n) as n×n complex skew-symmetric matrices, then we have an iden- tical embedding of so(i) ,→ so(n), by embedding so(i) in the top left hand corner.

Now, we note that we can also embedd the adjoint groups Gi = GL(i) or SO(i) into GL(n) or SO(n), via the embedding, " # g 0 g ,→ 0 Idn−i for g ∈ Gi. In both cases, we get a chain of inclusions of Lie subalgebras and thus of universal enveloping algebras

gi1 ⊆ g2 ⊂ · · · ⊂ gn ⇒

U(gi1 ) ⊆ U(g2) ⊂ · · · ⊂ U(gn) where gn = g = M(n) or so(n) and i1 = 1 for g = M(n) and i1 = 2 for g = so(n).

Now, let Z(gi) be the centre of U(gi). We have that S(gi) ⊂ S(g) and

U(gi) ⊂ U(g). We note that we have Gi module isomorphisms

Symi : S(gi) ' U(gi) 19 for 1 ≤ i ≤ n. We note that it is easy to see that by the PBW Theorem and the definition of symmetrization that Sym |gi = Symi. So in particular this yields that

S(gi) is a Poisson subalgebra of S(g). And since Gi ⊂ G is a subgroup, we have that Sym is also a Gi module map. Putting this together, we get

Gi GrSym = Ψ : S(gi) ' Gr(Z(gi)) ⊂ Gr(U(g)) (1.21)

Gi Thus, S(gi) is a commutative Poisson subalgebra of S(g). Now, we consider the subalgebra of S(g) generated by

Gi1 Gn S(gi1 ) , ··· ,S(gn) i.e. we consider the algebra:

Gi1 Gn Gi1 Gn J(g) = S(gi1 ) ··· S(gn) ' S(gi1 ) ⊗ · · · ⊗ S(gn)

We claim that this is a Poisson commuting subalgebra of S(g). We argue this inductively. We note that since Gr Sym : S(g)G ' Gr(Z(g)) by equation (1.20), we have that S(g)G is contained in the Poisson centre of S(g). Hence, it commutes

Gi Gn−1 with all of the subalgebras S(gi) for i1 ≤ i ≤ n. Now, we consider S(gn−1) , which via Gr Sym is isomorphic to Gr(Z(gn−1)). Observe that since U(gi) ⊂

U(gn−1) for 1 ≤ i ≤ n − 2, we have that Z(gn−1) commutes with U(gi) for 1 ≤

Gn−1 i ≤ n − 1. So that S(gn−1) Poisson commutes with S(gi) for 1 ≤ i ≤ n − 1. Proceeding in this fashion, we see that J(g) is a Poisson commutative subalgebra of S(g). We are now ready to define the Gelfand-Zeitlin algebra of U(g) and to see that J(g) is a classical analogue to the Gelfand-Zeitlin algebra in S(g). We define the Gelfand-Zeitlin algebra to be the associative subalgebra of U(g) generated by the centres Z(gi) ⊂ U(gi) ⊂ U(g). Thus, we define

GZ(g) = Z(gi1 ) ··· Z(gn) (1.22)

We observe that GZ(g) is an abelian subalgebra of U(g). One argues this inductively. The argument is completely analogous to the one we gave to see that

J(g) is Poisson commuting. We will do the first few steps. Since U(gi1 ) ⊂ U(g2), we have the centre of Z(g2) of U(g2) commutes with all of U(gi1 ), thus it commutes with Z(gi1 ). This gives that the associative algebra generated by Z(gi1 ) and Z(g2) 20

is abelian. Now, since U(gi1 ) ⊂ U(g2) ⊂ U(g3), Z(g3) commutes with Z(gi1 ) Z(g2) and since Z(g3) is abelian, the associative algebra Z(gi1 ) Z(g2) Z(g3) is abelian, and so on. Now, we have the following deep result (see Theorem 10.4.5 [Dix, 343]).

Theorem 1.6.1. Let S(g)G denote the G invariant polynomials on g∗, (here we are identifying S(g) with P (g∗)), then there exists an algebra isomorphism

Z(g) ' S(g)G

The formula for such an isomorphism involve twists of the symmetrization map described above (due to Harish-Chandra and simplified by Duflo). Thus, we can consider J(g) as an analogue of the Gelfand-Zeitlin algebra. We will see later that GZ and J(g) are free commutative algebras on the same number of generators, so that they are abstractly isomorphic.

Now, recall that we have that S(gi) is a Poisson subalgebra of S(g). Now g = M(n), so(n), being reductive, comes equipped with an invariant, non-degenrate ⊥ ⊥ form, namely the trace form. One can easily show that g = gi ⊕ gi where gi denotes the orthogonal compliment of gi with respect to the trace form. This in particular means that the restriction of the form to gi must be non-degenerate, since it is non-degenerate on g. Thus, we can use the trace form to establish ∗ an isomorphism gi ' gi as we did for g in section 1.3. Thus, we have that the isomorphism discussed in section 1.3 carries the subalgebra S(gi) ⊂ S(g) to the subalgebra P (gi) ⊂ P (g). Thus, we can naturally think of polynomials on gi as subalgebra of all polynomials on g. By constuction this is clearly a Poisson subalgebra, since S(gi) is a Poisson subalgebra of S(g). From now on, we identify

P (gi) with its image in P (g).

Now, we recall that we also embedd the adjoint groups Gi = GL(i) or SO(i) into GL(n) or SO(n), via the embedding, " # g 0 g ,→ 0 Idn−i for g ∈ Gi. Using this embedding, it is easy to see that the trace form on g is

Gi Gi-equivariant. Thus, it naturally maps the invariants S(gi) isomorphically onto 21

Gi P (gi) . Thus, we have an isomorphism

Gi1 Gn J(g) → P (gi1 ) ⊗ · · · ⊗ P (gn) (1.23)

Gi1 Gn So that P (gi1 ) ⊗· · ·⊗P (gn) is a Poisson commuting subalgebra of P (g). From

Gi1 Gn now, on we will refer to P (gi1 ) ⊗ · · · ⊗ P (gn) as J(g).

1.7 Commuting vector fields derived from the classical analogue of the Gelfand-Zeitlin algebra

We observed in the previous section that

Gi1 Gn Gi1 Gn S(gi1 ) ⊗ · · · ⊗ S(gn) ' P (gi1 ) ⊗ · · · ⊗ P (gn) = J(g) (1.24) is Poisson commutative. (Recall that i1 = 1 for g = M(n) and i1 = 2 for g = so(n).) For f, g ∈ J(g), we notice that by Corollary 1.5.1 we have that [ξf , ξg] =

ξ{f, g} = 0. Remark 1.7.1. Suppose f ∈ P (g)G then it is easy to see that the Hamiltonian G vector field generated by f, ξf is trivial. This follows from the fact that f ∈ P (g) is constant on an adjoint orbit and therefore ξf |Ox = 0, by Proposition 1.5.2.

Gi Let rk Gi denote the rank of the group Gi. Now, if we take fi,j ∈ P (gi) for i1 ≤ i ≤ n − 1, 1 ≤ j ≤ rk Gi to be a choice of fundamental adjoint invariants, then we see that we have a commutative Lie algebra of vector fields

L = span{ξfi,j |i1 ≤ i ≤ n − 1, 1 ≤ j ≤ rk Gi} (1.25)

Now, the upper bound on the dimension of L is the sum m = Pn−1 rk g . i=i1 i If we let d be the dimension of a regular adjoint orbit of G in g, then the amazing fact is that in the cases of g = M(n) or g = so(n) we have

m = d/2 22

Thus, we have potentially found a commuting system of symplectic vector fields on the adjoint orbits OX in g of dimension exactly half the dimension of a regular adjoint orbit. If we could integrate this Lie algebra, it should in theory allow us to construct polarizations of regular adjoint orbits in g.

Remark 1.7.2. We note that we could have done the same analysis with the Gelfand-Zeitlin algebra and the Poisson commuting Lie algebra J(g) for Lie al- gebras of Type Cl. (In this case, we would have gi = sp(2i, C) for 1 ≤ i ≤ n − 1.) We would also have obtained that the Lie algebra L in equation (1.25) was commu- Pn−1 tative. However, in this case m = i=1 rk gi is less than d/2. So this commuting system would not allows to construct polarizations of regular adjoint orbits. For example with the Lie algebra C5 (i.e. sp(10)), one computes that m = 10, but d/2 = 25. This indicates that there is no direct generalization to Lie algebras of

Type Cn. Moreover, no subalgebra of dimension d/2 of the commutative Lie algebra of vector fields V = {ξf |f ∈ J(g)} could be integrated to a group action which would have orbits of dimension d/2. This is because any subalgebra V 0 of V defines 0 a collection of subspaces Vx ⊂ Vx = {(ξf )x|f ∈ J(g)} ⊂ Tx(g) for x ∈ g, which are of dimension at most m. This follows from the fact that Vx = span{(ξqi )x|i ∈ I} where {qi}i∈I generate the algebra J(g). Thus, the orbits of group constructed from integrating the subalgebra V 0 could be of at most dimension m.

Let us recall what we mean by a regular or regular adjoint orbit and a polarization in this context.

Definition 1.7.1. Let g be a finite dimensional, reductive Lie algebra over C.

Then X ∈ g is regular if and only if its adjoint orbit OX is of maximal dimension dim g − rank g. Equivalently, X ∈ g is regular if and only if its centralizer is of minimal dimension rank g.

The following definition is adapted from ([Ki, pg 24]).

Definition 1.7.2. We let (M, ω) be a symplectic manifold. Let TM be the tangent bundle on M. A polarization P ⊂ TM is an integrable subbundle of the tangent bundle TM such that at each point m ∈ M, P (m) is a maximal isotropic subspace 23

⊥ (i.e. P (m) = P (m) ) of the symplectic vector space over C,(Tm(M), ωm). In particular this means that dim P = 1/2 dim M.

The integral submanifolds of P are called leaves of the polarization and are necessarily Lagrangian submanifolds of M. Thus, to be able to use the Lie algebra L defined in equation (1.25) to construct polarizations of regular adjoint orbtis, we would have to show essentially things:

1) The Lie algebra L has dimension d/2 and the points y ∈ OX for which dim L = d/2 form a submanifold of OX on which the assignment

y → span{(ξfi,j )y|i1 ≤ i ≤ n − 1, 1 ≤ j ≤ rk Gi} defines a distribution.

2) Every regular adjoint orbit Ox contains elements x for which the subspace of the tangent space span{(ξfi,j )x|i1 ≤ i ≤ n − 1, 1 ≤ j ≤ rk Gi} ⊂ T (Ox) is of maximal dimension d/2. Since, the Lie algebra L is commutative, the distribution would be inte- grable, but we also want to have natural coordinates on the leaves so that we can describe the geometry of the polarizations. We will see that we will actually be able to show the vector fields in L integrate to a global action of Cd/2 on g. We will also observe that the tangent space to a point x in one of these orbits is given by

span{(ξfi,j )x|i1 ≤ i ≤ n − 1, 1 ≤ j ≤ rk Gi}

In the case of g = M(n) 1) and 2) were proved by Kostant and Wallach in [KW]. In the case of g = so(n), we will show that the Lie algebra in equation (1.25) can be integrated to a group action on all of so(n), whose orbits of maximal dimension can be used to construct polarizations of certain regular adjoint orbits. However, it is still open if 2) is valid for g = so(n) and if one can use this Gelfand- Zeitlin system to construct polarizations of all regular adjoint orbits in so(n). Let us now turn our attention to the work of Kostant and Wallach in the case of g = M(n) which was the foundation of our current research. 2 The group A and the polarization of generic adjoint orbits in M(n)

2.1 Integration of commuting vector fields from Gelfand Zeitlin theory and the group A

We first need to introduce some terminology that will be used throughout the rest of the thesis. If X ∈ M(n) we call the i×i upper left-hand corner of X the i × i cutoff of X and denote in by Xi. We think of Xi ∈ M(n) via the embedding M(i) ,→ M(n) which sends Y ∈ M(i) to " # Y 0

0 0n−i

Recall the algebra J(g) defined in equation (1.24) in section 1.7.

G1 Gn J(g) = P (g1) ⊗ · · · ⊗ P (gn) ⊂ P (g) where gi = M(i) and Gi = GL(i). Recall that we think of Gi as embedded in GL(n) in the top left hand corner. We can define a commutative Lie algebra of vector fields by

V = {(ξf )|f ∈ J(M(n))}

24 25

By evaluating at X ∈ M(n), we get a collection of tangent spaces

X → VX = {(ξf )X |f ∈ J(M(n))}

In Remark 2.8 in [KW, pg 20] the authors observe that if J(M(n)) is generated by

(qi)i∈I , then VX is spanned by the tangent vectors (ξqi )X for i ∈ I. Thus, by our discussion in section 1.7, we have that

dim VX ≤ d/2

n
In the case of M(n) we have that d/2 = 2 . To integrate the Lie algebra L (n) in equation 1.25 to a specific action of C 2 , we choose a special set of generators j for J(M(n)). For 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ i let fi,j = tr(Xi ). Denote by a the commutative Lie algebra of vector fields L on M(n) for this choice of generators. i.e. we have that

a = span{ξfi,j |1 ≤ i ≤ n − 1, 1 ≤ j ≤ i}

Kostant and Wallach prove the following theorem in [KW, pgs 32-34].

Theorem 2.1.1. The Lie algebra a integrates to an action of a commutative com- (n) plex analytic group A ' C 2 on M(n) with Lie(A) = a. For X ∈ M(n), the tangent space to the A orbit of X is the subspace VX .

Remark 2.1.1. One can actually realize the A orbits more explicitly as the compo- sition of the commuting flows of the complete vector fields ξfi,j . One can compute that the vector field ξfi,j integrates to a global action of C on M(n) given by " #! exp(t j Xj−1) 0 Ad i · X 0 Idn−i

Since the vector fields ξfi,j commute, we can compose these flows in any order we (n) like to get an action of C 2 on M(n) whose orbits are the orbits of A.

Remark 2.1.2. We note that since each vector field ξf is tangent to an adjoint orbit

OX , we see that the action of A stabilizes the adjoint orbits. 26

Remark 2.1.3. One may ask what would happen if we had chosen a different set of generators for J(M(n)). For example, we could have chosen fi,j to be the coefficient j−1 of t in the characteristic polynomial for Xi. Then, one can show that in this case the Lie algebra L in (1.25) integrates to the action of an isomorphic group 0 0 0 A on M(n) and for X ∈ M(n), TX (A · X) = VX . The action of A commutes with the action of A and has the same orbit structure as that of the action of A on M(n). Since we are concerned with studying the geometry of these orbits, we lose nothing by choosing to study the action of the particular group A. For more details see Theorem 3.5 in [KW, pgs 34-35].

Since any A · X has tangent space equal to VX for any X ∈ M(n), we see n
that dim(A · X) ≤ 2 . We will now turn our attention to studying those elements of M(n) whose A orbits are of maximal dimension.

2.2 Strongly Regular Elements and Polarizations

We say that an element X ∈ M(n) is strongly regular if and only if its A n
orbit is of maximal dimension 2 . We denote the set of strongly regular elements of M(n) by M sreg(n). Note that n n dim A · X = ⇔ dim V = 2 X 2 Strongly regular is related to the term regular by the following theorem (see Theorem 2.7 in [KW, pg 19]). (Recall Definition 1.7.1).

sreg Theorem 2.2.1. X ∈ M (n) if and only if the tangent vectors (ξfi,j )x ∈ Tx(Ox) are linearly independent for 1 ≤ i ≤ n − 1, 1 ≤ j ≤ i and if and only if the differentials (dfi,j)x are linearly independent for all i, 1 ≤ i ≤ n and 1 ≤ j ≤ i.

Now, we recall a fundamental theorem of Kostant (see [K2, pg 382]).

Theorem 2.2.2. Let g be a semi-simple Lie algebra. Let φ1, ··· , φl (where l is the rank of g) be a basic set of adjoint invariants, then X ∈ g is regular if and only if

dφ1(X) ∧ · · · ∧ dφl(X) 6= 0 27

sreg Using Theorem 2.2.2, we see that if X ∈ M (n) each cutoff Xi is regular. However, this is not sufficient for X ∈ M sreg(n). We also note that the condition that df1,1(X) ∧ · · · ∧ dfn,n(X) 6= 0 is a Zariski open condition on M(n). Thus,

M sreg(n) is a Zariski open subset of M(n)

We now describe a canonical set of strongly regular elements.

Definition 2.2.1. A matrix X ∈ M(n) is said to be upper Hessenberg if it is of the form:

  a11 a12 ··· a1n−1 a1n    1 a ··· a a   22 2n−1 2n    X =  0 1 ··· a3n−1 a3n     ......   . . . . .    0 0 ··· 1 ann with aij ∈ C for i ≤ j. We denote the set of upper Hessenberg matrices by Hess.

Remark 2.2.1. Note that Hess is nothing other than the affine variety f + b where b is the standard Borel subalgebra of upper triangular matrices and f is the sum of negative simple root vectors.

The beautiful fact about Hessenberg matrices is that they provide a cross- section to the following regular map.

n(n+1) Φn : M(n) → C 2

given by

Φn(X) = (p1,1(X1), ··· , pi,j(Xi), ··· , pn,n(Xn))

(Where pi,j is the elementary symmetric polynomial of degree i − j + 1 in the n(n+1) −1 eigenvalues of Xi). For c ∈ C 2 we refer to Φn (c) as Mc(n). Note that

X,Y ∈ Mc(n) if and only if each of their cutoffs Xi and Yi have the same eigenvalues n(n+1) counting multiplicity. Thus, for c ∈ C 2 we write c = (c1, ··· , ci, ··· , cn) with 28

i ci ∈ C and think of ci as representing the coefficients of a monic polynomial of degree i (excluding the leading term). The theorem concerning the Hessenberg matrices is the following one (see Theorem 2.3 in [KW, pg 16]):

Theorem 2.2.3. The map Φ is surjective and its restriction to Hess is an alge- braic isomorphism. In particular the differentials dpi,j(X) for 1 ≤ i ≤ n, 1 ≤ j ≤ i are linearly independent for X ∈ Hess.

−1 (n+1) Remark 2.2.2. Thus, the fibre Φ (c) = Mc(n) for c ∈ C 2 contains a unique n+1
Hessenberg matrix. This says that if one is given any sequence of 2 complex numbers, then there exists a matrix X ∈ M with those numbers as eigenvalues of the cutoffs Xi for 1 ≤ i ≤ n, and moreover that matrix is unique if we insist that it is upper Hessenberg. The amazing fact is that there is no compatibility condition between the eigenvalues of neighbouring cutoffs. Compare this with interlacing condition that is required in Hermitian case (see [HJ, pg 185]).

Since the differentials dpi,j(X) for 1 ≤ i ≤ n, 1 ≤ j ≤ i are linearly independent, we have : Hess ⊂ M sreg(n)

Remark 2.2.3. Since the differentials dpi,j(X), 1 ≤ i ≤ n, 1 ≤ j ≤ i are linearly independent for X ∈ Hess, we have that the functions pi,j are algebraically in- dependent for 1 ≤ i ≤ n, 1 ≤ j ≤ i. Thus, we see that the Gelfand-Zeitlin Algebra GZ(g) of g = M(n) defined in equation 1.22 and the algebra J(M(n)) = P (M(1))GL(1) ⊗ · · · ⊗ P (M(n))GL(n) are both commutative algebras which are free n+1
on 2 generators and hence they are abstractly isomorphic as we promised in section 1.6.

As we will see below, it will be important to understand the interaction of the group A with the fibres Mc(n) of the map Φn. From the Poisson commutativity of the functions fi,j, it immediately follows that we have (see Proposition 3.6 in [KW, pg 35]).

(n+1) Proposition 2.2.1. The action of A stabilizes the fibres Mc(n) for any c ∈ C 2 . 29

Sketch of Proof: This proposition follows immediately from the fact that two functions on a sym- plectic manifold Poisson commute if and only if they are invariant under each others’ Hamiltonian flows (see Remark 1.4.1) in section 1.4. We recall that the A orbits are realized as the composition of the Hamiltonian flows of the vector fields

ξfi,j . Q.E.D.

We want in particular to understand the A orbits of maximal dimension in the fibre Mc(n). We denote

sreg sreg Mc(n) ∩ M (n) = Mc (n)

sreg and we note that Mc (n) is a Zariski open subset of Mc(n). Note that it is clear sreg that the action of A preserves the fibres Mc (n).

Now, by Remark 2.1.2, the adjoint orbit OX is a union of A orbits. Now, we can use the existence of strongly regular elements to construct polarizations of all generic adjoint orbits in M(n). The crucial theorem is the following one (see Theorem 3.36 in [KW, pg 53]).

sreg Theorem 2.2.4. Let X ∈ M(n) and let OX denote the set of strongly regular sreg sreg sreg elements in OX i.e. OX = OX ∩ M (n). Then OX is non-empty if and only sreg if X is regular, in which case OX is a Zariski open subset of OX and is therefore a symplectic manifold (in the complex sense). Moreover, the A orbits of dimension n
sreg 2 in OX are the leaves of a polarization of OX .

Sketch of Proof: sreg If OX is non-empty, then by Theorem 2.2.1 it is clear that X is regular. For the converse note that if X is regular then, it is similar to a companion matrix, which is an upper Hessenberg matrix and thus a strongly regular element in M(n).

The tangent space to each A orbit in OX is an isotropic subspace of the symplectic vector space Tx(OX ) by the Poisson commutativity of the functions fi,j sreg and by Proposition 1.5.2. Thus, for X ∈ OX , TX (A · X) is maximal isotropic, since it is isotropic and of dimension exactly half the dimension of the ambient 30

sreg n
manifold OX . Recall that 2 is exactly half the dimension of the regular adjoint 2 sreg orbit dim OX = n − n. Thus, the A orbits in OX are Lagrangian submanifolds sreg of OX and we have our desired polarization. Q.E.D.

Thus, we will turn our attention to studying the structure of the A orbits of maximal dimension in a given regular adjoint orbit OX . To do that, we need sreg to study the A orbit structure of the fibres Mc (n). The reason for this being that if X,Y ∈ M(n) are regular, then Y ∈ OX if and only if X,Y have the same characteristic polynomial. Thus, to describe the strongly regular elements in OX sreg n it is enough to describe Mc (n) for c = (c1, ··· , cn) with cn ∈ C determined by sreg OX . We can then get a description of the leaves of the polarization of OX by sreg describing the A orbit structure on the fibres Mc (n) for this particular choice of n cn ∈ C . sreg In [KW], the authors abstractly describe the fibres Mc (n) as smooth quasi-affine subvarities of M(n) with finitely many irreducible components which sreg coincide exactly with the A orbits in Mc (n). (see Propositions 3.10, 3.11 and Theorem 3.12 in [KW, pgs 37-38]).

sreg SN(c) sreg Theorem 2.2.5. Let Mc (n) = i=1 Mc,i (n) be the irreducible component sreg sreg decomposition of the variety Mc (n). Then each irreducible component Mc,i (n) sreg n
sreg is smooth in Mc (n) and is of dimension 2 so that Mc (n) is of pure dimension n
sreg 2 . Moreover the irreducible components Mc,i (n) are precisely the A orbits in sreg sreg Mc (n). Hence for X ∈ M (n), A · X is an irreducible, non-singular variety n
of dimension 2 .

sreg In [KW] the authors go on to determine the A orbit structure of Mc (n) (n+1) for c in a certain principal open subset of C 2 . The goal of much of our work sreg was to understand in more detail the A orbit structure of Mc (n) for any c ∈ C and to also come up with an algebraic parametrization for the A orbits in M sreg(n) and the larger set of matrices M(n) ∩ S, where S denotes the set of matrices each of whose cutoffs Xi for 1 ≤ i ≤ n − 1 are regular. Let us first, consider the generic matrices studied by Kostant and Wallach. 31

2.3 The action of A on generic matrices

Kostant and Wallach describe the orbit structure of the group A on a special set of matrices Ωn, consisting of matrices for which each Xi is regular diagonaliz- able (has distinct eigenvalues) and the cutoffs Xi and Xi+1 have no eigenvalues in n(n+1) i common. Recall that for c ∈ C 2 we write c = (c1, ··· , ci, ··· , cn) with ci ∈ C and think of ci as representing the coefficients of a monic polynomial of degree i (n+1) (excluding the leading term). We consider the set of c ∈ C 2 such that the poly- nomial represented by ci has distinct roots and the monic polynomials represented (n+1) by ci and ci+1 have no roots in common. We say that such a c ∈ C 2 satisfies the S eigenvalue disjointness condition. We note that Ωn = c Mc(n), where the union is taken over all c that satisfy the eigenvalue disjointness condition. One can show that Ωn is a Zariski principal open subset of M(n) (see Remark 2.16 in [KW, pgs

24-25]). The theorems concerning the action of A on Ωn proved by Kostant and Wallach are Theorems 3.23 in [KW, pg 45] and Theorem 3.28 in [KW, pg 49]. We combine the important facts of both results in a single statement below.

(n+1) Theorem 2.3.1. The elements of Ωn are strongly regular. Let c ∈ C 2 satisfy sreg the eigenvalue disjointness condition, then the fibre Mc(n) = Mc (n) is exactly × (n) one A orbit. As a variety Mc(n) ' (C ) 2 . Moreover, the fibre Mc(n) is a single × (n) orbit of a free, algebraic action of the torus (C ) 2 .

We were able to generalize this result to a larger class of matrices Θn, which contain Ωn as an open, dense subset. For matrices in Θn, we relax the condition that each cutoff Xi is diagonalizable, but still insist that it is regular. We also still insist that the spectra of two adjacent cutoffs have no intersection. Our result concerning Θn is the following theorem.

n(n+1) Theorem 2.3.2. The elements of Θn are strongly regular. For c ∈ C 2 such that ci and ci+1 represent coefficients of relatively prime monic polynomials, we sreg have Mc (n) = Mc(n) ∩ Θn is exactly one A orbit. Thus on Θn the upper Hessenberg matrices form a cross-section to the A action. Moreover, the fibre sreg sreg Mc (n) = A · X (X ∈ Mc (n)) is a single orbit of a free, algebraic action of the 32

complex, commutative connected algebraic group G = GX1 × GX2 × · · · × GXn−1 .

(Here GXi ⊂ Gl(i) denotes the group of the centralizer of Xi. We think of Gl(i) ,→ Gl(n) as embedded in the top left hand corner.)

2.4 The action of A on degenerate matrices

We now describe our results for the A orbit structure on an arbitrary fibre sreg Mc (n), which give descriptions of polarizations of all regular adjoint orbits in M(n). Recall from the previous section that we defined the set of generic matrices

Θn to be matrices whose cutoffs Xi are regular for 1 ≤ i ≤ n and with the property that the cutoffs Xi and Xi+1 do not share any eigenvalues. We now consider the A orbit structure on sets of matrices whose cutoffs are regular, but the spectra of two adjacent cutoffs Xi and Xi+1 intersect. Although such matrices can often be realized as simple deformations of matrices in Θn, their A orbit structure is radically different. The main results is:

n(n+1) i Theorem 2.4.1. Let c = (c1, c2, ··· , ci, ci+1, ··· , cn) ∈ C 2 with ci ∈ C . Sup- pose there are 0 ≤ ji ≤ i roots in common between the monic polynomials repre- sreg sented by ci and ci+1, 1 ≤ i ≤ n − 1. Then the number of A orbits in Mc (n) Pn−1 i=1 ji sreg is exactly 2 . Once again, we have that on Mc (n) the action of A has the same orbits of a free, algebraic action of a complex, commutative connected sreg algebraic group G = GX1 × GX2 × · · · × GXn−1 .(X ∈ Mc (n)).

Now, we have several corollaries to Theorem 2.4.1.

sreg Corollary 2.4.1. The action of A is transitive on Mc (n) if and only if c =

(c1, ··· , cn) with ci and ci+1 representing coefficients of relatively prime monic polynomials. Thus, Θn is the largest set of strongly regular matrices for which the sreg A action is transitive on the fibres Mc (n) over Θn.

sreg n−1 Corollary 2.4.2. The nilfibre M0 (n) contains 2 A orbits.

sreg We will see below in Chapter 6, section 6.2 that the nilfibre M0 (n) has a much richer structure than the corollary indicates. 33

2.5 Summary of the Γn Construction

Let us now discuss the construction that was invented to analyze the struc- n
ture of the A orbits of dimension 2 in the fibres Mc(n). Let S be the Zariski open subset of M(n) for which each i × i cutoff Xi is regular for 1 ≤ i ≤ n − 1. sreg The key idea is to parameterize the A orbits in Mc (n) and more generally in

Mc(n) ∩ S using a connected commutative algebraic group which is a subgroup of the product of centralizers GX1 × · · · × GXn−1 for X ∈ Mc(n) ∩ S. In order to construct this parametrization, we need to see that we if we are given X ∈ M(i+1), with Xi in regular Jordan form with an arbitrary characteristic polynomial that we choose the values of the last column and row of X so that X can have any monic polynomial of degree i + 1 as its characteristic polynomial. Theorem 2.2.3 indicates that this can be done for a Hessenberg matrix, but we will show that it can be accomplished for a much larger set of matrices. More specifically, we want to consider the following type of extension prob- lem. Suppose that we are given an (i + 1) × (i + 1) matrix of the form     λ1 1 ··· 0 y1,1   . .  .    0 λ .. .  .    1  0    . .  .    . .. 1  .         0 ······ λ1 y1,n1     .. .   . .         λr 1 ··· 0 yr,1    . .  .    0 λ .. .  .    r    0  . .  .   . .. 1 .           0 ······ λ y   r r,nr 

z1,1 ······ z1,n1 ··· zr,1 ······ zr,nr w with λj 6= λk for k 6= j. We want to choose the zi,j and yi,j so that this matrix has characteristic polynomial given by p(t) ∈ C[t] with p(t) monic and deg p(t) = i+1. We will see that such solutions can always be found regardless of the spectrum of the i × i cutoff or the choice of polynomial p(t). The set of such solutions will be referred to as the solution variety at level i and denoted by Ξi (Where we ci,ci+1 34

i think of ci ∈ C as denoting the coefficients of a monic polynmonial of degree i, similarly for ci+1.) We note that the group of the centralizer of the i × i cutoff in GL(i), G acts algebraically on the solution variety Ξi . We will then take i ci,ci+1 the orbits in Oi ⊂ Ξi of G that consist of regular elements of M(i + 1) for aj ci,ci+1 i n−1 1 ≤ i ≤ n − 2. At level n − 1 we can choose any orbits of Gn−1 in Ξcn−1,cn . We will then use these orbits to construct a regular map:

a1,a2,··· ,an−1 1 n−1 Γn : Oa1 × · · · × Oan−1 → Mc(n) ∩ S given by:

a1,a2,··· ,an−1 −1 −1 −1 Γn (x1, ··· , xn−1) = Ad(g1,2(x1) g2,3(x2) ··· gn−2,n−1(xn−2))xn−1 where gi,i+1(xi) conjugates xi into Jordan canonical form. The crucial theorem which we will prove that will allow us to describe and count the number of A n+1 sreg ( 2 ) orbits in Mc (n) for any c ∈ C is the following one.

a1,a2,··· ,an−1 Theorem 2.5.1. The image of the map Γn is exactly one A orbit in

Mc(n) ∩ S.

With this goal in mind, we now set out to study the extension problem and the solution varieties Ξi . ci,ci+1 3 Solution Varieties

3.1 Extension Problems

We consider the following extension problem which we will use to define the solution variety at level i,Ξi (see section 2.5). Recall that we are think- ci,ci+1 i ing of ci ∈ C as representing the coefficients of a monic polynomial of degree i−1 i i excluding the leading leading term (viz.if p(t) = z1 + z2t + ··· + zit + t , then ci = (z1, ··· , zi)). Suppose the polynomial represented by ci factors as

Qr nj Qs mj j=1(λj − t) and the one represented by ci+1 factors as j=1(µj − t) . Suppose we are given an (i + 1) × (i + 1) matrix of the form:     λ1 1 ··· 0 y1,1   . .  .    0 λ .. .  .    1  0    . .  .    . .. 1  .         0 ······ λ1 y1,n1     .. .   . .    (3.1)      λr 1 ··· 0 yr,1    . .  .    0 λ .. .  .    r    0  . .  .   . .. 1 .           0 ······ λ y   r r,nr 

z1,1 ······ z1,n1 ··· zr,1 ······ zr,nr w

(where λi 6= λj). Now, we want to find values for the zi,j and the yi,j such that the characterstic polynomial of the above matrix is equal to s s Y mj X (µj − t) where mj = i + 1 (3.2) j=1 j=1

35 36

We call the variety of all such solutions to this extension problem the so- lution variety at level i and denote it by Ξi . Notice that when we define the ci,ci+1 solution variety, we have inherently fixed an ordering of the Jordan blocks of the i×i cutoff Xi. To have a well-defined way of writing down elements of the solution variety Ξi , we introduce a lexicographical ordering in defined as follows. Let ci,ci+1 C z1, z2 ∈ C, we say that z1 > z2 if and only if Rez1 > Rez2 or if Rez1 = Rez2 then

Imz1 > Imz2. We assume, unless otherwise stated, that the eigenvalues of the i × i cutoff of an element in Ξi are in decreasing lexicographical order. ci,ci+1 We now show that Ξi is non-empty. The first step is to compute ex- ci,ci+1 plicitly the characteristic polynomial of the matrix in (3.1).

Proposition 3.1.1. The characteristic polynomial of the matrix in (3.1) is given by:

Qr nk (w − t) k=1,(λk − t) (3.3) h n −1 n −l i Pr nj Qr nk P j P j nj −1−l + j=1 (−1) k=1,k6=j(λk − t) l=0 j0=1 yj,j0+lzj,j0 (t − λj)

We prove Proposition (3.1.1) by first proving two special cases which we can then use in considering the general case. The first and simplest case is when the i × i cutoff of the matrix in (3.1) is a principal nilpotent Jordan block. We consider the following (i + 1) × (i + 1) matrix:   0 1 ··· 0 y1  . . .   0 0 .. . .     . . .   . .. 1 .  (3.4)      0 ······ 0 yi    z1 ······ zi w

The characteristic polynomial of the matrix in (3.4) is given in the following propo- sition.

Proposition 3.1.2. The characteristic polynomial of the matirx in (3.4) is given 37 by:

= (−1)i+1ti+1 + (−1)iwti

i i−1 Pi i i−2 Pi−1 i i−3 Pi−2 +(−1) t j=1 zjyj + (−1) t j=1 zjyj+1 + (−1) t ( j=1 zjyj+2) + ···

i P2 i +(−1) t j=1 zjyj+(i−2) + (−1) z1yi (3.5)

Proof: The proof proceeds by induction on i, the size of the principle nilpotent cutoff. One can easily work out the cases i = 2, 3 either by hand or with a computer algebra system. Thus, we may assume that the above formula holds for the case j = i, and we will show that it holds for i + 1. We evaluate the above determinant by expanding by cofactors along the last column in an elementary fashion. Consider the j − th cofactor for 1 ≤ j < i + 1.

−t 1 ··· 0 ...... 0 . 0 . . . −t 1

··· 0 0 −t 1 0 i+j+1 (−1) yj . . 0 −t .. .

...... 0 0 0 −t 1

. .. . . −t

z1 z2 ··· zj ······ zi Now, we evaluate the above cofactor by applying a sequence of elementary opera- tions. We move the j-th column to the position of the i-th column by interchanging it succesively with all columns Cl for j < l ≤ i. This results in the following de- 38 terminant

−t 1 ··· 0 0 ······ 0 ...... 0 . . . . . −t 0 0 ··· 0 1

0 ··· 0 −t 1 ······ 0 i+j+1 i−j (−1) yj(−1) . . (3.6) −t 1 .. .

...... 0 0 −t 1 0

. . −t 0

z1 z2 ··· zj+1 ······ zi zj

Now, we finally add the last column to the j-th column in (3.6) to get the following:

−t 1 ··· 0 0 ······ 0 ...... 0 . . . . . −t 1 0 ··· 0 1

0 ··· 0 −t 1 ······ 0 i+j+1 i−j (−1) (−1) yj . . (3.7) −t 1 .. .

...... 0 0 −t 1 0

. . −t 0

z1 z2 ··· zj + zj+1 ······ zi zj Now, we are finally in a position to use the inductive hypothesis on (3.7). Now, to use the inductive hypothesis, we note carefully that our matrix is of the form in the statement of Proposition 3.1.2 with w = zj + t, yj−1 = 1, yl = 0, for all l 6= j − 1 andz ˜l = zl for 1 ≤ l ≤ j − 1, z˜j = zj+1 + zj, z˜l = zl+1 for j + 1 ≤ l ≤ i − 1. We also note that we take ˜i = i − 1, since the we are now dealing with an i × i matrix and its (i − 1) × (i − 1) cutoff. Thus, applying the induction hypothesis, the above determinant becomes:

i+j+1 i−j i i i−1 i−1 i−1 i−2 (−1) yj(−1) [(−1) t + (−1) (zj + t)t + (−1) t (zj−1)+

i−1 i−3 i−1 i−4 i−2−(j−2) +(−1) t (zj−2) + (−1) t (zj−3) + ··· + t z1] 39

(where we let zk = 0 for k ≤ 0). Simplifying, we have that the above becomes:

i i−1 i−2 i−3 i−4 i−j (−1) yj[(zjt + t (zj−1) + t (zj−2) + t (zj−3) + ··· + t z1]

i i−1 i−2 i−3 i−4 i−j = (−1) [zjyjt + t (zj−1yj) + t (zj−2yj) + t (zj−3yj) + ··· + t z1yj]

Now, we have to treat the (i + 1) cofactor separately. However, this is the easiest cofactor to deal with. It is simply given by:

(w − t)(−t)i

So putting everything together, we get:

i j−1 i+1 i+1 i i i X X i−1−l (−1) t + (−1) wt + (−1) yjzj−lt (3.8) j=1 l=0 Now, to be able to connect this with the formula in the statement of this propo- sition, we have to interchange the order of summation in (3.8). Interchanging the order of summation in the last sum (3.8), we get

i−1 i X X i−1−l yjzj−lt (3.9) l=0 j=l+1 Substituting this into (3.8), we finally arrive at:

i−1 i i+1 i+1 i i i X X i−1−l (−1) t + (−1) wt + (−1) yjzj−lt (3.10) l=0 j=l+1 Consider the sum i X i−1−l yjzj−lt j=l+1 We can make a simple change of variables in the sum by putting j0 = j − l, then sum we get is: i−l X i−1−l yj0+lzj0 t j0=1 Now, making this change in (3.10), we see that we get:

i−1 i−l i+1 i+1 i i i X X i−1−l (−1) t + (−1) wt + (−1) yj0+lzj0 t l=0 j0=1 40

Comparing this equation with the one in the statement of Proposition 3.1.2, we see that we have reached the desired result. Q.E.D.

Now, using Proposition 3.1.2, we can easily find the characteristic polyno- mial of a matrix of the form (3.1), when the i × i cutoff is a single Jordan block, but not necessarily nilpotent.

Proposition 3.1.3. The characteristic polynomial of the matrix   λ 1 ··· 0 y1  . . .   0 λ .. . .     . . .   . .. 1 .       0 ······ λ yi    z1 ······ zi w is given by i−1 i−l i+1 i+1 i i i X X i−1−l (−1) (t − λ) + (−1) (w − λ)(t − λ) + (−1) yj0+lzj0 (t − λ) (3.11) l=0 j0=1 Proof: We are interested in computing the following determinantal equation:   λ − t 1 ··· 0 y1  . . .   0 λ − t .. . .     . . .  det  . .. 1 .       0 ······ λ − t yi    z1 ······ zi w − t We reduce this to the principal nilpotent case by making the change of variables t0 = t − λ ⇔ t = t0 + λ and w0 = w − λ. Then we use the result of Proposition 3.1.2 with t replaced by t0 to get that the above determinant is given by i−1 i−l i+1 0i+1 i 0 0i i X X 0i−1−l (−1) t + (−1) w t + (−1) yj0+lzj0 t = l=0 j0=1 i−1 i−l i+1 i+1 i i i X X i−1−l (−1) (t − λ) + (−1) (w − λ)(t − λ) + (−1) yj0+lzj0 (t − λ) l=0 j0=1 41

This completes the proof. Q.E.D.

Now, we can use Proposition 3.1.3 to prove Proposition 3.1.1. Proof of Proposition 3.1.1: Recall that we are interested in computing det(X − t Id) where X is the following matrix:     λ1 1 ··· 0 y1,1   . .  .    0 λ .. .  .    1  0    . .  .    . .. 1  .         0 ······ λ1 y1,n1     .. .   . .         λr 1 ··· 0 yr,1    . .  .    0 λ .. .  .    r    0  . .  .   . .. 1 .           0 ······ λ y   r r,nr 

z1,1 ······ z1,n1 ··· zr,1 ······ zr,nr w with λi 6= λj. We reduce it to the case of one Jordan block in the following manner. We consider the l-th cofactor corresponding to the j-th Jordan block of

Xi, where 1 ≤ l ≤ nj and 1 ≤ j ≤ r. We can write this cofactor as follows. We let Pj−1 kj = i=1 ni.

J − t λ1 . ..

Cj,l kj +l+i+1 (−1) yj,l . (3.12) ..

Jλr − t

z1 ··· zj ··· zr

Cj,l denotes the (nj − 1) × nj matrix given by removing the l-th row from the j-th

Jordan block of Xi − tIdi, Jλ1 − t denotes the n1 × n1 Jordan block of eigenvalue 42

nj λ1 − t, and zj ∈ C is the row vector given by zj = (zj,1, ··· , zj,nj ) (similarly for z1 ··· , zr and Jλr − t). Now, we claim that we can reduce the enormous determinant above to the following much more manageable nj × nj determinant.

r Y Cj,l l−nj +1 nk (−1) yj,l (λk − t) (3.13) k=1,k6=j zj,1 ··· zj,nj

To see this, one expands the determinant in equation (3.12) by cofactors starting with i − 1 st row. This yields the following determinant.

J − t λ1

...

Cj,l kj +l+i+1 (−1) (−1)(λr − t)yj,l . ..

Jλr − t

0 z1 ··· zj ··· zr

0 where the Jordan block for λr − t is now of size (nr − 1) × (nr − 1) and zr =

(zr,1, ··· , zr,nr−1). Now, we exapnd the above determinant by expanding by cofac- tors along the i − 2 row, etc. Continuing in this fashion and working our way up to the position of the matrix Cj,l, we arrive at the following determinant.

Pr kj +l+i+1 k=j+1 nr Qr nk (−1) (−1) yj,l k=j+1(λk − t)

  λ1 − t 1 ··· 0  . .   0 λ − t .. .   1   . .  0 . .. 1 (3.14)     0 ······ λ − t 1 . 0 ..

Cj,l

··· z1,1 ······ z1,n1 zj,1 ··· zj,nj

Pr n Notice that (−1)kj +l+i+1(−1) k=j+1 r = (−1)i−nj +l+i+1 = (−1)l−nj +1. Now, the determinant in (3.14) is evaluated the same way by expanding by cofactors first 43

along the row above the matrix Cj,l and then working up to the top left hand corner. A simple induction shows that this yields (3.13). Now, (3.13) we notice is

Qr nk the product of k=1,k6=j(λk − t) with the (l, nj + 1)-th cofactor in the expansion of the following determinant.   λj − t 1 ··· 0 yj,1  . . .   0 λ − t .. . .   j   . . .  det  . .. 1 .       0 ······ λj − t yj,nj   

zj,1 ······ zj,nj w − t

This is because (−1)l−nj +1 = (−1)l+nj +1. Thus, by equation (3.9) in the proof of Proposition 3.1.2 and using the change of variables used to deduce Proposition 3.1.3 the contribution from the j-th Jordan block of Xi −tIdi to the desired determinant is given by

r nj −1 nj −l nj Y nk X X nj −1−l (−1) (λk − t) yj,j0+lzj,j0 (t − λj) (3.15) k=1,k6=j l=0 j0=1

Now, we sum (3.15) over all the Jordan blocks, 1 ≤ j ≤ r to get:   r r nj −1 nj −l X nj Y nk X X nj −1−l (−1) (λk − t) yj,j0+lzj,j0 (t − λj)  (3.16) j=1 k=1,k6=j l=0 j0=1

Now, it remains only to consider the cofactor comming from the (i+1, i+1) entry, but this is easily caculated to be:

r Y nk (w − t) (λk − t) k=1 Adding this result to (3.16), we get the desired result at last.   r r r nj −1 nj −l Y nk X nj Y nk X X nj −1−l (w−t) (λk −t) + (−1) (λk − t) yj,j0+lzj,j0 (t − λj)  k=1 j=1 k=1,k6=j l=0 j0=1

Q.E.D. 44

Now, we want to find values for the zl,j and the yl,j so that the characterisitic polynomial in (3.3) is equal to one of the form

s s Y mj X (µj − t) where mj = i + 1 j=1 j=1

Let us call the above polynomial h and the one in (3.3) g. We can easily solve for w by insisting that the trace of the matrix (3.1) is Ps equal to j=1 mjµj. However, it is too difficult to solve for yi,j and the zi,j by equating coefficients of the polynomials g and h. We will instead equate certain derivatives of these polynomials, making use of the fact that i × i cutoff of the matrix in (3.1) is a regular Jordan form. In the next section we will study in detail the following system of equations:

(n1−1) (n1−1) g (λ1) = h (λ1)

(n1−2) (n1−2) g (λ1) = h (λ1) . .

g(λ1) = h(λ1) . . . (3.17) .

(nr−1) (nr−1) g (λr) = h (λr)

(nr−2) (nr−2) g (λr) = h (λr) . .

g(λr) = h(λr)

Recall that nj is the size of the Jordan block corresponding to the eigenvalue λj in the i × i cutoff of the matrix in (3.1).

Remark 3.1.1. Let us note that solving the above systems of equations is both Ps Pr necessary and sufficient for g = h, so long as we set w = j=1 mjµj − k=1 nkλk. Then we note that the polynomial q = g − h is of at most degree i − 1, since both g and h have leading coefficient (−1)i+1 and we have chosen w so that the trace Ps of the matrix in (3.1) is j=1 mjµj. Now, suppose that we have a polynomial 45 q ∈ C[t] such that

q(λ1) = 0 0 q (λ1) = 0 . .

(n1−1) q (λ1) = 0

n1 Then it is easy to see that (t − λ1) must divide q. Thus, if the equations in

Qr ni (3.17) are satisfied, q = g − h is divisible by the product i=1(t − λi) , since

λi 6= λj for i 6= j. Now, in our case q = g − h has degree at most i − 1. But, Pr j=1 nj = i so that q = 0 and g = h.

3.2 The generic solution variety

We preserve the notation of that last section. Now, we discuss how we can use the above system of equations (3.17) to describe the solution variety Ξi in ci,ci+1 various cases. We first describe the most generic case where λk 6= µj for any j, k.

Consider first the equations involving only λ1, i.e. consider:

(n1−1) (n1−1) g (λ1) = h (λ1)

(n1−2) (n1−2) g (λ1) = h (λ1) . (3.18) .

g(λ1) = h(λ1)

Where we recall that g is the characteristic polynomial of the matrix in (3.1), which we restate here for the convenience of the reader.   r r r nj −1 nj −l Y nk X nj Y nk X X nj −1−l (w−t) (λk −t) + (−1) (λk − t) yj,j0+lzj,j0 (t − λj)  k=1 j=1 k=1,k6=j l=0 j0=1

A closer inspection of the equations in (3.18) shows that this system only involves coefficients from the first Jordan block. That is to say that the LHS of the system of equations in (3.18) involves only the variables z1,j and y1,j for 1 ≤ j ≤ n1. This is because all of the other terms in the polynomial g are multiplied by a factor

n1 (k) of (t − λ1) and therefore evaluate to 0 in g (λ1) for 0 ≤ k ≤ n1 − 1. Thus, in 46 solving the system in (3.18), we need now only consider the following terms from g: " r n1−1 n1−l # Y nk n1 X X n1−1−l (λk − t) (−1) y1,j0+lz1,j0 (t − λ1) (3.19) k=2 l=0 j0=1

n1 n1−j Qr nr We notice that the coefficient in (3.19) of (−1) (t−λ1) k=2(t−λk) is given by the j − th row of the matrix product:     z1,1 z1,2 ··· z1,n1 y1,1  . .   .   0 z .. .   .   1,1     . .  ·  .  (3.20)  . .. z   .   1,2   

0 ··· 0 z1,1 y1,n1

Note that the equation g(λ1) = h(λ1) yields

r s n1 Y nj Y mk (−1) z1,1y1,n1 (λj − λ1) = (µk − λ1) j=2 k=1

s Q (µ − λ )mk n1 k=1 k 1 ⇒ z1,1y1,n = (−1) 1 Qr nj j=2(λj − λ1)

× ⇒ z1,1, y1,n1 ∈ C (3.21) since λj 6= λ1, 2 ≤ j ≤ r and µk 6= λ1, 1 ≤ k ≤ s. 0 0 Now, the equation g (λ1) = h (λ1) allows us to solve for y1,n1−1 uniquely as a regular function of z1,1 and z1,2 ∈ C, since λ1 6= λj. We now proceed by induction to see that we can solve for y1,n1−s uniquely as a regular function of z1,1, ··· , z1,s+1. We first observe that the s − th derivative of

" r n1−1 n1−l # n1 Y nk X X n1−1−l (−1) (λk − t) y1,j0+lz1,j0 (t − λ1) k=1,k6=1 l=0 j0=1 evaluated at t = λ1 only involve coefficients of terms of the form

q Qr nk (t − λ1) k=1,k6=1(λk − t) for q ≤ s. Thus, the s-th derivative consists of terms of the form z1,j and y1,n1−j+1 for 1 ≤ j ≤ s + 1. Now, by induction we can assume that we have solved for y1,n1−j+1 for 1 ≤ j < s+1 as uniquely as a regular function × of z1,1 ∈ C and z1,2, ··· , z1,j ∈ C. So that we need only solve for y1,n1−s. Now, the 47

triangular nature of the system in (3.20) allows us to see that y1,n1−s first occurs

n1 Qr nk s × in the coeffiecient of (−1) k=2(λk − t) (t − λ1) with coefficient z1,1 ∈ C . Thus, it first appears when one takes the s − th derivative and it has coefficient

n1 Qr nk × (−1) s ! k=2(λk − λ1) z1,1. Now, since z1,1 ∈ C and λ1 6= λk for k 6= 1 then using the induction hypothesis and the triangular nature of the system in (3.20), we see that we can solve for y1,n1−s uniquely as a function of the z1,1, ··· , z1,s+1 with z1,s+1 ∈ C.

We can argue analogously for the other λk replacing the system in (3.18) by the analogous system for λk.

Remark 3.2.1. Notice that we could have interchanged the roles of the z1,j and × the y1,j and argued in a similar manner. Now, suppose that we take y1,n1 ∈ C .

n1 n1−j Qr nr Then the coefficient of (−1) (t − λ1) k=2(λk − t) can be represented as the n1 − j + 1 − th column of the matrix product.   y1,n1 y1,n1−1 ··· y1,1  .. .  h i  0 y1,n . .  z ······ z  1  1,1 1,n1  . .   . .. y   1,n1−1 

0 ··· 0 y1,n1

Thus, we can use the same induction argument to get that z1,s+1 can be solved uniquely as a regular function of y1,n1 , ··· , y1,n1−s with

× y1,n1 ∈ C and y1,n1−1, ··· , y1,n1−s+1 ∈ C

We have now proven the following theorem.

Theorem 3.2.1. The generic solution variety Ξi is non-empty. Moreover, we ci,ci+1 have actually shown that Ξi ' ( ×)r × i−r ⊂ i. Where the isomorphism is ci,ci+1 C C C given by Φ:Ξi → ( ×)r × i−r ci,ci+1 C C

is given by

Φ(A) = (z , z , ··· , z , ··· , z ) for A ∈ Ξi 1,1 1,2 r,1 r,nr ci,ci+1 48

Proof: An X ∈ Ξi ⊂ M(i + 1) is of the form ci,ci+1     λ1 1 ··· 0 y1,1   . .  .    0 λ .. .  .    1  0    . .  .    . .. 1  .         0 ······ λ1 y1,n1     .. .   . .         λr 1 ··· 0 yr,1    . .  .    0 λ .. .  .    r    0  . .  .   . .. 1 .           0 ······ λ y   r r,nr 

z1,1 ······ z1,n1 ··· zr,1 ······ zr,nr w

× r i−r where the bottom row z = (z1,1, z1,2, ··· , zr,1, ··· , zr,nr ) ∈ (C ) ×C is arbitrary and the yi,j are regular functions in z determined in our work above. Thus the map Φ is bijective. It is clearly regular and the inverse is easily seen to be regular. This completes the proof. Q.E.D

Now, we observe that the group of the centralizer of the i × i cutoff, which we denote by G acts regularly on Ξi by conjugation. Here we are thinking of i ci,ci+1 Gl(i) ,→ Gl(i + 1) via the embedding " # Y 0 Y,→ 0 1

We now show:

Proposition 3.2.1. G acts simply transitively on Ξi and thus we have Ξi i ci,ci+1 ci,ci+1

' Gi as varieties.

Proof: We first show that G acts freely on Ξi . To show this we have to analyze the i ci,ci+1

Gi action a little more closely. Now, since Xi is a regular Jordan block, we have 49

that Gi, as an algebraic group, is a product Gi = GJ1 × · · · × GJr where GJk is the centralizer of the Jordan block corresponding to λk. Elementary linear algebra tells us that GJk consists of invertible upper triangular Toeplitz matrices.   ak,1 ak,2 ··· ak,nk  . .   0 a .. .  G =  k,1  Jk  . .   . .. a   k,2  0 ······ ak,1

× with ak,1 ∈ C and ak,j ∈ C for 2 ≤ j ≤ nk. It is easy to see that when Gi acts on Ξi via conjugation it is acting via the standard representation of G on the ci,ci+1 i last column of X and the dual representation on the bottom row. In this case, it is enough to consider the action of Gi on the last column. Since Gi = GJ1 ×· · ·×GJr the standard action on the left column is the standard diagonal action. i.e. we have:       a1,1 a1,2 ··· a1,n1 y1,1   . .    .    0 a .. .    .    1,1      . . 0   .    . ..    .    . a1,2    .         0 ······ a   y   1,1   1,n1   .   .   ..  ·  .             ar,1 ar,2 ··· at,nr   yr,1        .. .    .    0 ar,1 . .    .   0        . ..    .    . . ar,2    .        0 ······ ar,1 yr,nr which reduces to r matrix products of the form     aj,1 aj,2 ··· aj,nj yj,1  . .   .   0 a .. .   .   j,1     . .   .   . .. a   .   j,2   

0 ······ aj,1 yj,nj for 1 ≤ j ≤ r. Now, we suppose that g ∈ G is in the stabilizer of X ∈ Ξi . i ci,ci+1 50

Then we get r matrix equations of the form       aj,1 aj,2 ··· aj,nj yj,1 yj,1  . .   .   .   0 a .. .   .   .   j,1       . .   .  =  .   . .. a   .   .   j,2     

0 ······ aj,1 yj,nj yj,nj

× Now, we have to make use of the fact that aj,1, yj,nj ∈ C . The above matrix equation becomes the following system of linear equations

Pnj k=1 aj,kyj,k = yj,1 Pnj −1 k=1 aj,kyj,k+1 = yj,2 . . (3.22) Pnj −l+1 j=1 aj,kyj,k+l−1 = yj,l . .

aj,1yj,nj = yj,nj

Now, we solve the above system easily using induction. The last equation in the system clearly gives us that aj,1 = 1, since yj,nj 6= 0. Now, assume that we have been able to show that aj,1 = 1, aj,2 = 0, ··· , aj,nj −l+2 = 0. We show that this forces anj −l+1 = 0. Consider the equation:

nj −l+1 X aj,kyj,k+l−1 = yj,l k=1 Using our inductive assumption, this equation becomes:

a1,jyj,l + aj,nj −l+1yj,nj = yj,l

Since, aj,1 = 1, we have aj,nj −l+1yj,nj = 0. Again since yj,nj 6= 0, we must have aj,nj −l+1 = 0. Repeating this argument for each 1 ≤ j ≤ r, we easily see that we have g ∈ Stab(X) ⇒ g = Id. Thus, G acts freely on Ξi . So that each G orbit in Ξi is of dimen- i ci,ci+1 i ci,ci+1 sion i. Now, we have that Gi being the centralizer of a regular element in M(i) is a connected algebraic group so that each G orbit in Ξi is an irreducible i ci,ci+1 51 variety of dimension i. This implies that each orbit must be open since Ξi is ci,ci+1 itself an irreducible variety of dimension i by Theorem 3.2.1. Hence, Gi must act transitively on Ξi , since any two non-empty subsets in an irreducible variety ci,ci+1 must intersect non-trivially. This completes the proof of the theorem. Q.E.D.

We now turn our attention to investigating the structure of the solution variety Ξi when we allow for overlaps between the roots of the polynomials ci,ci+1 represented by ci and ci+1.

3.3 Ξi in the case of overlap ci,ci+1

Now suppose that λ = (λ1, ··· , λr) are eigenvalues of the i × i cutoff and we want to extend this to an (i + 1) × (i + 1) matrix whose eigenvalues are given by (µ1, ··· , µs). Now, suppose that for 1 ≤ j ≤ min(r, s), we have that λ1 =

µ1, ··· , λj = µj. Without loss of generality, we can assume that the overlaps occur in this manner. Now, an inspection of the equations in (3.17) shows that we must have the following equtions holding:

z y = 0 for 1 ≤ k ≤ j k,1 k,nk (3.23) × zk,1yk,nk ∈ C for j < k ≤ r

Now, we claim that we can find 2j disjoint subsets of the solution variety on which Gi acts simply transitively. Suppose without loss of generality that in × (3.23) we choose y1,n1 = 0 and z1,1 ∈ C . Then we claim that we can still solve the system in (3.17) for the y1,j as regular functions of the z1,j. This follows from our discussion in section 3.2. Similarly from Remark 3.2.1 in 3.2 we see that the same × holds if we had chosen instead z1,1 = 0 and y1,n1 ∈ C . The same argument works for all 1 ≤ k ≤ j. Thus, the solution variety Ξi contains at least 2j disjoint ci,ci+1 quasi-affine subvarieties, determined by choosing either zk,1 = 0 or yk,nk = 0 for each k, 1 ≤ k ≤ j but not both 0. We need a notation for these subvarieties.

First, we define an index ai that can take two values ai = U, L (U for upper and

L for lower). Then Za1,··· ,aj is the subvariety determined as follows. Consider the 52

conditions in (3.23). If for k, 1 ≤ k ≤ j, we choose the subvariety with zk,1 6= 0, we think of this as being the lower subvariety at position k and assign ak the value L.

Conversely, if we had chosen yk,nk 6= 0, we would think of this as being the upper subvariety at position k and assign ak = U. In this way it is clear that we get all j of the 2 subvarieties. The first observation about the subvarieties Za1,··· ,aj is the following

× r i−r Proposition 3.3.1. The subvarieties Za1,··· ,aj are isomorphic to (C ) × C as varieties. Thus, each Za1,··· ,aj is a smooth, irreducible affine variety of dimension i.

Proof: The proof is completely analogous to the proof of Theorem 3.2.1, except one has to modify slightly the definition of the mapping Φ : Ξi → ( ×)r × i−r in ci,ci+1 C C that proof. Now, for Za1,··· ,aj we define an isomorphism Φa1,··· ,aj : Za1,··· ,aj → (C×)r × Ci−r given as follows.

Φa1,··· ,aj (Z) = (x1, ··· , xj, xj+1, ··· , xr)

for Z ∈ Za1,··· ,aj where xk = (zk,1, ··· , zk,nk ) if ak = L or xk = (yk,1, ··· , yk,nk ) if ak = U and with xk = (zk,1, ··· , zk,nk ) for j < k ≤ r. With this definition of

Φa1,··· ,ak and our work above in Remark 3.2.1, the same argument as we used in the proof of Theorem (3.2.1) shows us that this map is the desired isomorphism. This completes the proof. Q.E.D

We again have in this case that the centralizer of the i × i cutoff Gi acts on Ξi . Now, we claim that we have the following proposition: ci,ci+1

Proposition 3.3.2. Gi preserves each of the above subvarieties and acts simply transitively on them.

Proof:

We note first that it is easy to see that Gi preserves each one of these subvarieties.

We show that Gi must act freely on the subvariety Za1,··· ,aj . Then we note that by 53

Proposition 3.3.1 gives us that each Za1,··· ,aj is an irreducible variety of dimension i. Thus, any Gi orbit in Za1,··· ,aj must in fact be open and the action of Gi must be transitive.

The proof that the action of Gi is free is very similar to the proof of Propo- sition 3.2.1 in the generic case. If Za1,··· ,aj has al = U for some 1 ≤ l ≤ j, then to see that component of the stabilizer of z ∈ Za ,··· ,a in GJ must be trivial, we 1 j λl use the exactly same argument as in the proof of Proposition 3.2.1. If we have, on the other hand, that al = L, then we argue that g ∈ Gi stabilizes z if and only −1 if g stabilizes z. Then since g acts on the bottom row of z ∈ Za1,··· ,aj via right translation by the inverse, a totally analogous argument to the one in Proposition −1 3.2.1, gives that the component of g in GJ must be trivial. More explicitly, we λl can consider the matrix equation   aj,1 aj,2 ··· aj,nj  .. .  h i  0 aj,1 . .  h i z ··· z   = z ··· z j,1 j,nj  . .  j,1 j,nj  . .. a   j,2  0 ······ aj,1

× × with zj,1 ∈ C , zj,l ∈ C for l ≥ 2 and aj,1 ∈ C , aj,l ∈ C for l ≥ 2. Equating the (1, 1) entries on both the LHS and the RHS, we see that we × have the equation zj,1 aj,1 = zj,1 ⇒ aj,1 = 1, since zj,1 ∈ C . Now, we can proceed inductively as we did in the proof of Proposition 3.2.1. The equation for the (1, k) entry is (for k ≤ nj):

zj,1 aj,k + zj,2 aj,k−1 + ··· + zj,k−1 aj,2 + zj,k aj,1 = zj,k

By induction, we have that this equation becomes:

zj,1 aj,k + zj,k = zj,k ⇒ zj,1 aj,k = 0

× which yields that aj,k = 0, since zj,1 ∈ C . Thus, by induction the component of −1 g in GJ is trivial. This completes the proof. λk Q.E.D. 4 The Γn Construction

4.1 Structure of Stabilizer groups Stab(X) for X ∈ Ξi ci,ci+1

i In this chapter, we adopt the following notation. For ci ∈ C , we think of ci as representing coefficients of a monic polynomial of degree i (excluding the leading term). Then Ξi ⊂ M(i + 1) is the solution variety at level i defined ci,ci+1 in the previous chapter with the i × i cutoff of X ∈ Ξi having characteristic ci,ci+1 polynomial prescribed by ci and X has characteristic polynomial given by ci+1.

Qr nj Let us say that the polynomial represented by ci is of the form j=1(t − λj) and the roots of the one represented by ci+1 are µ1, ··· , µs. Then we recall that X ∈ Ξi is a matrix of the form: ci,ci+1

    λ1 1 ··· 0 y1,1   . .  .    0 λ .. .  .    1  0    . .  .    . .. 1  .         0 ······ λ1 y1,n1     .. .   . .    (4.1)      λr 1 ··· 0 yr,1    . .  .    0 λ .. .  .    r    0  . .  .   . .. 1 .           0 ······ λ y   r r,nr 

z1,1 ······ z1,n1 ··· zr,1 ······ zr,nr w

54 55

Let Gi denote the centralizer of the i × i cutoff of the matrix in (4.1). Recall that G acts algebraically on Ξi . To understand the structure of the orbits i ci,ci+1 O ⊂ Ξi of G on Ξi , we need to understand the possible structure of the ci,ci+1 i ci, ci+1 isotropy groups for this action. We recall that Gi is an abelian connected algebraic group, which is the product of groups Qr G where G denotes the centralizer j=1 Jλj Jλj of the Jordan block corresponding to the eigenvalue λj as above. We also recall that the action of G is the diagonal action of the product Qr G on the last i j=1 Jλj column of X ∈ Ξi and the dual action on the last row of X. Recall also that ci,ci+1 G ' × × nj −1. To that affect we have the following Lemma. Jλj C C

Lemma 4.1.1. Let X ∈ Ξi be as above, and let Stab(X) ⊂ G be the isotropy ci,ci+1 i group or stabilizer of X under the action of G on Ξi . Then, without loss of i ci,ci+1 generality, we have that

q r Y Y Stab(X) = G × U (4.2) Jλj j j=1 j=q+1 where U ⊂ G is a unipotent Zariski closed subgroup (possibly trivial) for some j Jλj q, 0 ≤ q ≤ r.

Proof:

For ease of notation, we let GJ = GJ . Recall from the proofs of Propositions λk k × × 3.2.1 and 3.3.2 that if for some k, 1 ≤ k ≤ r we have yk,nk ∈ C or z1,k ∈ C , then the component of Stab(X) in GJk is trivial. Thus, we may as well assume that we have

zk,1 = yk,nk = 0 for all k.

We consider the component of Stab(X) in GJk . Suppose that we have yk,i 6= 0, but for i < l ≤ nk, we have that yk,l = 0. Then we consider the matrix equation:

Ak · y = y 56

where Ak ∈ GJk is the matrix given by   ak,1 ak,2 ··· ak,nk  . .   0 a .. .   k,1  Ak =  . .   . .. a   k,2  0 ······ ak,1

× nk with ak,1 ∈ C , ak,l ∈ C for 2 ≤ l ≤ nk, and y ∈ C is the column vector T y = (yk,1, ··· , yk,i, 0, ··· , 0) . This gives rise to a system of linear equations which one can solve easily by back substitution. The equation for the i th row is:

ak,1yk,i = yk,i

× Since yk,i ∈ C , this forces ak,1 = 1. So that the stabilizer, in GJk consists of unipotent, upper triangular matrices. We note that the component of Stab(X) in

GJk is automatically a Zariski closed subgroup, since it is the intersection of the closed subgroups Stab(X) and Gk. A similar computation produces the result if we had assumed that we had instead yj,l = 0 for all l and zk,i 6= 0 , but zk,l = 0 for 1 ≤ l < i.

The other case to consider is the extreme case in which one has yk,l =

0 for all l and zk,l = 0 for all l. In this case it is easy to check that the component of

Stab(X) in GJk is exactly GJk itself. Repeating this analysis for each k, 1 ≤ k ≤ r and after possibly reordering the Jordan blocks of Xi, we get the desired result. Q.E.D. We have an immediate corollary to the Lemma.

Corollary 4.1.1. For any X ∈ Ξi , the isotropy group of X, denoted by ci,ci+1 Stab(X) under the action of G on Ξi is connected. i ci,ci+1

Proof: Upon reordering the eigevalues, we can always assume that Stab(X) has the form given in (4.2) in Lemma 4.1.1. This proves the result, since unipotent algebraic groups are always connected and the groups GJj are all connected, being central- izers of regular elements in M(nj). 57

Q.E.D.

Using Lemma 4.1.1, we can prove a basic structural result about the group G when it acts on the variety Ξi . We now show that as an algebraic group i ci,ci+1 over , it is a direct product of an isotropy group of an element in Ξi and a C ci,ci+1 connected Zariski closed subgroup.

Theorem 4.1.1. Let X ∈ Ξi . Let Stab(X) ⊂ G denote the isotropy group of ci,ci+1 i X under the action of G on Ξi . Then as an an algebraic group i ci,ci+1

Gi = Stab(X) K

With K ⊂ Gi a Zariski closed, connected subgroup (possibly trivial) and K ∩ Stab(X) = {e} so that as an algebraic group, we have

Gi ' Stab(X) × K

Proof:

For the purposes of this proof we denote by G the group Gi and by H the group Stab(X). Without loss of generality, we assume Stab(X) is as given in (4.2). Let g = Lie(G) and let h = Lie(H). Now, by Lemma 4.1.1, we know that h is given by q r M M h = g ⊕ n (4.3) Jλj j j=1 j=q+1 Where g is the Lie algebra of the abelian algebraic group G and n = Lie(U ) Jλj Jλj j j is a Lie subalgebra of nnj , the strictly upper triangular matrices in M(nj). For convenience we denote g by g and G by G . Now, we want to make a special Jλj j Jλj j choice of a Lie algebra compliment to h in g. That is to say that we want to find a Lie subalgebra V ⊂ g such that

g = h ⊕ V

Where the direct sum is a direct sum of Lie algebras. Because g is abelian, we need only choose V to be a vector space compliment to h and it is automatically a Lie algebra compliment. We choose a vector space compliment V as follows. Let 58

Lr V = j=q+1 mj with mj a vector space compliment to nj in gj. i.e. gj = nj ⊕ mj for q + 1 ≤ j ≤ r. Then g = h ⊕ V .

Now, we claim that we can choose the compliments mj so that V is actually an algebraic subalgebra of g. This means that there exists a unique Zariski closed connected subgroup K ⊂ G with Lie(K) = V . Now, since each subalgebra gj ⊂ g ss n n is abelian, we have a decomposition into subalgebras gj = gj ⊕ gj with gj being ss the set of nilpotent elements of gj and gj being the set of semi-simple elements of n ss gj. Note that gj and gj are actually subalgebras, since the sum of two commuting nilpotent endomorphisms is nilpotent and the sum of two commuting semi-simple endomorphisms is semi-simple. Now, since nj ⊂ gj consists of nilpotent endomor- n n phisms, it must be a subalgebra of gj . Now, since gj is abelian, we can again find n a vector space compliment to nj in gj which is a nilpotent Lie subalgebra. Let us n call this compliment nej, so that we have gj = nj ⊕ nej. So, we now choose our mj ss to be the subalgebra mj = gj ⊕ nej.

We would now like to see that mj so defined are algebraic Lie subalgebras of gj, as this will give us that V is an algebraic subalgebra of g. We first note that nej is algebraic. This follows from the following general Lemma. Lemma 4.1.2. Let n ⊂ M(n) be the subalgebra of nilpotent upper triangular matrices. Let v ⊂ n be a subalgebra, then v is algebraic.

Proof: For simplicity, we prove the result for v is abelian. The proof can be modified to give the result for any Lie algebra v consisting of nilpotent linear transformations. We consider the regular map

exp : v → GL(n)

Let N ⊂ GL(n) be the subgroup of unipotent upper triangular matrices. We note that the map exp : n → N is an algebraic isomorphism, since it has regular inverse log : N → n. Now, v ⊂ n is closed, so the image of exp(v) = T is a subgroup that is Zariski closed and connected. We claim that Lie(T ) = v. To see this, we use Theorem 1.3.2. in [GW, pg 36] which states that if G ⊂ Gl(n) is a Zariski closed subgroup and A ∈ M(n) 59 is nilpotent, then A ∈ Lie(G) ⇔ exp(A) ∈ G. Now, let X ∈ v, then X is nilpotent, and exp(X) ∈ T , so X ∈ Lie(T ). Thus v ⊂ Lie(T ). For the reverse inclusion, let X ∈ Lie(T ), then X is nilpotent since T ⊂ N. Thus, we have that exp(X) ∈ T , so that exp(X) = exp(v) with v ∈ v. Taking logarithms, we have log(exp(X)) = log(exp(v)) ⇒ X = v. Thus, Lie(T ) ⊂ v. This completes the proof of the lemma. Q.E.D.

Remark 4.1.1. Note that the above Lemma actually holds for any Lie subalgebra v of M(n) consisting of nilpotent linear transformations, since by conjugation we may always assume that v ⊂ n.

Now, in our case we have that nej is a Lie algebra consisting of nilpotent upper triangular matrices, since gj consists of upper triangular matrices. Thus, by the above lemma it is an algebraic subalgebra. Let Nfj be the unique Zariski closed connected subgroup of Gj with Lie(Nfj) = nej (i.e. Nej = exp(enj)). Now, the ss × subalgebra nej ⊕ gj is the Lie algebra of the closed, connected subgroup C × Nfj, × where C is the semi-simple part of the abelian algebraic group Gj. Indeed, note × × ss Lr that Lie(C × Nfj) = Lie(C ) ⊕ Lie(Nfj) = gj ⊕ nej = mj. Thus, V = j=q+1 mj is an algebraic subalgebra of g. Namely, it is the Lie algebra of the Zariski closed, × connected subgroup of G given by K = Kq+1 × · · · × Kr with Kj = C × Nfj ⊂ Gj. Now, we have our candidate for the connected subgroup K in the statement of the theorem. First, we show that K ∩ H = {e}. The argument proceeds in two steps.

(A) K ∩ H is discrete (i.e. finite).

(B) K ∩ H is unipotent. Since any unipotent group is connected, a finite unipotent group must be trivial. We get (A) easily from the fact that we have constructed K with Lie(K) = V . Now, we know for algebraic subgroups of H,K ⊂ GL(n) that we have Lie(K ∩ H) = Lie(K) ∩ Lie(H), see Corollary 1.2.12 in [GW, pg 31]. However in our case 60

Lie(K) ∩ Lie(H) = V ∩ h = 0 since V is a Lie algebra compliment to h. Thus dim(Lie(K ∩ H)) = dim K ∩ H = 0 ⇒ K ∩ H is finite. This gives (A). Qr Now, (B) follows directly from the fact that K ⊂ j=q+1 Gj. Thus K ∩H ⊂ Qr Qr j=q+1 Gj ∩ H = j=q+1 Uj by equation (4.2). Thus, K ∩ H must be unipotent. So now, we have (B). Thus, we have shown that K ∩ H = {e} is trivial. Thus, as an abstract group, it is easy to see that HK ' H × K. We want to see that HK ' H × K as an algebraic group. First, we note that HK is a Zariski closed subgroup of G. HK is the image of H × K under the morphism φ : H × K → G given by φ((h, k)) = h k. Hence, HK is a constructible subset of G. The fact that HK is closed follows from the following Proposition (see [Hum 1,pg 54]) for a proof.

Proposition 4.1.1. Let T be a constructible subgroup of an algebraic group G0, then T = T .

Hence, HK is a closed subgroup of G. Now, it is easy to see that HK ' H × K as algebraic groups. Note that by Corollary 4.1.1 the group H = Stab(X) is connected. In our work above, we have seen that K is connected. The map that achieves the isomorphism φ :(h, k) → h k is clearly a bijective homomorphism of connected algebraic groups, so it must be an isomorphism of algebraic groups (see Corollary 11.1.3 in [GW, pg 467]). So we have that dim HK = dim(H × K) = dim H + dim K. We can now show easily that G = HK. From that last paragraph we have that HK is a connected, algebraic subgroup of G of dimension dim(H)+dim(K) = dim(h) + dim(V ). But since we have chosen V so that g = h ⊕ V , we have that dim(H) + dim(K) = dim(G). Thus HK is a closed, irreducible subvariety of di- mension dim(G) of the irreducible variety G. Hence, we must have that G = HK. Thus, G = HK ' H ×K as algebraic groups as desired. This completes the proof of the theorem. Q.E.D. 61

4.2 The Γn maps

Now, using Theorem 4.1.1, we can define a parameterization of certain A orbits in the fibres Mc(n). Let S ⊂ M(n) be the set of matrices such that each cutoff Xi for 1 ≤ i ≤ n − 1 is regular. We note that Mc(n) ∩ S naturally has the structure of a quasi-affine variety, since S is Zariski open in M(n). Indeed, S is given by the non-vansishing of the i-forms dφi,1 ∧ · · · ∧ dφi,i for 1 ≤ i ≤ n − 1 (here the φi,j are fundamental adjoint invariants for M(i)). We note that by Theorem sreg 2.14 in [KW, pg 23] that Mc (n) ⊂ Mc(n) ∩ S so that Mc(n) ∩ S is non-empty.

Our goal is to parameterize the A orbits in Mc(n) ∩ S and through that n+1 sreg ( 2 ) to describe the A orbits in Mc (n). Recall that we write c ∈ C as c = i (c1, ··· , ci, ··· , cn) with ci ∈ C and think of ci as representing the coefficients of a monic polynomial of degree i (excluding the leading term). Let G act on Ξi , i ci,ci+1 the solution variety at level i, as in the previous sections. Let G = G1 ×· · ·×Gn−1. We will show that the A orbits are given by orbits of certain subgroups of G acting sreg on certain analytic submanifolds of M(n). On Mc (n) the action of A will be realized by an algebraic action of the full group G.

In order to define the mappings that parameterize A orbits in Mc(n) ∩ S, we need to be able to conjugate elements of a G orbit O ⊂ Ξi into Jordan i ci,ci+1 form for 1 ≤ i ≤ n − 2 in an algebraic manner. In other words, we need to find a function from the smooth variety O → Gl(i + 1) given by X → g(X) with the property that Ad(g(X)) · X is in Jordan form. We can do this precisely because of Theorem 4.1.1. Fix an X ∈ O as above. Then we know that O' Gi/Stab(X). But by Theorem 4.1.1, we have that Gi = Stab(X) K with K a connected, closed subgroup of Gi. This means that O' K and is an orbit under K with K acting freely. More specifically, given Y ∈ O, Y = k · X for a unique k ∈ K and the function O → K given by Y → k is an algebraic isomorphism (See Theorem 25.1.2 in [TY, pg 387]). (We note that the connected group K acts simply transitively on O, so that we can apply part (iv) of the above theorem to the orbit map k → k · X, which is a bijective K-equivariant morphism of affine varieties which are both homogenous K-spaces). 62

Now, if we think of O as being the K orbit of X, then we can first fix a g ∈ GL(i+1) such that Ad(g)·X is in Jordan canonical form. Then for any Y ∈ O with Y = k · X for a unique k ∈ K, the matrix g k−1 ∈ Gl(i + 1) conjugates Y into Jordan form and the function Y → g k−1 is regular on O by our observations above. Now, suppose we are given G orbits in Ξi , Oi = K · x ' K with i ci,ci+1 ai ai ai ai x ∈ Ξi for 1 ≤ i ≤ n − 1, with Oi consisting of regular elements of M(i + 1) ai ci,ci+1 ai for 1 ≤ i ≤ n − 2. We can then define the following regular map:

a1,··· ,an−1 Γn : Ka1 × · · · × Kan−1 → Mc(n) ∩ S,→ M(n) given by:

a1,··· ,an−1 Γn (k1, ··· , kn−1) = (4.4) −1 −1 −1 Ad(k1g1,2(xa1 ) k2g2,3(xa2 ) ··· kn−2gn−2,n−1(xan−2 ) kn−1)xan−1 where g (x ) ∈ Gl(i + 1) conjugates the base point of the orbit Oi , x into i,i+1 ai ai ai Jordan canonical form. We now make a few observations about the mapping defined in (4.4). First, we need the following definition.

Definition 4.2.1. Let X ∈ M (n) ∩ S. We say that z ∈ Ξi ⊂ M(i + 1) is a c ci,ci+1 projection of X into the solution variety Ξi at level i if z is the (i + 1) × (i + 1) ci,ci+1 cutoff of a matrix of the form Ad(g) · X with g ∈ Gl(i) ,→ Gl(n) is such that

(Ad(g)·X)i = Ad(g)·Xi is in Jordan canonical form with eigenvalues in decreasing lexicographical order.

a1,··· ,an−1 Remark 4.2.1. We claim that X ∈ ImΓn if and only if there exists a pro- jection into the solution variety at level j, z ∈ Ξi such that z ∈ Oj . j ci,ci+1 j aj Note that the sufficiency is trivial from the definition of the mapping

a1,··· ,an−1 Γn . To see the necessity, we first make the following observation. If we have Ad(g)Y = B = Ad(h)Y ⇔ Ad(hg−1)B = B ⇔ hg−1 ∈ GL(n)B ⇔ h = kg , k ∈ GL(n)B with GL(n)B the centralizer of B in GL(n). So that if two ma- trices put a given matrix into the same Jordan canonical form then these matrices 63 are in the same right coset of the centralizer of the Jordan form. Thus, if there exists a projection z that is in the orbit Oj in Ξj , then any projection into j aj cj ,cj+1 the solution variety at level j must be in the orbit Oj . Now, if Y ∈ M (n) ∩ S aj c satisfies the property that any projection zj into the solution variety at level j is in the orbit Oj , then Y ∈ O1 ⊂ Ξ1 . Thus, [Ad(g (x ))k−1 · Y ] ∈ O2 for aj 2 a1 c1,c2 1,2 a1 1 3 a2 a unique k1 ∈ Ka1 . (We show the uniqueness for a general kj below. Exactly the same argument shows the uniqueness of k1). Now, by induction we can assume that we have that the (j + 1) × (j + 1) cutoff of

−1 −1 Z = Ad(gj−1,j(xaj−1 ))Ad(kj−1) ··· Ad(g1,2(xa1 ))Ad(k1 ) · Y is in the orbit Oj for unique choice of k , ··· , k ∈ K , ··· ,K respectively. aj 1 j−1 a1 aj−1 Since the (j + 1) × (j + 1) cutoff of Z is in Oj , then there exists a k ∈ K such aj j aj that the (j + 2) × (j + 2) cutoff of Ad(g (x ))Ad(k−1) · Z is in Oj+1 . This j+1,j aj j aj+1 −1 is because [Ad(gj+1,j(xaj ))Ad(kj ) · Z]j+2 is a projection of Y into the solution variety at level j + 1.

Now, we claim that k1, ··· , kj are the unique elements of Ka1 , ··· ,Kaj respectively such that the (j + 2) × (j + 2) cutoff of

0 −1 −1 −1 Z = Ad(gj+1,j(xaj ))Ad(kj )Ad(gj−1,j(xaj−1 ))Ad(kj−1) ··· Ad(g1,2(xa1 ))Ad(k1 )·Y is in the orbit Oj+1 . aj+1

Indeed, suppose that we have another set of elements t1, ··· , tj in

Ka1 , ··· ,Kaj respectively such that the (j + 2) × (j + 2) cutoff of

−1 −1 −1 T = Ad(gj+1,j(xaj ))Ad(tj )Ad(gj−1,j(xaj−1 ))Ad(tj−1) ··· Ad(g1,2(xa1 ))(t1 ) · Y is in the orbit Oj+1 . Then, we have that there exists an element k ∈ K such aj+1 j+1 aj+1 that 0 Zj+2 = Ad(kj+1)Tj+2 (4.5) But since k centralizes the (j + 1) × (j + 1) cutoff of an element in Oj+1 , the j+1 aj+1 (j + 1) × (j + 1) cutoffs of both sides of the equation (4.5) are equal. This yields

−1 −1 −1 Ad(gj+1,j(xaj ))Ad(kj )Ad(gj−1,j(xaj−1 ))Ad(kj−1) ··· Ad(g1,2(xa1 ))Ad(k1 ) · Yj+1 =

−1 −1 −1 Ad(gj+1,j(xaj ))Ad(tj )Ad(gj−1,j(xaj−1 ))Ad(tj−1) ··· Ad(g1,2(xa1 ))Ad(t1 ) · Yj+1 64

(since gj+1,j(xaj ) ∈ GL(j + 1)). Simplifying, the above becomes:

−1 −1 −1 Ad(tj)Ad(kj )Ad(gj−1,j(xaj−1 ))Ad(kj−1) ··· Ad(g1,2(xa1 ))Ad(k1 ) · Yj+1 =

−1 −1 = Ad(gj−1,j(xaj−1 ))Ad(tj−1) ··· Ad(g1,2(xa1 ))Ad(t1 ) · Yj+1 (4.6) Notice that the LHS of equation (4.6) is in the orbit Oj . Thus, by the the aj induction assumption applied to the RHS of equation (4.6), we must have t1 = k1, t2 = k2, ··· , tj−1 = kj−1. Thus, equation (4.6) becomes:

−1 Ad(tj kj ) · Zj+1 = Zj+1 ⇒

−1 tj (kj) ∈ Stab(Zj+1) ∩ Kaj = Stab(xaj ) ∩ Kaj

But since Kaj ∩ Stab(xaj ) is trivial, we get kj = tj. This completes the inductive step.

Thus, by induction, we can conclude that there exists unique k1, ··· , kn−2 ∈

Ka1 , ··· ,Kan−2 respectively so that the element

−1 −1 yan−1 = Ad(gn−2,n−1(xan−1 ))Ad(kn−2) ··· Ad(g1,2(xa1 ))Ad(k1 ) · Y

n−1 is in the orbit Oan−1 . Thus, there exists an unique kn−1 ∈ Kan−1 such that

−1 a1,··· ,an−1 Ad(kn−1) · yan−1 = xan−1 which is the base point for the map Γn . We conclude that there exist unique k1, ··· , kn−1 in Ka1 , ··· ,Kan−1 respectively such

a1,··· ,an−1 that Y = Γn (k1, ··· , kn−1). Note that the argument for the necessity in Remark 4.2.1 lets us see that

a1,··· ,an−1 Γn is injective, since the kj ∈ Kaj that we produce at each step of the procedure are unique. For the remainder of the section, we work mainly in the analytic category. The reason being that the group A is an analytic group, so that to analyze the

a1,··· ,an−1 relationship between ImΓn and the action of A we have to use anayltic techniques. We now consider the topology on subsets of Cn induced by the Eu- clidean distance, and we refer to it as the the analytic topology. Let us observe at this point that everything we have proven so far, goes over to the analytic cate- gory. This is because smooth varieties naturally have complex analytic structure 65 and morphisms between them are holomorphic maps (see [Shaf, Ch 7,8]). Also if a group G is a connected algebraic group over C, then it is naturally a complex Lie group and connected in the Lie group topology (see Theorem 11.1.2 [GW, pg

480]). Thus the groups Gi, Stab(xai ), and Kai can all be considered as connected complex analytic groups that act holomorphically on the analytic submanifold Oi ai of M(n). (In what follows below the word “continuous ” means continuous in the analytic topology). The argument in Remark 4.2.1 also tells us how to construct the inverse

a1,··· ,an−1 map as a map ImΓn → Ka1 × · · · × Kan−1 . We want to see that the inverse

a1,··· ,an−1 map is continuous in the analytic topology when ImΓn is given the subspace topology. We can see that in the following way. We work inductively as in Remark

a1,··· ,an−1 4.2.1. Given Y ∈ ImΓn , we have that Y2 = k1 · xa1 for unique k1 ∈ Ka1 .

Now, we realize that k1 is obtained from Y by composing the continuous functions 1 Y → Y2 → k1. The last arrow is a continuous because on the orbit Oa1 = Ka1 · xa1 the function k1 · xa1 → k1 is biholomorphic, since it is an algebraic isomorphism. −1 Thus, the function Y → Ad(g1,2(xa1 )Ad(k1 )) · Y is continuous as a function from

a1,··· ,an−1 ImΓn → M(n). This starts off the induction. (Recall that the matrix g1,2(xa1 ) is independent of Y ). 1) We assume inductively that the map;

a1,··· ,an−1 Φ0 : ImΓn → M(n)

given by

−1 −1 Y → Ad(gj,j−1(xaj−1 )kj−1) ··· Ad(g1,2(xa1 )k1 ) · Y

is continuous and that kl ∈ Kal for 1 ≤ l ≤ j − 1 have already been computed as continuous functions of Y . Where k1, ··· , kj−1 are the unique elements of −1 −1 Ka1 , ··· ,Kaj−1 respectively such that Ad(gj,j−1(xaj−1 )kj−1) ··· Ad(g1,2(xa1 )k1 ) · Y is in the orbit Oj as in Remark 4.2.1. aj (2) It then follows that the map ImΓa1,··· ,an−1 → Oj given by n aj

−1 −1 Φ1(Y ) = [Ad(gj,j−1(xaj−1 )kj ) ··· Ad(g1,2(xa1 )k1 ) · Y ]j+1 66 is continuous. Now, Φ (Y ) ∈ Oj = k · x for unique k as in Remark 4.2.1. Now, 1 aj j aj j we use again that the canonical map Oj → K is a biholomorphism and hence aj aj continuous to see that we can obtain kj as a continuous function of Y . (3) Now, we note that we have a natural holomorphic map on the product.

Φ3 : GL(j + 1) × M(n) → M(n)

given by

(g, Z) → Ad(g) · Z (4) We get a continuous map

a1,··· ,an−1 Φ4 : ImΓn → Gl(j + 1) × M(n)

given by

−1 Y → (gj+1,j(xaj )kj (Y ), Φ0(Y ))

(5) Lastly we note that the compositions of the continuous maps Φ3 and

Φ4 is the map

−1 −1 Ψ(Y ) = Φ3 ◦ Φ4 = Ad(gj+1,j(xaj )kj ) ··· Ad(g1,2(xa1 )k1 ) · Y

This completes the inductive step. And we thus get that all ki are continu-

a1,··· ,an−1 ous functions of Y ∈ ImΓn and by our discussion in Remark 4.2.1 Y =

a1,··· ,an−1 Γn (k1, ··· , kn−1). Thus, the inverse map is continuous in the analytic topol- ogy.

4.3 The analytic structure of ImΓn

We retain the notation of the last section. Our goal is to show that the

a1,··· ,an−1 subsets ImΓn of Mc(n) ∩ S are A orbits in Mc(n) ∩ S. To that end, we need

a1,··· ,an−1 to see that the sets ImΓn are embedded complex submanifolds of M(n). From our work in the last section we have seen that the map

a1,··· ,an−1 Γn : Ka1 × · · · × Kan−1 → Mc(n) ∩ S,→ M(n) 67 defined in equation (4.4) is a topological embedding in the analytic category. To

a1,··· ,an−1 see that ImΓn is an embedded complex submanifold of M(n), we need to

a1,··· ,an−1 see that the map Γn is an immersion. To that effect, we have the following proposition.

Proposition 4.3.1. The map

a1,··· ,an−1 Γn : Ka1 × · · · × Kan−1 → Mc(n) ∩ S,→ M(n) defined by

−1 −1 −1 Ad(k1g1,2(xa1 ) k2g2,3(xa2 ) ··· kn−2gn−2,n−1(xan−2 ) kn−1)xan−1

a1,··· ,an−1 is an immersion (see equation (4.4)). Thus ImΓn is a complex embedded submanifold of M(n).

Proof: The proof proceeds by explicitly computing the differential of the map

a1,a2,··· ,an−1 Γn . We will also be able to use this computation in the next section to

a1,a2,··· ,an−1 see that tangent space to the image of Γn is a subspace of VX = {ξfi,j |1 ≤ j i ≤ n − 1, 1 ≤ j ≤ i} with fi,j(X) = tr (Xi ), which we recall is precisely the tangent space to the A orbit A · X.

a1,a2,··· ,an−1 For ease of notation, we denote Γn simply by Γn. Let X ∈ M(n) =

Γn(id1, ··· , idn−1) with idj ∈ Kaj denoting the identity element. We observe that

−1 −1 −1 Γn(k1, ··· , kn−1) = Ad(k1g1,2(xa1 ) k2g2,3(xa2 ) ··· kn−2gn−2,n−1(xan−2 ) kn−1)xan−1

with xan−1 given by

xan−1 = Ad(gn−2,n−1(xan−2 )) ··· Ad(g1,2(xa1 )) · X Making this substitution the above becomes:

−1 −1 −1 Ad(k1g1,2(xa1 ) k2g2,3(xa2 ) ··· kn−2gn−2,n−1(xan−2 ) kn−1)Ad(gn−2,n−1(xan−2 )) ···

Ad(g1,2(xa1 )) · X (4.7) 68

Now, let Y ∈ ImΓn, so that Y = Γn(k1, ··· , kn−1). Now, we can define a map ΨY : Ka1 × · · · × Kan−1 → ImΓn to be a Γn map but “ based ” at Y instead of at X. i.e. ΨY would be defined as

−1 −1 ΨY (ke1, ··· , k]n−1) = Ad(ke1)Ad(k1)Ad(g1,2(xa1 ) )Ad(ke2)Ad(k2)Ad(g2,3(xa2 ) ) ···

−1 Ad(gn−2,n−1(xan−2 ) )Ad(k]n−1)yan−1 (4.8) n−1 Where yan−1 is a projection of Y into the orbit Oan−1 and is given by

−1 −1 yan−1 = Ad(gn−2,n−1(xan−2 ))Ad(kn−2) ··· Ad(g1,2(xa1 ))Ad(k1 ) · Y (4.9)

as computed in Remark 4.2.1. Now, using that yan−1 = Ad(kn−1)xan−1 , we can write that above as

−1 −1 Ad(ke1k1g1,2(xa1 ) ke2k2g2,3(xa2 ) ··· (4.10) −1 ··· kgn−2kn−2gn−2,n−1(xan−2 ) kgn−1kn−1gn−2,n−1(xan−2 ) ··· g1,2(xa1 )) · X

Comparing equations (4.10) and (4.7), we see that we have

ΨY (ke1, ··· , kgn−1) = Γn(ke1 k1, ··· , kgn−1 kn−1)

So that we have the relation:

Ψy = Γn ◦ Lk (4.11)

where k ∈ Ka1 × · · · × Kan−1 is given by k = (k1, ··· , kn−1) and Lk denotes left translation by k. Thus, taking differentials at the identity in Ka1 × · · · × Kan−1 we obtain

(dΨY )id = (dΓn)k ◦ (dLk)id (4.12)

By equation (4.12), we see that, since Lk is a differomorphism, (dΓn)k is of full rank if and only if (dΨY )id has full rank. Thus, to compute the rank of (dΓn)k we just compute the rank of (dΨY )id, which is much easier, since we only have to worry about differentiating at the identity in Ka1 × · · · × Kan−1 . For ease of notation, in the remainder of the proof we write gi,i+1 for gi,i+1(xai ). 69

Now, we focus on differentiating the map ΨY at the identity. Using equation

(4.9), we can write ΨY as

−1 −1 −1 −1 −1 Ad(ke1k1g1,2ke2k2g2,3 ··· gn−2,n−1kgn−1gn−2,n−1kn−2 ··· g1,2k1 ) · Y

Let Lie(Kai ) have basis {αi,1, ··· , αi,s(i)} with s(i) = dim Lie(Kai ) ≤ i.

Because the group Kai is abelian and connected as a complex analytic group, we Ps(i) note that we can represent ki ∈ Kai as exp( j=1 ti,jαi,j) for ti,j ∈ C. Thus, to compute the differential of ΨY at the identity, we need to consider the differentials

∂ −1 Ps(i) −1 |t=0 Ad(exp(t1,1α1,1)k1g ··· exp( ti,jαi,j)kig ··· ∂ti,j 1,2 j=1 i,i+1 (4.13) −1 Ps(n−1) −1 −1 ··· gn−2,n−1 exp( j=1 tn−1,jαn−1,j)gn−2,n−1kn−2 ··· g1,2k1 ) · Y

(where t is the tuple t = (ti,j)1≤i≤n−1, 1≤j≤s(i)) Now, using the fact that we are differentiating at the identity, the above becomes (in coordinates):

−1 −1 −1 −1 −1 ad(k1g1,2k2 ··· gi−1,iαi,jkigi,i+1ki+1 ··· kn−2gn−2,n−1gn−2,n−1kn−2 ··· (4.14) −1 −1 −1 ··· ki+1gi,i+1ki gi−1,i ··· g1,2k1 ) · Y

The above simplifies to

−1 −1 −1 −1 −1 vi,j = ad(k1g1,2k2g2,3 ··· ki−1gi−1,iαi,jgi−1,iki−1 ··· g1,2k1 ) · Y

−1 −1 (with vi,j ∈ T (M(n))Y ). Now, let hi = gi−1,iki−1 ··· g1,2k1 ∈ Gl(i) ,→ Gl(n) and note that the i × i cutoff of Ad(hi) · Y is in Jordan canonical form by construction. −1 Now αi,j ∈ Lie(Kai ) ⊂ gi with gi = Lie(Gi). Hence hi αi,jhi ∈ GL(i) centralizes the i × i cutoff of Y . With this notation the above equation becomes:

−1 vi,j = ad(hi αi,jhi) · Y (4.15)

Now, we suppose by way of contradiction that the vi,j are dependent. We may assume that we have a strictly increasing sequence of indices i1 < i2 < ··· < it so that the following linear relation holds.

ad(h−1α h ) · Y + ··· + ad(h−1α h ) · Y = 0 (4.16) i1 fi1 i1 it fit it 70

Pjm × where αi = clαi,l ∈ Lie(Ka ) with cl ∈ and {j1, ··· , jm} fj l=j1 i C ⊂ {1, 2 ··· , s(i)} a strictly increasing sequence. We observe that equation (4.16) implies that

t X h−1α h ∈ M(n)Y (4.17) ij fij ij j=1 Y with M(n) the centralizer of Y ∈ M(n). We also claim that since ij > i1 for all j > 1, we have t X [h−1α h ,Y ] = 0 (4.18) ij fij ij i1+1 j=2 We use an argument very similar to the one in the proof of Theorem 2.14 in [KW, pg 24]. To see (4.18) we have to show that for j ≥ 2, we have

[h−1α h ,Y ] = 0 (4.19) ij fij ij i1+1

⊥ ⊥ This follows from the fact that M(n) = M(ij)⊕M(ij) (where M(ij) denotes the orthogonal compliment of M(ij) with respect to the trace form). The component of Y in M(i ) is clearly Y , but we observed above that [h−1α h ,Y ] = 0. Thus, j ij ij fij ij ij since ad M(ij) stabilizes the components of the above decomposition, we have that [h−1α h ,Y ] ∈ M(i )⊥. Now, since i ≥ i + 1, we have that M(i )⊥ ⊂ ij fij ij j j 1 j ⊥ M(i1 + 1) . But then we get easily equation (4.19) and thus equation (4.18) by linearity. Equations (4.17) and (4.18), then imply that

[h−1α h ,Y ] = 0 i1 fi1 i1 i1+1

But we have that h−1α h ∈ M(i ) so that the above becomes: i1 fi1 i1 1

[h−1α h ,Y ] = 0 i1 fi1 i1 i1+1

But this becomes [α , Ad(h ) · Y ] = 0 (4.20) fi1 i1 i1+1

But yi1+1 = Ad(hi1 ) · Yi1+1 is a projection of Y ∈ ImΓn into the solution variety

i1 i1 Ξ and it must therefore lie in the orbit O by Remark 4.2.1. Thus, αi1 ∈ ci1 ,ci1+1 ai1 f Lie(K )∩Lie(Stab(y )) = Lie(K )∩Lie(Stab(x )). Where the last equality ai1 i1+1 ai1 ai1 71

follows because the group Gi is abelian so that Stab(xai ) = Stab(yai ) ⊂ Gi for all i. But, we have chosen K such that Lie(K ) ∩ Lie(Stab(x )) = 0 (see ai1 ai1 ai1 Pjm Theorem 4.1.1 above). Thus, αi = 0 ⇒ clαi,l = 0 ⇒ cl = 0 for all l, a f1 l=j1 contradiction. Thus, the tangent vectors vi,j in equation (4.15) are linearly inde- pendent. Since Y ∈ ImΓn was chosen arbitrarily this gives us that we have (dΓn)k is of full rank by equation (4.12). Hence Γn is an immersion. We have already observed that it is a topological embedding in the analytic topology, so that ImΓn is a complex embedded submanifold of M(n), as desired. This completes the proof. Q.E.D.

Remark 4.3.1. In the above proof, we have used that Lie(Stab(xai )) = {Z ∈ gi| [Z, xai ] = 0}. We can see this in the following way. We again work in the analytic category. Since Stab(xai ) is a Lie group over C, we have that Z ∈ gi, Z ∈

Lie(Stab(xai )) if and only if exp(t Z) ∈ Stab(xai ) for all t ∈ C. We first suppose that Ad(exp(tZ)) · xai = xai , then differentiating at t = 0 yields ad(Z) · xai = 0.

Conversely, suppose that we have ad(Z) · xai = 0, then we note that for any t ∈ C we have that Ad(exp(tZ)) · xai = exp(ad(tZ)) · xai = xai .

a1,··· ,an−1 Remark 4.3.2. With this analytic structure on ImΓn , the inverse map

a1,··· ,an−1 −1 (Γn ) defined in the previous section is easily seen to be holomorphic, as

a1,··· ,an−1 Γn has non-singular differential and is bijective onto its image, so it is thus a biholomorphism.

a1,··· ,an−1 Now, we note that on the submanifold ImΓn , we have a holomorphic action of the analytic group K = Ka1 × · · · × Kan−1 . Given as follows:

a1,a2,··· ,an−1 −1 If (Γn ) (Y ) = (k1, ··· , kn−1) then

0 0 a1,a2,··· ,an−1 0 0 (k1, ··· , kn−1) · Y = Γn (k1k1, ··· , kn−1kn−1) (4.21)

This action is easily seen to be a holomorphic action as follows. We write for k = (k1, ··· , kn−1) ∈ K = Ka1 × · · · × Kan−1 . We consider the following holomorphic maps. 72

a1,a2,··· ,an−1 Φ1 : K × ImΓn → K × K

given by

a1,a2,··· ,an−1 −1 ((k1, ··· , kn−1),Y ) → ((k1, ··· , kn−1), (Γn ) (Y ))

Φ2 : K × K → K

given by

0 0 0 0 ((k1, ··· , kn−1), (k1, ··· , kn−1)) → (k1k2, ··· , kn−1kn−1)

a1,a2,··· ,an−1 Φ3 = Γn

Then the action map is given by the composition Φ3 ◦ Φ2 ◦ Φ1. Since the action map is a composition of holomorphic maps, we have that the action is holomorphic.

a1,a2,··· ,an−1 Remark 4.3.3. We note that the action of K on ImΓn is a free and tran- sitive action.

a1,a2,··· ,an−1 Our goal is to see that for X ∈ ImΓn the A orbit of X is the same as its K orbit.

4.4 The tangent space to ImΓn

Our goal in this section is to show that A acts on the manifold

a1,a2,··· ,an−1 ImΓn . The following result from differential geometry will be of use later in this section.

Proposition 4.4.1. Let M be a complex (or real differentiable) manifold. Let N ⊂ M be a submanifold. Let V be a complete holomorphic (resp smooth) vector

field on M with the property that for each p ∈ NVp is tangent to N i.e. Vp ∈ 73

dipTp(N) (with i : N,→ M denoting the inclusion). Then the integral curve of V starting at p ∈ N lies in N.

This is a standard consequence of the uniqueness theorem for ordinary differential equations.

a1,··· ,an−1 Thus, if we can show that for X ∈ ImΓn the tangent space

a1,··· ,an−1 TX (ImΓn ) = VX , then Proposition (4.4.1) will give us that A acts on

a1,··· ,an−1 ImΓn . During the proof of Proposition 4.3.1, we have computed the tangent space

a1,a2,··· ,an−1 to the embedded submanifold ImΓn ⊂ M(n). We can see this as follows.

a1,a2,··· ,an−1 Recall the map ΨY defined in equations (4.8) and (4.10) for Y ∈ ImΓn ,

a1,a2,··· ,an−1 Y = Γn (k1, ··· , kn−1). In the course of that proof we computed the differential of the map ΨY at the identity element in K = Ka1 × · · · × Kan−1 . But

a1,··· ,an−1 we note that this map is simply the orbit map θY : K → ImΓn at Y (see equation (4.11)). Thus, differentiating θY = ΨY at the identity, should give us

a1,··· ,an−1 TY (K · Y ) = TY (ImΓn ). Now, we computed the image of the differential of

θY in equation (4.15) to be

−1 vi,j = ad(hi αi,jhi) · Y for 1 ≤ i ≤ n − 1, 1 ≤ j ≤ dim(K ) (with {α ··· , α } a basis for ai i,1 i,dim(Kai ) −1 Lie(Kai )). Recall that hiYi+1hi is a projection of Y into the solution variety Ξi at level i and thus is in the orbit Oi . We observe that Lie(K ) ⊂ g = ci,ci+1 ai ai i Lie(G ). Recalling that Lie(G ) is the centralizer of the i×i cutoff of x ∈ Ξi , i i ai ci,ci+1 −1 we see that we have that since hi ∈ Gl(i) ,→ Gl(n), hi αi,jhi centralizes the i × i cutoff of Y . Now, recall that Y ∈ Mc(n) ∩ S, so that the i × i cutoff of Y is regular. Thus, the centralizer of Yi in M(i) is spanned by the powers i−1 {Idi,Yi, ··· ,Yi }. We recall Theorem 2.13 in [KW, pg 23] which states that the subspace VX = {(ξf )X |f ∈ J(M(n))} given by the Hamiltonian fields from the analogue Gelfand-Zeitlin algebra is equal to (in coordinates):

VX = span{[z, X]|z ∈ ZX }

Pn−1 for X ∈ M(n). Here ZX is defined to be i=1 ZXi , where ZXi is the associative subalgebra of M(i) ,→ M(n) generated by Idi and Xi. Thus, we have that vi,j ∈ VY 74 and thus

a1,··· ,an−1 TY (ImΓn ) ⊂ VY (4.22)

We want to show that the inclusion in (4.22) is actually equality. To see

a1,··· ,an−1 that the subspace VX ⊂ TX (M(n)) is tangent to the manifold ImΓn , we define the following map

a1,··· ,an−1 ΦY : G1 × G2 · · · × Gn−1 → ImΓn

given by

(h1, ··· , hn−1) →

−1 −1 −1 → Ad(h1k1g1,2(xa1 ) h2k2g2,3(xa2 ) ··· hn−2kn−2gn−2,n−1(xan−2 ) hn−1) · yan−1 (4.23) with hi ∈ Gi, where (k1, ··· , kn−1) ∈ Ka1 × · · · × Kan−1 is fixed and yan−1 is a n−1 projection of Y into the orbit Oan−1 and is given by

−1 −1 yan−1 = Ad(gn−2,n−1(xan−2 ))Ad(kn−2) ··· Ad(g1,2(xa1 ))Ad(k1 ) · Y

a1,··· ,an−1 with Y = Γn (k1, ··· , kn−1). First, we observe that from Remark 4.2.1 ΦY

a1,··· ,an−1 actually maps into ImΓn . Since for Z ∈ ImΦY a projection into the solution variety at level j is given by

−1 −1 −1 −1 [Ad(gj,j−1(xaj−1 )kj−1hj−1) ··· Ad(g1,2(xa1 )k1 h1 ) · Z]j

−1 = [Ad(hj)Ad(kj)Ad(gj,j+1(xaj ) ) ··· Ad(hn−1)yan−1 ]j is in the orbit given by Oj for all 1 ≤ j ≤ n − 1. Now, note that Φ : G × · · · × aj Y 1

a1,··· ,an−1 a1,··· ,an−1 Gn−1 → Γn is a holomorphic map into the embedded submanifold Γn .

This follows easily from the fact that ΦY is a holomorphic map into M(n) and since

a1,··· ,an−1 ImΓn is an embedded submanifold, it follows automatically that the map is

a1,··· ,an−1 holomorphic into ImΓn (see Corollary 8.25 in [Lee, pg 191]). We also note that ΦY is a submersion, since its restriction to the subgroup Ka1 × · · · × Kan−1 of

a1,··· ,an−1 G1 ×· · ·×Gn−1 is the map ΨY = Γn ◦Lk defined in the proof of Proposition 75

4.3.1 in equation (4.8) which by construction is a diffeomorphism onto its image.

(Recall Lk is left translation by k = (k1, ··· , kn−1)). The upshot is then that we can compute the tangent space to Y by differentiating ΦY at the identity. This computation is totally analogous to the one done in the proof of Propo- sition 4.3.1 in equations (4.13) and (4.14). The same computations as in those equations, now produces the following result.

−1 a1,··· ,an−1 wi,j = ad(hi γi,jhi) · Y ∈ TY (ImΓn ) (4.24)

Where hi ∈ Gl(i) is as in equation (4.15) in the proof of Proposition 4.3.1. But now

γi,j ∈ gi = Lie(Gi) and is part of a basis {γi,1, ··· γi,i} for gi. Thus the elements of the form −1 {hi γi,jhi|1 ≤ j ≤ i} form a basis for the centralizer in M(i) of the i × i cutoff of Y for 1 ≤ i ≤ n − 1.

a1,··· ,an−1 Since Y ∈ ImΓn ⊂ Mc(n) ∩ S, we have that

span{wi,j|1 ≤ i ≤ n − 1, 1 ≤ j ≤ i} = span{[z, Y ]|z ∈ ZY } = VY

a1,··· ,an−1 as in the beginning of the section. Since Y ∈ ImΓn is arbitrary, we have proven the following result.

Proposition 4.4.2. The the commuting Lie algebra of vector fields {ξfi,j |1 ≤ i ≤

j a1,··· ,an−1 n−1, 1 ≤ j ≤ i} (where fi,j = tr(Xi )) is tangent to the submanifold ImΓn .

a1,··· ,an−1 Thus A acts on ImΓn . Proof: The only statement that remains to be proven is the last statement, but this follows from the fundamental result in differential geometery in Proposition 4.4.1 and the fact that the A action is obtained by composing that action of the commuting

flows of the commuting vector fields ξfi,j . Q.E.D.

4.5 The submanifolds ImΓn and A orbits

a1,··· ,an−1 In the last section, we showed in Proposition 4.4.2 that each ImΓn is

a1,··· ,an−1 a union of A orbits. Now let Y ∈ ImΓn and consider the A orbit of Y , A·Y . 76

a1,··· ,an−1 We know that the dim A·Y = dim VY = dim ImΓn . (Recall from [KW] that

a1,··· ,an−1 a1,··· ,an−1 for X ∈ M(n), TX (A·X) = VX .) Thus for Y ∈ ImΓn , A·Y ⊂ ImΓn

a1,··· ,an−1 is a submanifold of the same dimension as ImΓn , hence it must be open.

a1,··· ,an−1 Thus, ImΓn is a disjoint union of open A orbits, so we write

a1,··· ,an−1 a ImΓn = A · X(i) i∈I

a1,··· ,an−1 a1,··· ,an−1 with X(i) ∈ ImΓn . We note that A · X(j), j ∈ I is given by ImΓn \ ` ` ( i6=j, i∈I A·X(i)), since the union is disjoint. But i6=j, i∈I A·X(i) is open, so that A·X(j) is closed. Thus A·X(j) is both open and closed in the connected manifold

a1,··· ,an−1 a1,··· ,an−1 ImΓn . Hence since A · X(j) 6= ∅, we must have ImΓn = A · X(j) is exactly one A orbit. We have now reached our goal and have:

a1,··· ,an−1 Theorem 4.5.1. The embedded complex submanifold ImΓn is exactly one

A orbit in Mc(n) ∩ S. Moreover, every A orbit in Mc(n) ∩ S is given as the image

a1,··· ,an−1 of a mapping of the form Γn where ai denotes a choice of Gi orbit in the solution variety Ξi at level i. ci,ci+1

Proof: Only the second statement needs to be addressed. It follows easily from the fact

a1,··· ,an−1 that Mc(n)∩S is covered by sets of the form ImΓn for some choice of orbits ai (see Remark 4.2.1). Q.E.D. We have an immediate corollary.

Corollary 4.5.1. The A orbits in Mc(n) ∩ S are irreducible Zariski constructible subsets of M(n) that are isomorphic as analytic manifolds to a product of abelian connected algebraic groups over C, which is a subgroup of the group G1 ×· · ·×Gn−1.

Remark 4.5.1. The first statement of the corollary appears already in Theorem 3.7 in [KW, pg 36]. However, in that reference the authors only establish a bijection in the case of strongly regular elements (see Theorem 3.14 in [KW, pg 40]). This result is thus a strengthening of Theorem 3.7 and gives a partial generalization of

Theorem 3.14 to elements in Mc(n) ∩ S which are not strongly regular. 77

sreg 4.6 The Γn construction and Mc (n)

n
Recall that if X ∈ M(n) is such that dim A · X = 2 , then X is said to be strongly regular and the strongly regular elements in the fibre Mc(n) are denoted sreg sreg by Mc (n). Theorem 2.4 in [KW, pg 23] states that Mc (n) ⊂ Mc(n)∩S. Thus,

sreg a1,··· ,an−1 by Theorem 4.5.1, an A orbit in Mc (n), must be an image of a map ImΓn for some choice of orbits a ∈ Ξi . i ci,ci+1

a1,··· ,an−1 Theorem 4.6.1. The submanifold ImΓn is an A orbit of maximal dimen- sion n
if and only if the orbit Oi ⊂ Ξi is of maximal dimension i, in which 2 ai ci,ci+1 case G acts freely on Oi . i ai

Proof: By Lemma 4.1.1 Oi is of maximal dimension i if and only if Stab(x ) is trivial ai ai and hence Gi acts freely. In this case Kai = Gi and is of dimension i so that

a1,··· ,an−1 n
a1,··· ,an−1 the manifold ImΓn has dimension 2 . Thus ImΓn is an A orbit of n
dimension 2 .

a1,··· ,an−1 n
a1,··· ,an−1 Conversely, if ImΓn is of dimension 2 , then since ImΓn = Pn−1 i=1 dim Kai ⇒ dim Kai = i for all i. This follows from the fact that dim Kai ≤ i for all i. Thus, Oi is of dimension i. ai Q.E.D.

a1,··· ,an−1 In the remainder of this section ImΓn denotes a strongly regular A orbit. Now, in the case where X ∈ M sreg(n) is strongly regular A · X is a smooth and irreducible varierty (see Theorem 3.12 in [KW, pg 48]). We want to see that

a1,··· ,an−1 in this case Γn is actually an isomorphism of varieties. This follow from the following corollary to Zariski’s Main Theorem (see [TY, pg. 229]).

Proposition 4.6.1. Let u : X → Y be a bijective morphism of irreducible varities. If Y is normal, then u is an isomorphism.

a1,··· ,an−1 Now, we have observed that we have a regular map Γn : G1 × · · · ×

a1,··· ,an−1 Gn−1 → A · X. Where X is a fixed point in the image of Γn . We have

a1,··· ,an−1 observed that Γn is bijective. Since non-singular varieties are normal, and 78

the groups Gi for 1 ≤ i ≤ n − 1 are all connected, the above corollary to Zariski’s

a1,··· ,an−1 main theorem gives us that Γn is actually an isomorphism of varieties. To summarize we have:

a1,··· ,an−1 Γn a1,··· ,an−1 G1 × · · · × Gn−1 ' ImΓn (4.25) where the isomorphism is actually biregular. Now, recall the action of G that we defined in equation (4.21). Now, the maps defining the action map can be

a1,··· ,an−1 −1 all be seen to be morphisms, since (Γn ) is a morphism. Thus we have

a1,··· ,an−1 an algebraic action of G on ImΓn which has the same orbit structure as sreg the A action. Now, by Theorem 3.12 in [KW,pg 38] the A orbits in Mc (n) sreg are the irreducible components on Mc (n) and since they are disjoint they are sreg sreg both open and closed in Mc (n) (in the Zariski topology on Mc (n)). Following sreg the notation of [KW], we index these components by Mc,i (n) = A · X(i) with sreg sreg sreg X(i) ∈ Mc (n). Now, we have regular maps φi : G × Mc,i (n) → Mc (n)

a1,··· ,an−1 sreg given by the action of G on ImΓn . We note that the sets G × Mc,i (n) are sreg (Zariski) open in the product G × Mc (n) and are disjoint. Thus the morphisms

φi glue to a unique morphism

sreg sreg Φ: G × Mc (n) → Mc (n)

such that

Φ| sreg = φ G×Mc,i (n) i

sreg The morphism Φ gives us an algebraic action of the group G on Mc (n) whose sreg orbits are the orbits of A in Mc (n). We have thus proved the following theorem.

sreg Theorem 4.6.2. On Mc (n) the orbits of the group A are given by orbits of an algebraic action of the connected abelian algebraic group G = G1 × · · · × Gn−1 with sreg Gi = GXi for X ∈ Mc (n). (Recall that GXi denotes the centralizer of Xi in GL(i).)

sreg Now that we have developed a way to construct A orbits in Mc (n), we n+1 sreg ( 2 ) can count the number of A orbits in Mc (n) for any c ∈ C and explicitly 79

(n+1) describe the orbits. However, first we have to see that for any c ∈ C 2 that the solution variety Ξi has G orbits Oi which consist of regular elements of ci,ci+1 i M(i + 1) on which Gi acts freely. We will actually be able to show that if Oi is an orbit in Ξi on which G acts freely, then it must consist of regular elements of ci,ci+1 i M(i + 1). This will be the content of the next chapter. 5 Regular elements and maximal orbits of the solution variety Ξi ci,ci+1

i We first consider the case where ci ∈ C represents a monic polynomial of the form (t − λ)i. So that the i × i cutoff of X ∈ Ξi is a single Jordan block ci,ci+1 given by   λ 1 ··· 0  . .   0 λ .. .     . .   . .. 1    0 ······ λ r and we let λi+1 = (λi+1,1, ··· λi+1,r) ∈ C be chosen arbitrarily. Then we saw from our work in section 3.2 that the solution variety Ξi contains a G orbit on ci,ci+1 i which the Gi action is free. This orbit contains an element of the form

  λ 1 ··· 0 f1(1, ··· , 0)  . . .   0 λ .. . .     . . .   . .. 1 .  (5.1)      0 ······ λ fi(1)    1 ······ 0 w This solution is clearly a regular element in M(i + 1). We have an obvious choice i+1 of a cyclic vector, ei, where ei denotes the i − th standard basis vector in C . Now, the difficulty comes in showing that more general solution varieties consist of regular elements.

80 81

Suppose now that we have a solution variety like the one considered in equation (3.1). For example consider an element of M(i + 1) of the form

    λ1 1 ··· 0 y1,1   . .  .    0 λ .. .  .    1  0    . .  .    . .. 1  .         0 ······ λ1 y1,n1     .. .   . .         λr 1 ··· 0 yr,1    . .  .    0 λ .. .  .    r    0  . .  .   . .. 1 .           0 ······ λ y   r r,nr 

z1,1 ······ z1,n1 ··· zr,1 ······ zr,nr w with λi 6= λj, but otherwise they are allowed to be arbitrary. We suppose that the Jordan block corresponding to the eigenvalues λj is of size nj. In sections 3.2 and 3.3 we showed that if we are given any monic polynomial of degree i + 1, say p(t), then we can choose the zi,j and yi,j in the the above matrix so that it has characteristic polynomial the given polynomial. We also observed that we can find sets of solutions to the extension problem on which Gi (the centralizer of the i × i Jordan block) acts simply transitively. We claim now that if O ⊂ Ξi is such ci,ci+1 an orbit of G in Ξi then O consists of regular elements of M(i + 1). i ci,ci+1 We show this in a slightly indirect fashion using the description of strongly regular orbits in M(i + 1) developed in Chapter 4, section 4.6. (Recall that we denote by S the set of elements in M(i + 1) such that each cutoff Xj for 1 ≤ j ≤ i (i+1) is regular.) We consider the fibre Mc(i+1)∩S where c ∈ C 2 is such that cj, cj+1 for 1 ≤ j ≤ i−2 represent coefficients of relatively prime monic polynomials and cj j for 1 ≤ j ≤ i − 1 represents coefficients of polynomial (t − µj) . That is to say that j X ∈ Mc(i + 1) has that the characteristic polynomial of Xj is equal to (t − µj) for 1 ≤ j ≤ i − 1, and µj 6= µj+1 for 1 ≤ j ≤ i − 2. Now, we let ci correspond to the characteristic polynomial of the i × i cutoff of the matrix in (3.1). (i.e. the

Qr nj polynomial represented by ci factors as j=1(t − λj) ). We also let ci+1 be the 82 coefficients (excluding the leading term) of the polynomial p(t). By our work in Chapter 4, section 4.2, we can use the orbit of O ⊂ Ξi to ci,ci+1 construct a Γi+1 mapping onto a subset of Mc(i+1)∩S. Since for the construction of such a Γi+1 mapping, we recall that we only need the elements of the solution varieties at levels below i to be regular. Only elements in Ξj for 1 ≤ j ≤ i − 1 cj ,cj+1 that are conjugated into Jordan form. So now, we have a map:

Γa1,a2,··· ,ai : O1 × · · · × Oi−1 × O → M (i + 1) ∩ S i+1 a1 ai−1 c where each orbit Oj ⊂ Ξj is an orbit of G on which the G action is free. aj cj ,cj+1 j j We know that for 1 ≤ j ≤ i − 2 that there is only one choice of orbit given in section 3.2. However, for Oi−1 we can choose a G orbit of the form (5.1). Now, ai−1 i i+1
by Theorem 4.6.1, the image of the above map is an A-orbit of dimension 2 , so that it consists of strongly regular elements. Now, we recall Thm 2.14 in [KW, pg 23] that X is strongly regular implies that Xm is regular for all 1 ≤ m ≤ i + 1.

a1,a2,··· ,ai We recall the definition of Γi+1 :

a1,a2,··· ,ai −1 −1 −1 Γi+1 (h1, ··· , hi) = Ad(h1g1,2(xa1 ) h2g2,3(xa2 ) ··· hi−1gi−1,i(xai−1 )hi)xai where g (x ) conjugates the “base point ” of the orbit Oj , x into Jordan j,j+1 aj aj aj canonical form for 1 ≤ j ≤ i − 1 and (h1, ··· , hi) ∈ G1 × · · · × Gi.

We see that any y ∈ O ⊂ M(i + 1) (y = Ad(hi) · xai for unique hi ∈ Gi) is a1,a2,··· ,ai sreg conjugate to an X ∈ ImΓi+1 . Since X ∈ M (n) we have that X is regular so that xai must be regular. To summarize we have now proven the following proposition.

Proposition 5.0.2. Let O ∈ Ξi be a G orbit of maximal dimension i, on ci,ci+1 i which Gi necessarily acts freely by Lemma 4.1.1, then the elements of O are regular elements of M(i + 1). sreg 6 Counting A orbits in Mc (n)

We preserve the notation of section 3.3. We can use our results in the sreg previous Chapters to count A orbits in arbitrary fibres in Mc (n). The main result is the following.

n(n+1) Theorem 6.0.3. Let c = (c1, c2, ··· , ci, ci+1, ··· , cn) ∈ C 2 . Suppose there are

0 ≤ ji ≤ i roots in common between the monic polynomials represented by ci and Pn−1 sreg i=1 ji ci+1. Then the number of A orbits in Mc (n) is exactly 2 . We have that sreg on Mc (n) the orbits of A are the orbits of an algebraic action of the complex, commutative connected algebraic group G = G1 × G2 × · · · × Gn−1.

i Qr nj Proof: We suppose that ci ∈ C are the coefficients of the polynomial i=1(t−λj) so that elements in the solution variety Ξi are of the form (3.1). Then we know ci,ci+1 by equations from section 3.3

zk,1yk,nk = 0 for 1 ≤ k ≤ ji × zk,1yk,nk ∈ C for j < k ≤ r

ji that there are at least 2 Gi orbits of maximal dimension in the solution variety Ξi at level i. (We can suppose without loss of generality that the overlaps occur ci,ci+1 in this order). Where each orbit is given by a choice of either yk,nk 6= 0 or zk,1 6= 0 for 1 ≤ k ≤ j . Now, I claim that these are all the orbits of dimension i in Ξi . i ci,ci+1

Without loss of generality, suppose that we take z1,1 = y1,n1 = 0. If there are solutions in this case, it is easy to see that Gi stabilizes them. However, it is easy to see that they would have a non-trivial isotropy group in Gi that would contain

83 84 the subgroup:     1 0 ··· c     .. .     0 1 . .     .  0    . ...     . 0        0 ······ 1     .   ..         1 0 ··· 0      .. .     0 1 . .    0      . ..     . . 0       0 ······ 1 where c ∈ C. Thus, if such solutions exist they could not belong to orbits of maximal dimension. Hence there are exactly 2ji orbits of dimension i in Ξi . ci,ci+1 We recall that by Proposition 5.0.2 these orbits all consist of regular elements of Pn−1 j M(i + 1). Thus, we can construct 2 i=1 i mappings of the form

a1,··· ,an−1 sreg Γn : G1 × · · · × Gn−1 → Mc (n) with a a choice of orbit Z with b = U, L as in section 3.3. By Theorem i b1,··· ,bji l n
4.6.1 the images of such mappings are precisely the A orbits of dimension 2 in

sreg a1,··· ,an−1 Mc (n). Now, recall from Remark 4.2.1, that X ∈ ImΓn if and only if every projection z into the solution variety at level j is in the orbit Oj ∈ Ξi j aj ci,ci+1

a1,··· ,an−1 for each j, 1 ≤ j ≤ n − 1. It follows that the images of the different Γn P ji sreg mappings are disjoint. Thus, we have precisely 2 orbits in Mc (n). The last statement of the theorem then follows directly from Theorem 4.6.2. This gives us the desired result. Q.E.D.

Now, we have several important corollaries.

sreg Corollary 6.0.1. The action of A is transitive on Mc (n) if and only if c = (c1, ··· , cn) with ci and ci+1 representing coefficients of relatively prime poly- nomials. 85

Proof: The proof is immediate from Theorem 6.0.3.

sreg n−1 Corollary 6.0.2. The nilfibre M0 (n) contains 2 A-orbits.

Proof: This follows again from Theorem 6.0.3, since there is now exactly one overlap at each level.

6.1 The new set of generic matrices Θn

Corollary 6.0.1 allows us to identify the set of matrices in M(n) on which the action of A is transitive. This expands the results of Kostant and Wallach in

[KW]. Let Ri= set of regular elements of M(i). We define a set of matrices

Θn = {X ∈ M(n)|Xi ∈ Ri, σ(Xi) ∩ σ(Xi+1) = ∅, 1 ≤ i ≤ n − 1}

Here σ(Xi) denotes the spectrum of Xi. It follows from Remark 2.16 in [KW, pgs 24-25], that Θn ⊂ M(n) is Zariski open. The result concerning Θn is the n(n+1) following. First, we need some extra terminology. We say that c ∈ C 2 satisfies i the eigenvalue disjointness condition if c = (c1, ··· , cn) with ci ∈ C such that ci and ci+1 represent coefficients of relatively prime monic polynomials.

Theorem 6.1.1. The elements of Θn are strongly regular. Thus, for X ∈ Θn, Xi n(n+1) is regular for all 1 ≤ i ≤ n. For c ∈ C 2 satisfying the eigenvalues disjointness sreg condition, we have Mc (n) = Mc(n) ∩ Θn is exactly one A orbit. Thus, on Θn the upper Hessenberg matrices form a cross-section to the A action.

Proof: Only the first statement needs to be proven. The second automatically follows from the first. To see that elements of Θn are strongly regular, note that if X ∈ Θn, (n+1) then X ∈ Mc(n) ∩ S where c ∈ C 2 satisfies the eigenvalues disjointness con- dition. Thus, X ∈ ImΓa1,··· ,an−1 where a = Oi = Ξi is the unique orbit of n i ci,ci+1 G in Ξi which is necessarily of dimension i by Proposition 3.2.1. But then i ci,ci+1

a1,··· ,an−1 a1,··· ,an−1 n
X ∈ ImΓn with dim ImΓn = 2 . Thus by Theorem 4.5.1, X is 86

sreg sreg strongly regular. Thus Θn ⊂ M (n) and Mc(n) ∩ Θn ⊂ Mc (n). The rest of the Theorem follows from Theorem 4.5.1 and Corollary 6.0.1. Q.E.D.

(n+1) Corollary 6.1.1. For c ∈ C 2 with c satisfying the eigenvalues disjointness condition, we have that

sreg Mc (n) ' G1 × · · · × Gn−1 as algebraic varieties.

Proof: n+1 ( 2 ) sreg This follows immediately from the fact that for such c ∈ C , Mc (n) is exactly one A orbit and then the isomorphism follows by equation (4.25). Q.E.D.

Remark 6.1.1. This theorem and its corollary generalize Theorems 3.23 and 3.28 in [KW, pgs 45, 49]. In these theorems the authors prove that analogous result for a Zariski open subset Ωn ⊂ Θn where each cutoff Xi is regular semi-simple. (n+1) Remark 6.1.2. We note that for a matrix X ∈ Mc(n) where c ∈ C 2 satisfies the eigenvalue disjointness condition its strictly upper triangular part is determined by its strictly lower triangular part. This follows directly from the form the solution varieties Ξi in section 3.2 and the definition of the mappings Γa1,··· ,an−1 . ci, ci+1 n

sreg 6.2 The nilfibre M0 (n)

The nilfibre carries with it some very interesting structure beyond the result i in Corollary 6.0.2. Consider the solution variety Ξ0,0. i.e. Consider extending an 87

(i + 1) × (i + 1) matrix of the form:   0 1 ··· 0 y1  . . .   0 0 .. . .     . . .   . .. 1 .       0 ······ 0 yi    z1 ······ zi w

i to an (i+1)×(i+1) principal nilpotent. Ξ0,0 has exactly two Gi orbits of dimension i.   0 1 ··· 0 0  . . .   0 0 .. . .    i  . . .  O =  . .. 1 .  L      0 ······ 0 0    z1 ······ zi 0

× with z1 ∈ C and zj ∈ C , 2 ≤ j ≤ i. And   0 1 ··· 0 y1  . . .   0 0 .. . .    i  . . .  O =  . .. 1 .  U      0 ······ 0 yi    0 ······ 0 0

× with yi ∈ C and yj ∈ C , 1 ≤ j ≤ i − 1.

n−1 a1,a2,··· ,an−1 This allow us to construct 2 mappings of the form Γn where i i i ai = OU , OL is a choice of Gi orbit of dimension i in Ξ0,0. By Theorem 4.5.1, the sreg image of each such mapping gives us an A orbit in M0 (n). sreg We want to see that the closure of an A orbit in M0 (n) is a nilradical of a Borel subalgebra of M(n). The main tool we use here is Gerstenhaber’s theorem, which states the following (see [Ger]):

Theorem 6.2.1. (Gerstenhaber) 88

n
A linear space L of nilpotent matrices in M(n) that has dimension 2 is a nilradical of a Borel subalgebra b ⊂ M(n) which is conjugate to the standard Borel by some member w ∈ Sn, the Weyl group of M(n).

We use this theorem to prove the following theorem.

sreg n (2) Theorem 6.2.2. Let A · X be an orbit in M0 (n). Then A · X ' C both as a variety and as a vector space over C. More explicitly, if the A orbit is given

a1,a2,··· ,an−1 i i by Γn where ai = OU or OL, then A · X is the set of matrices of the following form    b  1    .    Xi−1 .    X : Xi =     bi−1      0 0 

i−1 i−1 with bj ∈ C if ai−1 = OU or if ai = OL then we would have     Xi−1 0  X : Xi =    b1 ··· bi−1 0  with bj ∈ C.

Before proving this theorem, let us consider an example in the case of M(3).

Example 6.2.1. Suppose we have chosen a2 = U and a1 = L. The description in the above theorem would then give us that   0 0 a   A · X =  b 0 c    0 0 0

Note that A · X ' C3 both as a vector space and as a variety.

Proof (of Theorem 6.2.2):

Without loss of generality, let us assume that we have made the choice an−1 = U.

The case of an−1 = L is very similar. The proof proceeds by induction on n, the case

a1,··· ,an−1 n = 1 being trivial. It follows from the definition of the Γn maps in Chapter 89

(n−2) 4 that (A · X)n−1 = (An−1 · Xn−1) where An−1 ' C 2 is the group generated by integrating the vector fields {ξfi,j , 1 ≤ i ≤ n − 2, 1 ≤ j ≤ i} on M(n − 1). More

a1,··· ,an−1 specifically, one can see directly from the definition of the map Γn that

a1,··· ,an−1 a1,··· ,an−2 sreg Im(Γn )n−1 = ImΓn . Let X ∈ M0 (n). Now, if we consider the

Zariski continuous map X → Xn−1, we get (A · X)n−1 ⊆ (A · X)n−1 = An−1 · xn−1 . Now, we will use a dimension argument. We first define a set of matrices C as all matrices of the form   b1  .   An−1 · Xn−1 .       bn−1    0 0 n
with bi ∈ C. Now C is an irreducible variety of dimension 2 , since n−1
dim An−1 · Xn−1 = 2 and An−1 · Xn−1 is irreducible by Theorem 3.7 in [KW, n−1
pg 36]. (Note that dim An−1 · Xn−1 = 2 , since Xn−1 is clearly strongly regular if X is.) Now, we note from our work above that A · X ⊂ C , since (A · X)n−1 ⊂

An−1 · Xn−1 and the obvious fact from the Γn construction above that in this case of an−1 = U we have:   b1  .   M(n − 1) .    A · X ⊂    bn−1    0 0

But then, by Theorem 3.7 in [KW, pg 36], we again have that A · X is a n
closed, irreducible subvariety of C of dimension 2 . Thus, by basic properties of dimension, we must have A · X = C. But by induction An−1 · Xn−1 is a linear space with the form described in the theorem. And hence C, being the product of linear spaces is linear and has exactly the form described in the theorem. Q.E.D. sreg Now, we know that the closure of each orbit in M0 (n) is exactly the sreg nilradical of a Borel subalgbera. We adopt the following notation. If X ∈ M0 (n)

a1,a2,··· ,an−1 i i and A · X is given by the image of Γn with ai = OU , OL, then we set

A · X = na1,··· ,an−1 . We would now like to find the element of the Weyl group that 90

− conjugates the nilradical na1,··· ,an−1 into the standard nilradical, n (i.e. the strictly lower triangular matrices). First, let us see how to do it in the Example 6.2.1 that we considered above.

Example 6.2.2. Recall that we chosen a2 = U and a1 = L and the closure we obtained was   0 0 a   A · X =  b 0 c    0 0 0

One can check that the matrix that conjugates n− into A · X is the permutation matrix representing the permutation σ = (132). We note that σ = (12)(13) is the product of the long elements for Type A2 and for Type A3, respectively.

Now, we give an explicit recipe for finding the permutation σ that conju- − sreg gates n into the closure of any A orbit in M0 (n). First, we develop a techinque to conjugate one of the two standard nilradicals (n− or n+) into the closure of sreg + an A orbit in M0 (n). (n denotes the strictly upper triangular matrices). The theorem is the following:

a1,a2,··· ,an−1 Theorem 6.2.3. Let the A orbit be given by the image of the map Γn i i where ai = OU or OL. (To abbreviate the notation, we say ai = L or U). If + an−1 = U, the following procedure produces a permutation σ that conjugates n − into na1,··· ,an−1 and if an−1 = L it produces a permutation which conjugates n into na1,··· ,an−1 .

Step 1) Start with the identity permutation σ0 ∈ Sn.

Step 2) Counting down from n − 1, find the first i1 such that ai1 6= an−1. (Note − + that if no such i exists then A · X = n if an−1 = L or A · X = n if an−1 = U.

In either case, take σ to be the indentity permutation σ0.) Now take σi1 = wi1,0 σ0, where wi1,0 is the long element in Si1+1.

Step 3) Starting with σi1 , ai1 repeat steps 1 and 2.

Step 4) This procedure must eventually stop at some index ik < n, i.e. a1 = a2 =

··· = aik . Then take σ = σik . 91

Furthermore, the permutation constructed in this manner is the unique per- − + mutation that maps either n or n into na1,··· ,an−1 .

Proof: The proof is by induction on n, noting that the case n = 2 is trivial. Without − loss of generality let us suppose that an−1 = L (so that we start with n ). Now − suppose that ai = U, but aj = L, for i < j ≤ n − 1. Then we conjugate n by the long element wi,0 ∈ Si+1. We note that since a permutation Sσ acts on Ei,j via −1 SσEi,jSσ = Eσ(i),σ(j), wi,0 permutes the elements Ek,l for i + 2 ≤ k ≤ n, 1 ≤ l ≤ − k − 1 among themselves, since k − 1 ≥ i + 1. Thus Ad(wi,0) · n is a matrix of the form   Ai+1 0    ∗     .   ∗ ..    ∗ 0 where Ai+1 ∈ M(i + 1) is strictly upper triangular. We think of ∗ as representing the remaining rows which are still in strictly lower triangular form. Now, by

Theorem 6.2.2, X ∈ na1,··· ,an−1 has the form     b1   .     na1,··· ,ai−1 .      0        bi          0 0     ∗ 0     .   ∗ ..    ∗ ∗ 0

Now, by Theorem 6.2.2, it is clear that the (i+1)×(i+1) cutoff of the above matrix is just a nilradical of the form na1,··· ,ai ⊂ M(i + 1), i.e. is just Ai+1 · xi+1 where i (2) Ai+1 ' C is the group generated by the vector fields {ξfl,j , 1 ≤ l ≤ i, 1 ≤ j ≤ l} on M(i + 1). Thus, we can apply the induction hypothesis to find a permutation

τi ∈ Si+1 with τi = wj1,0 ··· wjk,0 as given in the statement of the theorem such that + + Ad(τi)ni+1 = na1,··· ,ai with ni+1 the strictly upper triangular matrices in M(i + 1). 92

Now, we notice again that τi ∈ Si+1 permutes the the elements Ek,l for i + 2 ≤ k ≤ − n, 1 ≤ l ≤ k − 1 amongst themselves. Thus, Ad(τiwi+1,0) · n = na1,··· ,an−1 . This completes the induction and the existence part of the proof of the theorem. The uniqueness follows easily from that fact that if we have for example σn−σ−1 = τn−τ −1, then σ−1τn−τ −1σ = n−. By properties of the Weyl group (see Lemma 2.5.5 in [GW, pg 97]) we must have σ = τ (Similarly for n+). Q.E.D.

− Corollary 6.2.1. To conjugate n into na1,··· ,an−1 we take σ as in the theorem if an−1 = L and if an−1 = U, we take σ˜ = σwn,0 where σ is given by the theorem and wn,0 ∈ Sn is the long element.

Remark 6.2.1. If we start with n−, then we note that what we are doing to get to na1,··· ,an−1 is either choosing or not choosing to conjugate by the long element of the Weyl group at the various levels M(i) for 1 ≤ i ≤ n − 1. Thus in this way we are n−1 selecting 2 special elements of the Weyl group for Type An that parameterize n−1 sreg the 2 A orbits in M0 (n). 7 Gelfand-Zeitlin Theory in

Types Bl and Dl

7.1 Preliminaries

We briefly recall our discussion in sections 1.6 and 1.7. We realize the complex orthogonal Lie algebra so(n) as n × n skew symmetric matrices over C. We can think of so(i) as embedded in so(n) in the top left-hand corner

" # X 0 X,→ 0 0n−i for X ∈ so(i). For X ∈ so(n), we refer to the i × i skew-symmetric matrix in the upper left-hand corner of X as the the i×i cutoff of X and denote it by Xi ∈ so(i). We also think of SO(i) ,→ SO(n) via the embedding

" # g 0 g ,→ 0 Idn−i for g ∈ SO(i). (Recall in this case that SO(i) consists of i × i orthogonal matrices with complex entries). Let P (so(n)) denote the algebra of polynomials on so(n). Recall from section 1.6 that P (so(i)) ⊂ P (so(n)) is a Poisson subalgebra. This allowed us to define a classical analogue to the Gelfand-Zeitlin algebra of U(so(n)) in P (so(n)), J(so(n)) = P (so(2))SO(2) ⊗ · · · ⊗ P (so(n))SO(n) which is Poisson commutative. We saw further in section 1.7 that using J(so(n)),

93 94 we could define a commutative Lie algebra of Hamiltonian vector fields:

V = {ξf |f ∈ J(so(n))}

SO(i) Choosing a set of generators for the algebra J(so(n)), {fi,j ∈ P (so(i)) | 2 ≤ i ≤ SO(i) n − 1, 1 ≤ j ≤ rk i} (where rk i denotes the rank of so(i) and fi,j ∈ P (so(i)) is a fundamental adjoint invariant), we obtained a commutative Lie algebra of vector fields

L = {ξfi,j |2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} (7.1) of dimension at most d/2, where d is the dimension of a regular adjoint orbit in so(n). Using the Lie algebra V , we can then define a collection of subspaces:

X → VX = {(ξf )X |f ∈ J(so(n))} ⊂ TX (so(n))

We now show that the Lie algebra L integrates to a global action of Cd/2 d/2 on so(n). If we let B = C and X ∈ so(n), then we will show TX (B · X) = VX . Note that

VX = span{(ξfi,j )X |2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} (7.2) c.f. Remark 2.8 in [KW, pg 20].

7.2 The integration of vector fields derived from the classical analogue of the

Gelfand-Zeitlin algebra for Types Bl and Dl

To integrate the Lie algebra L in equation (7.1), we need to see that each vector field ξfi,j integrates to a global action of C on so(n). This will follow from the following proposition.

Proposition 7.2.1. Let p ∈ P (so(i))SO(i) where P (so(i)) is viewed as a Poisson subalgebra of P (so(n)) as in section 1.6. The the vector field ξp integrates to global action of C on so(n) given by the formula:

t · X = Ad exp(−tdp(Xi)) · X (7.3) 95 for t ∈ C,X ∈ so(n).

Proof: Recall that for any reductive Lie algebra g, and any polynomial f on g we think of the differential of df(X) at X ∈ g as an element of g via the formula d | f(X + tZ) = β(df(X),Z) dt t=0 where Z ∈ g is arbitrary and β(·, ·) is the g-equivariant form on g. We now claim that p viewed as a polynomial on so(n) has differential dp(X) = dp(Xi) ∈ so(i) ,→ so(n) given by " # dp(Xi) 0 0 0

d for X ∈ so(n). To see this consider dt |t=0 p(X + tY ) for Y ∈ so(n). Now, we recall from section 1.6 that so(n) = so(i) ⊕ so(i)⊥ where so(i)⊥ is the orthogonal compliment of so(i) with respect to the trace form on so(n). If we write Y ∈ so(n) ⊥ with respect to this decomposition as Y = Yi + Yi , we see that p(X + tY ) = p(Xi + tYi) for any t ∈ C. Thus,

d dt |t=0 p(X + tY ) =

d dt |t=0 p(Xi + tYi) =

tr(dp(Xi)Yi) = (7.4)

⊥ tr(dp(Xi)Yi + dp(Xi)Yi ) =

tr(dp(Xi)Y ) as desired. By equation (1.9), we have

−[dp(Xi),X] (ξp)X = (∂ )X (7.5) for X ∈ so(n). Now, it is well known that for p ∈ P (so(i))SO(i), dp(Z) ∈ so(i)Z for Z ∈ so(i) (where so(i)Z denotes the centralizer in so(i) of Z) (see [K1, pg 1]). 96

Using this fact, we can show that

θ(t, X) = Ad(exp(−t dp(Xi))) · X is the integral curve for the vector field ξp starting at X ∈ so(n). To see this, we compute the differential to the curve θ(t, X) at an arbitrary t0 ∈ C.

d dt |t=t0 Ad(exp(−t dp(Xi)) · X =

d dt |t=t0 (exp(t ad(−dp(Xi)))) · X =

ad(−dp(Xi))(exp(t0 ad (−dp(Xi))) · X)

Now, if we let

Y = exp(t0 ad (−dp(Xi)) · X = Ad(exp(−t0 dp(Xi))) · X = θ(t0,X), then we note that since −dp(Xi) centralizes Xi, we have that (θ(t, X))i = θ(t, X)i for all t ∈ C. In particular, we have Yi = Xi. We thus obtain that

ad(−dp(Xi)) (exp(−t0 ad(dp(Xi))) · X) = ad(−dp(Yi)) · Y = (ξp)Y

(in coordinates) by equation (7.5), which verifies the claim. This completes the proof of the theorem. Q.E.D Now, we can use Proposition 7.2.1 to integrate the Lie algebra L in equation (7.1).

The fact that any two vector fields ξf , ξg with f, g ∈ J(so(n)) commute easily allows us to integrate L to an action of Cd/2. Recall that we denote by d the dimension of a regular adjoint orbit in so(n). Note that we have ( ) l2 − l if n = 2l d/2 = l2 if n = 2l + 1

We now have the following analogue of Theorem 3.4 in [KW, pg 34].

Theorem 7.2.1. Let L be the commutative Lie algebra of vector fields given by:

L = {ξfi,j |2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} 97

Then L integrates to an action of a group B = Cd/2 on so(n). If X ∈ so(n) and B · X denotes its B orbit, then TX (B · X) = VX where VX ⊂ TX (so(n)) is the subspace given in equation (7.2).

Proof:

By Proposition 7.2.1 we have that each field ξfi,j integrates to a global action of C on so(n). Since the fields {ξfi,j |2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} pairwise commute the composition of the corresponding flows defines a group action of Cd/2 on so(n). All that remains to be seen is that the tangent space to these orbits is the subspace VX . This follows from the standard fact that orbits of a group action are submanifolds with tangent space spanned by the vector fields given by the infinitesimal action of the Lie algebra. Q.E.D.

Remark 7.2.1. We note that the action of B stabilizes adjoint orbits by Proposition

(4.4.1), since the vector fields ξfi,j for 2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i are tangent to adjoint orbits.

Remark 7.2.2. One notes that group B that we obtain above in Theorem 7.2.1 is SO(i) obtained by fixing a set of generators {fi,j ∈ P (so(i)) | 2 ≤ i ≤ n − 1, 1 ≤ j ≤ SO(2) SO(n−1) SO(i) rk i} for the ring P (so(2)) ×· · ·×P (so(n−1)) with fi,j ∈ P (so(i)) a fundamental adjoint invariant. Choosing a different set of generators for J(so(n)), 0 0 SO(i) {fi,j|1 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} with fi,j ∈ P (so(i)) a fundamental adjoint 0 invariant, defines a different Lie algebra L = {ξ 0 |2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i}. fi,j By Theorem 7.2.1, the Lie algebra L0 integrates to an action of Cd/2 on so(n). One can show that Theorem 3.5 in [KW, pg 34] remains valid in the orthogonal case, so that the two actions of Cd/2 commute with one another and have the same orbit structure on so(n). Since we are concerned with studying the geometry of these orbits, we lose nothing by choosing a specific set of invariants fi,j. We will discuss a convenient choice of invariants below.

We say that X ∈ so(n) is strongly regular if its B orbit is of maximal dimension d/2 (just as in the case of M(n)). The set of strongly regular elements 98 is denoted by sosreg(n). Note that we have:

sreg X ∈ so (n) ⇔ dim B · X = d/2 ⇔ dim VX = d/2

Now, one can easily show that Proposition 2.6 and Theorem 2.7 in [KW, pg 19] carry over to the orthogonal case.

sreg Theorem 7.2.2. X ∈ so (n) if and only if the differentials (dfi,j)x are linearly independent for 2 ≤ i ≤ n and 1 ≤ j ≤ rki. Thus, the set of strongly regular elements in so(n) is Zariski open.

Remark 7.2.3. sosreg(n) is Zariski open, as in the case of M(n), but we do not know apriori that it is not empty, as we do not have an analogue of Theorem 2.3 in [KW, pg 16] (i.e. Hessenberg matrices) as in the case of M(n). We will show in later sections that sosreg(n) is non-empty.

Now, to fix an action of Cd/2, we make a choice of generators of the Poisson commtutative algebra J(so(n)). We take as fundamental invariants for so(2l) the following polynomials:

2j pj(X) = tr(X ) for 1 ≤ j ≤ l − 1 (7.6)

pl(X) = Pfaff(X)

For so(2l + 1, C), we take

2j pj(X) = tr(X ) for 1 ≤ j ≤ l (7.7)

Now, to be able to write down this action of Cd/2 by composing the com- muting actions of C in equation (7.3), we need to know the differentails of the 2j functions tr(Xi ) and Pfaff(Xi). The differential of the former is easily computed 2j−1 to be 2j Xi so that equation (7.3) becomes in this case:

2j−1 t · X = Ad(exp(−t 2j Xi )) · X

Obtaining a formula for the differential of the polynomial Pfaff(Xi) is more difficult and is addressed in the next section. The actual formula will not be used in the following sections, but it is derived for completeness. 99

7.2.1 The differential of the Pfaffian

We now compute the differential of Pfaff(Xi). Remark 7.2.4. We note that by our discussion in the proof of Proposition 7.2.1 (see equation (7.4)) that it suffices to consider the case n = 2l and compute d(Pfaff(X)).

To compute dPfaff(X), we use the fact that det(X) = Pfaff(X)2 and com- pute d(det)(X) on the Zariski open set of non-singular elements in so(2l). Sup- d posing that X ∈ so(2l) is invertible, we consider dt (det(X + tY )|t=0). Since X is invertible this becomes d det(X) (det(I + tX−1Y )| ) = det(X)tr (X−1Y ) (7.8) dt t=0 Where the last equality follows from the fact that the differential of the determinant at the identity it is the identity. But now we claim that (7.8) is nothing more that tr(A(X)Y ) where A(X) denotes the classical adjoint matrix. This follows from the definition of A(X) which gives X−1 = det(X)−1A(X). Thus, for X ∈ so(2l) non-singular, we have that d(det)(X) = A(X). Now, we want to observe that A(X) is skew-symmetric for X ∈ so(2l). It will suffice to prove the claim for X ∈ so2l invertible, because being skew- symmetric is a polynomial condition on the entries of A(X), which are themselves globally defined regular functions on so(2l). Suppose that X ∈ so(2l) invertible. We have that det(X) Id = XA(X) . We transpose this equation and use the fact that X is skew-symmetric to get det(X) Id = −A(X)t X. Setting these two equations equal and canceling X yields A(X) = −(A(X))t, as desired. Now, for X ∈ so(2l) is non-singular, we have that d(Pfaff)(X) is given by:

d(det(X))(Y ) = 2Pfaff(X)d(Pfaff)(X)(Y )

1 hence dPfaff(X) = 2 A(X)/Pfaff(X)

Thus, on a Zariski dense set in so(2l), we have 1 dPfaff(X)Pfaff(X) = A(X) (7.9) 2 100

Now, the entries of Q = dPfaff(X) are all polynmial functions on so(2l). Specif- ∂g ically Qij = −2 |X (where vij is the basis dual to the basis Eij − Eji of so(2l) ∂vij and g = Pfaff(X)). Since the polynomials in (7.9) are globally defined and agree on a Zariski dense set, they must agree on all of so(n). We claim that this implies that Pfaff(X) divide the entries of the matrix A(X) for any X ∈ so(2l). This follows from the following fact see Scholium B.2.12 in [GW, page 630].

Proposition 7.2.2. The polynomial g = Pfaff(X) on so(2l) is irreducible as a polynomial in {xij|1 ≤ i < j ≤ 2l}.

Now, using (7.9) one has that Pfaff(X) = 0 implies ∆ji(X) = 0 for all i, j

(where ∆ji(X) denotes the 2l − 1 × 2l − 1 minor of X obtained by deleting the j th row and the i th column.). One can also obtain this fact directly without using (7.9). Simply observe that the rank of a 2l × 2l skew-symmetric matrix is always even. By Proposition 7.2.2 g = Pfaff is irreducible. So that if we have g(X) =

0 implies ∆ji(X) = 0, then g | ∆ji(X) by unique factorization. Thus, g(X) =

Pfaff(X) is, up to associates, an irreducible factor of ∆ij for all i, j. Now, to obtain a more explicit description of d(Pfaff(X)), we want to figure out what the product of the remaining irreducible factors of ∆ij are. We claim that we have

∆ij = kij Pfaff(X) Pfaff(X(i, j)) (7.10) where X(i, j) denotes the 2l − 2 × 2l − 2 skew symmetric matrix obtain from X by deleting both the i-th row and column and the j-th row and column of X and × kij ∈ C . Note that Pfaff(X(i, j)) is irreducible as an element of the polynomial ring C[xij]1≤i

Proposition 7.2.2 that it is clearly irreducible in the subring C[xkm]1≤k

Pfaff(X(i, j)) = f g with f, g ∈ C[xij]1≤i

Pfaff(X(i, j)) = 0 implies ∆ij(X) = 0 (7.11) 101

Since then we have Pfaff(X(i, j)) Pfaff(X)|∆ij(X). Equation (7.10) then follows from the fact that the degree of ∆ij(X) is 2l − 1, the degree of Pfaff(X) is l, and the degree of Pfaff(X(i, j)) is l − 1. Now, we turn our attention to showing equation (7.11). We note that by the skew-symmetry of the matrix A(X), it is enough to show (7.11) for i < j and hence (7.10) for i < j. Simply take kji = −kij in (7.10). We make use of a rank argument. The essential fact to keep in mind is that the rank of a 2k × 2k skew-symmetric matrix is even. Let Zij be the 2l −1×2l −1 matrix obtained from

X ∈ so(2l) by deleting the i the row and j − th column so that ∆ij = det(Zij).

Then one notes that X(i, j) is obtained from Zij by deleting the i-th column and the j −1-th row. Now, suppose the rank of X(i, j) is k ≤ 2l −2. Then consider the matrix Cij of size 2l − 1 × 2l − 2 which is obtained by reinserting the j − 1 the row of Zij without the entry in the i-th position in the j − 1-th position. It can have at most rank k + 1. This follows clearly from the fact that if {v1, ··· , vk} is a basis for the row space of X(i, j) and w ∈ C2l−2 is the reinserted row then the dimension of the span of {v1, ··· , vk} ∪ {w} can be at most k + 1. Now, we obtain Zij from

Cij by reinserting the i-th column of Zij in Cij (in the i-th position). The same argument again shows that the rank of Zij can be at most k+2. Now, suppose that Pfaff(X(i, j)) = 0. This implies that X(i, j) is singular. Since X(i, j) ∈ so(2l − 2), this gives us that rk X(i, j) ≤ 2l − 4, thus rk Zi,j ≤ 2l − 2. But Zij is of size

2l − 1 × 2l − 1. Thus Zij is singular, and we have ∆ij(X) = det Zij = 0 as desired. Thus, we have equation (7.11) for i < j. We have now proven the following result.

Proposition 7.2.3.

[−dPfaff(X),X] [−1/2A(X)/Pfaff(X),X] (ξPfaff(X))X = (∂ )X = (∂ )X for X ∈ so(n) where A(X) is the classical adjoint of X. Moreover

(A(X)/Pfaff(X))ij = ∆ji(X)/Pfaff(X) = kjiPfaff(X(i, j)) where Pfaff(X(i, j)) is the Pfaffian of the 2l − 2 × 2l − 2 skew-symmetric matrix obtained from X by deleting both the i-th column and row and the j-th column and × row and kij ∈ C . 102

We now have the following immediately corollary which follows from Re- mark 7.2.4.

Corollary 7.2.1.

[−dPfaff(Xj ),X] [−1/2A(Xj )/Pfaff(Xj ),X] (ξPfaff(Xj ))X = (∂ )X = (∂ )X (7.12) with

(A(Xj)/Pfaff(Xj))kl = ∆lk(Xj)/Pfaff(Xj) = clkPfaff(Xj(k, l))

× and clk ∈ C (for X ∈ so(n) j < n even).

7.3 The generic matrices for the orthogonal Lie algebras

We want to generalize the result that Kostant and Wallach obtained for certain regular semi-simple matrices in M(n) to the orthogonal case (see Theorems 3.23, 3.28 in [KW, pg 45, 49]). First, we define the following regular map:

Φ: so(n) → Cd/2+rk n

Φ(X) = (f2,1(X2), f3,1(X3), ··· , fn,rk n(X))

SO(i) d/2+rk n with fi,j ∈ P (so(i)) a fundamental adjoint invariant. For c ∈ C , −1 let so(n)c = Φ (c).

Proposition 7.3.1. The action of B preserves the fibres so(n)c.

Proof: The proof here is exactly the same as the proof of Proposition 3.6 in [KW, pg

35]. Since the functions {fi,j| 2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} all Poisson commute, fi,j is constant along the Hamiltonian flow of fk,l (see [Lee, pg 489]). Thus, fi,j is invariant under the action of the group B. Hence, the fibres soc(n) are invariant under the action of B. Q.E.D. 103

We define Ωn ⊂ so(n) to be the set of matrices where each cutoff Xi is regu- lar semi-simple, and the spectrum of Xi and Xi+1 have no intersection. We say that d/2+rk n c ∈ C satisfies the eigenvalue disjointness condition if c = (c1, ··· , ci, ··· , cn) rk i with each ci ∈ C representing a regular semi-simple element element of so(i) (up to the action of the Weyl group, Wi) and the elements represented by ci and ci+1 have no eigenvalues in common. (Here we are making use of the canonical isomorphism

rk i Φi : hi/Wi → C given by

Φi(X) = (fi,1(X), ··· , fi,rk i(X)) where hi ⊂ so(i) is a Cartan subalgebra, Wi is the Weyl group and fi,j are funda- mental adjoint invariants for so(i)). For c ∈ Cd/2+rk n satisfying the eigenvalue disjointness condition we claim that we have that soc(n) ⊂ Ωn. This follows from the following general result.

Proposition 7.3.2. Let g be a semi-simple Lie algebra of rank l and let φ1, ··· , φl be a basic set of adjoint invariants. Let h ⊂ g be a Cartan subalgebra and let hreg denote the set of regular elements in h. Let X ∈ g and suppose that there exists a reg Y ∈ h such that φ1(X) = φ1(Y ), ··· , φl(X) = φl(Y ), then X ∈ G · Y (where G denotes the adjoint group of g ). In particular X is regular semi-simple.

Proof:

We write X ∈ g in its Jordan decomposition, X = Xs + Xn where Xs and

Xn are the semi-simple and nilpotent parts of X respectively. Now, one knows that G for any invariant polynomial φ ∈ P (g) , we have that φ(X) = φ(Xs) (see [K2, pg

360]). Then we have that φi(Y ) = φi(Xs) for all fundamental invariants φi. Thus, since both Y and Xs are semi-simple, we have that Xs = g · Y for some g ∈ G.

Hence Xs must be regular semi-simple. But then from the Jordan decomposition of X, we know [Xs,Xn] = 0 so that Xn ∈ Cg(Xs). But since Xs is regular semi- simple Cg(Xs) = h where h is the unique Cartan subalgbera containing Xs. This 104

gives us that Xn = 0 and that X = Xs = g · Y , which is the desired result. Q.E.D

With this proposition, we also clearly have Ω = ∪ d/2+rk n so (n) with c n c∈C c satisfying the eigenvalues disjointness condition. We note also that the action of

B preserves the set Ωn. This is clear by the argument in the proof of Proposition

(7.3.1), which states that each function fi,j is invariant under the action of B. d/2+rk n So that if X ∈ soc(n) with c ∈ C satisfying the eigenvalue disjointness condition, then B · X ∈ soc(n) ⊂ Ωn. Our goal in the next several sections will be to describe the action of B on the set Ωn. We will accomplish this by considering an analogous construction to the Γn construction that we had for M(n). As in the case of M(n), the first step will be to describe the structure of the solution varieties for this generic situation.

7.4 Generic solution varieties for orthogonal Lie algebras

First, we recall the form of regular semi-simple elements in Types Bl and

Dl. Consider first the case of type Bl i.e. so(2l + 1). In this case, we can choose a Cartan subalgebra consisting of matrices of the form:   0 a1    −a 0   1     0 a2       −a2 0    (7.13)  ..   .     0 a   l     −a 0   l  0 with ai ∈ C. We will refer to the Cartan subalgebra in (7.13) as the standard Cartan subalgebra in so(2l + 1). Now, regular semi-simple elements in this particular Cartan h are exactly the elements which do not have repeated eigenvalues. i.e. the elements of the 105

form (7.13) with ai 6= 0 and ai 6= (+/−) aj for i 6= j. To see this, we must recall the following. Let g be a semi-simple Lie algebra over C and h ⊂ g be a Cartan subalgebra. We recall that X ∈ g is regular if and only if dp1(X) ∧ dp2(X) ∧ · · · ∧ G dpl(X) 6= 0, with pj ∈ P (g) fundamental adjoint invariants (see Theorem 9 [K2, pg 382]). Now, let qi = pi|h for 1 ≤ i ≤ l and X ∈ h. We claim that we have

dq1(X) ∧ dq2(X) ∧ · · · ∧ dql(X) 6= 0

if and only if (7.14)

dp1(X) ∧ dp2(X) ∧ · · · ∧ dpl(X) 6= 0

(Here we are thinking of the dpi(X) and dqi(X) as covectors. )The sufficiency is obvious. To see the necessity, note that if X ∈ h is regular, and gX denotes the centralizer of X ∈ g, then gX = h, see ([CM, pgs 20-21]). Now let Φ(g, h) be a system of roots for g relative to h. Now, since gX = h, we must have that α(X) 6= 0 for all α ∈ Φ(g, h). But now, we make use of the fact that [see Hum 2, pg 69]: Y dq1(X) ∧ dq2(X) ∧ · · · ∧ dql(X) = k α(X) (7.15) α∈Φ+(g,h) where k ∈ C× and Φ+(g, h) denotes the set of positive roots with respect to the chosen Cartan h. Thus, dq1(X) ∧ dq2(X) ∧ · · · ∧ dql(X) 6= 0, completing the proof of the claim. Now, we let g = so(2l + 1) and suppose that X ∈ h is regular where h ⊂ so(2l + 1) is the Cartan subalgebra consisting of elements of the form (7.13). We want to understand what the RHS of equation (7.15) means in this case. To see the root system for a Lie algebra of type Bl, we have to consider another realization of so(2l + 1). We let g0 = so(2l + 1) thought of as matrices which are skew-symmetric about the skew diagonal. In other words, g0 consists of matrices in M(2l + 1) which are skew-adjoint with respect to the symmetric bilinear form defined by the matrix S ∈ M(2l + 1) where S is given by:   0 0 sl   S =  0 1 0  (7.16)   sl 0 0 106

with sl being the l×l matrix with ones down the skew-diagonal and zeros elsewhere. Let h0 ⊂ g0 be the standard Cartan. This means that h0 consists of diagonal matrices which are skew-symmetric about the skew-diagonal i.e. matrices of the form

diag[x1, ··· , xl−1, xl, 0, −xl, −xl−1, ··· , −x1] with xi ∈ C. Then one computes that the right side of equation (7.15) is the polynomial l Y 2 2 f(X) = 2 l! x1 ··· xl (xj − xi ) (7.17) 1≤i

f(Z) 6= 0 ⇔

Z has distinct eigenvalues ⇔

⇔ X has distinct eigenvalues A simple calculation shows that the eigenvalues of the matrix in (7.13) are

(+/−)aj ı, 0 for 1 ≤ j ≤ l. Thus, the condition that ai 6= 0 and ai 6= (+/−) aj for i 6= j is exactly the condition that the matrix in (7.13) have distinct eigenvalues. 107

The case of type Dl (i.e. so(2l)) is very similar. In this case, we choose our Cartan subalgebra h to be matrices of the form:   0 a1    −a1 0       0 a2       −a2 0  (7.18)  .   ..       0 a   l  −al 0 with ai ∈ C for 1 ≤ i ≤ l. We will refer to (7.18) as the standard Cartan subalgebra in so(2l). Again the regular semi-simple elements are the elements with distinct eigen- values i.e. the elements of the form (7.18) for which ai 6= (+/−)aj for i 6= j. We can argue this exactly as we did in type Bl. We note that if we realize so(2l) as 2l × 2l matrices which are skew-symmetric about the skew-diagonal, then the standard Cartan is given by diagonal matrices of the form

diag[x1, x2, ··· , xl, −xl, ··· , −x2, −x1] with xi ∈ C, and in this case, one computes that the determinant (7.15) is given by (cf. [Hum 2, pg. 68]):

l−1 Y 2 2 2 (l − 1)! (xj − xi ). 1≤i

In the case of orthogonal Lie algebras so(n) there are two different types of extension problems to consider. We have to consider extending from Type Dl to

Type Bl and from Type Bl to Type Dl+1. 108

7.4.1 Extending from Type Bl to Type Dl+1

We consider the following 2l + 2 × 2l + 2 skew-symmetric matrix.   0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22     .. .   . .  (7.19)      0 al zl1     −a 0 z   l l2     0 zl+1    −z11 −z12 −z21 −z22 · · · −zl1 −zl2 −zl+1 0 with ai ∈ C for 1 ≤ i ≤ l and ai 6= 0 for all i and ai 6= (+/−)aj for i 6= j. We want to choose the zij and zl+1 in the matrix in (7.19) so that it has characteristic polynomial given by l+1 Y 2 2 (t + bi ) i=1 with bi 6= 0, bi 6= (+/−)bj and bi 6= (+/−)aj for i 6= j. To do this we first have to find the characteristic polynomial of the matrix in (7.19). To that affect we have the following proposition.

Proposition 7.4.1. The characteristic polynomial of the matrix in (7.19) is given by the following polynomial.

l " #! l l X 2 2 2 Y 2 2 2 Y 2 2 2 Y 2 2 (zi1 + zi2) t (t + aj ) + zl+1 (t + ai ) + t (t + ai ) (7.20) i=1 j=1, j6=i i=1 i=1

Proof: 109

We are interested in computing the following determinant.   t −a1 −z11    a t −z   1 12     t −a2 −z21       a2 t −z22     .. .  det  . .  (7.21)      t −al −zl1     a t −z   l l2     t −zl+1    z11 z12 z21 z22 ··· zl1 zl2 zl+1 t

We expand by cofactors down the last column, keeping careful track of the signs involved. We consider the cofactor corresponding to −zj1. After eliminating the row containing −zj1 (i.e. the 2(j − 1) + 1-th row), we get the following:

2j−1+2l+2 (−1) (−zj1)×

  t −a1      a1 t     t −a   2     a t   2   .   ..  (7.22)     det  t −a   j−1     aj−1 t 0 0       0 aj t     ..   .       t    z11 z12 z21 z22 ········· zj1 zj2 zl+1

Now, we want to compare this cofactor to the one from −zj2, since some of the 110 terms cancel.

2j+2l+2 (−1) (−zj2)×

  t −a1      a1 t     t −a   2     a t   2   .   ..  (7.23)     det  t −a   j−1     aj−1 t 0 0       0 t −aj     ..   .       t    z11 z12 z21 z22 ········· zj1 zj2 zl+1 Now, we expand the 2l + 1 × 2l + 1 determinant in (7.22) by cofactors along the 2j − 1-th column and the one in (7.23) by cofactors along the 2j-th column. The claim is that the terms involving aj from (7.22) and from (7.23) cancel and the remaining terms are the desired ones. We concentrate first on the terms involving the aj. Consider (7.22). We get the following determinant

2j−1+2l+2 (−1) (−zj1)aj×

  t −a1    a1 t       t −a2       a2 t     ..   .      det  t −aj−1     a t 0 0   j−1     0 0 t −a   j+1   .   ..   aj+1 t     t    z11 z12 z21 z22 ········· zj2 zj+11 ········· zl+1 (7.24) 111

Now, we look at the cofactor from −aj in (7.23). We get:

2j−1+2j (−1) (−zj2)(−aj)×

  t −a1    a1 t       t −a2       a2 t     ..   .      det  t −aj−1     a t 0 0   j−1     0 0 t −a   j+1   .   ..   aj+1 t     t    z11 z12 z21 z22 ········· zj1 zj+11 ········· zl+1 (7.25) Finally, we evaluate the determinant in (7.24) and (7.25) by expanding by cofactors along the 2j − 1-th column. For (7.24), we get the following:

2j−1 2l+2j−1 (−1) (−zj1)aj(−1) zj2 det(X) where X ∈ M(2l − 1) is the 2l − 1 × 2l − 1 matrix given by taking the matrix the 2l + 1 × 2l + 1 cutoff of the matrix in (7.21) and removing the 2j − 1 and 2j rows and columns. For (7.25), we get the following:

2j−1+2l (−zj2)(aj)(−1) zj1 det(X)

One observes that the above two equations cancel one another.

Now, it only remains to compute the cofactors in (7.22) comming from zj1 and the one in (7.23) comming from zj2. First, consider (7.22): We get 112

2j−1 2j−1+2l+1 (−1) (−zj1)(−1) (zj1)×

  t −a1    a t   1   .   ..       t −aj−1       aj−1 t  det      t     t −a   j+1     aj+1 t     ..   .    t

And from (7.23), we get:

2j+2l+2 2l+1+2j (−1) (−zj2)zj2(−1) ×

  t −a1    a t   1   .   ..       t −aj−1       aj−1 t  det      t     t −a   j+1     aj+1 t     ..   .    t

These two determinants evaluate to:

" l # 2 2 2 Y 2 2 (zj1 + zj2) t (t + ai ) (7.26) i=1, i6=j 113

In evaluating the determinant in (7.21), we now only have two more co- factors to consider, namely the one from zl+1 and the one in the 2l + 2 × 2l + 2 position. Let us first consider the one from zl+1. It is easily seen to be:

l 2 Y 2 2 zl+1 (t + ai ) (7.27) i=1 The final cofactor is also easily seen to be:

l 2 Y 2 2 t (t + ai ) (7.28) i=1 Adding the cofactors from (7.26) for 1 ≤ j ≤ l to (7.27) and (7.28). We arrive at

l " l #! l l X 2 2 2 Y 2 2 2 Y 2 2 2 Y 2 2 (zj1 + zj2) t (t + ai ) + zl+1 (t + ai ) + t (t + ai ) (7.29) j=1 i=1, i6=j i=1 i=1 which is the desired result at last. Q.E.D. Now, we actually want to see that we can use this result to extend from the given semi-simple orbit in so(2l+1) to an arbitrary semi-simple orbit in so(2l+2). Recall that we want to choose the zij and zl+1 so that the matrix in (7.19) has charteristic polynomial given by l+1 Y 2 2 (t + bi ) (7.30) i=1 with bi 6= 0, bi 6= (+/−)bj and bi 6= (+/−)aj for i 6= j. If we can solve this extension problem, then the matrix in (7.19) will be similar in O(2l + 2) to a matrix of the form   0 b1    −b1 0       0 b2       −b2 0   .   ..       0 b   l+2  −bl+2 0 114

with bi ∈ C for 1 ≤ i ≤ l + 2 and satisfying bi 6= 0, bi 6= (+/−)bj and bi 6=

(+/−)aj for i 6= j. Let us call the polynomial in (7.29) f(t) and the one in (7.30) g(t). Now f(t), g(t) are both polynomials of degree 2l + 2. However, since both polynomials are monic and since they are both characteristic polynomials of matrices with zero trace, their difference h(t) = f(t) − g(t) is a polynomial of degree at most 2l. Now, we want to find conditions on the zij and zl+1 such that h(t) ≡ 0. If we can choose the zij and zl+1 such that there exist 2l + 1 distinct complex numbers x1, ··· , x2l+1 with the property that f(xi) = g(xi) for 1 ≤ i ≤ 2l + 1, then, since f − g is of degree at most 2l, we would be forced to have f − g ≡ 0 ⇒ f = g.

We choose for our xi the distinct 2l+1 eigenvalues of the 2l+1×2l+1 cutoff of the matrix in (7.19). This means that we take x1 = a1ı, x2 = −a1ı, ··· x2l−1 = alı, x2l = −alı, x2l+1 = 0. Now, if we substitute y = (+/−)aiı into the equation f(t) = g(t), for 1 ≤ i ≤ l, we obtain: Ql+1 (b2 − a2) f(y) = g(y) ⇔ z2 + z2 = j=1 j i (7.31) i1 i2 2 Ql 2 2 −(ai ) j=1, j6=i(aj − ai ) for 1 ≤ i ≤ l. We note that this is defined precisely because our assumption of regularity on the 2l + 1 × 2l + 1 cutoff gives us that ai 6= (+/−)aj and ai 6= 0. Now, by our hypothesis on the disjointness between the eigenvalues of the adjacent cutoffs, we actually have the RHS of (7.31) is non-zero, so we can rewrite (7.31) as: 2 2 2 f(y) = g(y) ⇔ (zi1 + zi2) = di (7.32)

r Ql+1 2 2 j=1(bj −ai ) with di 6= 0, di = 2 Ql 2 2 . Observe that di only depends upon the −(ai ) j=1, j6=i(aj −ai ) values of the aj and the bk.

Now, to find the condition on zl+1, we substitute t = 0 into the equation f(t) = g(t) to get: Ql+1 2 2 i=1 bi f(0) = g(0) ⇔ zl+1 = Ql 2 i=1 ai (7.33) Ql+1 i=1 bi hence zl+1 = (+/−) Ql i=1 ai

Once again the RHS of (7.33) is defined since ai 6= 0 for 1 ≤ i ≤ l. We note that we have some ambiguity in the choice of zl+1. We will see that the choice 115

of sign for zl+1 corresponds to the choice of sign of the Pfaffian of the matrix in (7.19) Now, we note that if (7.32) and (7.33) are satisfied then we have that h(xi) = f(xi) − g(xi) = 0 for 1 ≤ i ≤ 2l + 1 ⇒ h(t) ≡ 0 ⇒ f(t) = g(t) for all t. And we solved the extension problem. Let us make some observations about the structure of the solutions. For now, let us fix a choice of sign for zl+1. We want to consider the equation (7.32) for i = 1, ··· , l. Consider the closed subvariety of C2 defined by:

2 2 2 2 Xi = {(zi1, zi2) ∈ C |zi1 + zi2 = di 6= 0}

We claim that Xi is isomorphic to SO(2) (thought of as 2 × 2 complex orthogonal matrices). Note also that we clearly have an action of G = SO(2) on Xi given as follows. Let g ∈ SO(2) so that " # u −w g = w u

2 2 with w + u = 1. Then g acts on Xi by left translation (thinking of the elements of Xi as column vectors). " # " # z z u − wz g · i1 = i1 i2 zi2 zi1w + uzi2

2 2 2 2 2 2 2 2 Now, we observe (zi1u − wzi2) + (zi1w + uzi2) = zi1(u + w ) + zi2(u + w ) = 2 2 2 2 2 zi1 + zi2 = di , since u + w = 1. Thus, we have a regular action of SO(2) on Xi.

With this action on Xi, we have a G-equivariant isomorphism, Φi from Xi to G, where the action on G is given by left translation. The map is given by " #! " # zi1 1 zi1 −zi2 Φi = (7.34) zi2 di zi2 zi1

An easy computation shows us that this map is G-equivariant. Consider " #! zi1 Φi g · . By our above work this is just zi2 " #! " # zi1u − wzi2 1 zi1u − wzi2 −(zi1w + uzi2) Φi = zi1w + uzi2 di zi1w + uzi2 zi1u − wzi2 116

But we have that " #! " # " # zi1 u −w 1 zi1 −zi2 g · Φi = zi2 w u di zi2 zi1 which becomes " # 1 zi1u − wzi2 −(zi1w + uzi2)

di wzi1 + uzi2 uzi1 − wzi2

Thus, we have Φi is a G-equivariant isomorphism. The upshot of this discussion is that the set of solutions to the extension problem outlined at the beginning of the section are isomorphic to SO(2)l (once we have fixed a choice of sign for zl+1) and the centralizer of the 2l + 1 × 2l + 1 cutoff acts diagonally by l copies of the standard left regular representation of SO(2) on itself with this identification. This follows immediately from the observation that the centralizer of the 2l + 1 × 2l + 1 cutoff of the matrix in (7.19) is simply given by   SO(2)    SO(2)     .  G2l+1 =  ..  (7.35)      SO(2)    1

Now, we want to observe that once we have fixed a choice of sign for zl+1 that

G2l+1 acts freely on the set of solutions given by (7.32) and (7.33) and hence such solutions form one G2l+1 orbit. Denote such a set of solutions by Z ⊂ so(2l + 2).

First of all, note that from the description of G2l+1 in (7.35), it is easy to see that the action preserves the exact value of zl+1, so G2l+1 acts on Z. The fact that the action is free now follows easily from our above description of the action and the fact that the action of SO(2) on itself by left translation is free. Now, to see that the group acts transtively, we observe that Z forms an irreducible variety of dimension 2l that is isomorphic to SO(2l) by our above work. Hence, since l G2l+1 ' SO(2) acts freely any G2l+1 orbit is open in the set of solutions. Since two non-empty open subsets of an irreducible variety must intersect, we have that the action of G2l+1 on Z is transitive. 117

Now, let us discuss the significance of the choice of sign of zl+1 in equation (7.33). Let X be the matrix in (7.19). Then by Proposition (7.4.1) the determinant of this matrix is simply l 2 Y 2 det(X) = zl+1 ai i=1 (Since the degree of the characteristic polynomial in this case is even, we have Ql that the constant term is simply det(X).) Thus, Pfaff(X) = (+/−)zl+1 i=1 ai. Ql Suppose first that Pfaff(X) = zl+1 i=1 ai. Then choosing the positive sign in Ql+1 equation (7.33), gives Pfaff(X) = i=1 bi. We claim that the matrix in equation (7.19) is now similar in SO(2l + 2) to a regular semi-simple element of the form:   0 b1    −b1 0       0 b2       −b2 0     ...       0 b   l+1  −bl+1 0

Ql+1 with bi 6= (+/−)bj. Note that the Pfaffian of such a matrix is exactly i=1 bi and we have chosen the zij and the zl+1 so that the matrix X now takes the same values on the fundamental adjoint invariants

2 4 2l ψ1(X) = tr(X ), ψ2(X ), ··· , ψl(X) = tr(X ), ψl+1(X) = Pfaff(X) as this regular semi-simple element. Now, the claim follows from Proposition 7.3.2.

Now, suppose that we choose the negative sign for zl+1 in (7.33). Then Ql+1 Pfaff(X) = − i=1 bi. This means that we have chosen the extension so that the matrix X is conjugate via an element of SO(2l + 2) to the element in so(2l + 2) 118 given by   0 −b1    b1 0       0 b2       −b2 0   .   ..       0 b   l+1  −bl+1 0 Ql If we suppose instead that Pfaff(X) = −zl+1 i=1 ai, then choosing the pos- Ql+1 itive sign in equation (7.33) gives Pfaff(X) = − i=1 bi and choosing the negative Ql+1 sign in (7.33) gives Pfaff(X) = i=1 bi, so that the situation is reversed. The point is that we can choose a sign in equation (7.33) so that we can extend from any regular semi-simple orbit so(2l + 1) to any regular semi-simple orbit in so(2l + 2) satisfying the eigenvalue disjointness condition. We now summarize all that we have shown in the following theorem.

Theorem 7.4.1. Given a matrix of the form (7.19) i.e.   0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22     .. .   . .       0 al zl1     −a 0 z   l l2     0 zl+1    −z11 −z12 −z21 −z22 · · · −zl1 −zl2 −zl+1 0 with the 2l + 1 × 2l + 1 cutoff regular semi-simple, we can choose the zij and zl+1 such that this matrix has characteristic polynomial given by

l+1 Y 2 2 (t + bi ) i=1 119

with the bi satisfying the disjointness condition that bi 6= 0, bi 6= (+/−)bj, bi 6=

(+/−)aj. Moreover, we can choose the sign of zl+1 such that the Pfaffian of the Ql+1 Ql+1 matrix in (7.19) is either i=1 bi or − i=1 bi. Thus, we can extend from any reg- ular semi-simple orbit in so(2l + 1) to any regular semi-simple orbit in so(2l + 2) satisfying the disjointness condition. Moreover, each of these extensions is iso- l l morphic as a variety to the torus SO(2) and the centralizer G2l+1 = SO(2) acts simply transitively on the extensions.

7.4.2 Extending from Type Dl to Type Bl

We consider the following 2l + 1 × 2l + 1 skew-symmetric matrix.

  0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22    (7.36)  .. .   . .     0 a z   l l1     −a 0 z   l l2  −z11 −z12 −z21 −z22 · · · −zl1 −zl2 0 with ai ∈ C for 1 ≤ i ≤ l and ai 6= 0 for all i and ai 6= (+/−)aj for i 6= j.

We want to choose the zij so that this matrix has characteristic polynomial given by: l Y 2 2 t (t + bi ) (7.37) i=1 with bi ∈ C satisfying the conditions that bj 6= (+/−)ak, bi 6= (+/−)bj, and bi 6= 0. To do this we first have to find the characteristic polynomial of the matrix in (7.36). To that affect we have the following proposition.

Proposition 7.4.2. The characteristic polynomial of the matrix in (7.36) is given by: " l l l # Y 2 2 X 2 2 Y 2 2 t (t + ai ) + (zi1 + zi2) (t + ai ) (7.38) i=1 i=1 j=1, j6=i 120

Proof: We can actually derive the result quickly using Proposition 7.4.1. We note that Proposition 7.4.1 implies that the determinant   t −a1 −z11    a t −z   1 12     t −a2 −z21       a2 t −z22     .. .  det  . .       t −al −zl1     a t −z   l l2     t −zl+1    z11 z12 z21 z22 ··· zl1 zl2 zl+1 0 is given by:

l " l #! l l X 2 2 2 Y 2 2 2 Y 2 2 2 Y 2 2 (zj1 + zj2) t (t + ai ) + zl+1 (t + ai ) + t (t + ai ) j=1 i=1, i6=j i=1 i=1

If we set zl+1 = 0 above then   t −a1 −z11    a t −z   1 12     t −a2 −z21       a2 t −z22     .. .  det  . .  (7.39)      t −al −zl1     a t −z   l l2     t 0    z11 z12 z21 z22 ··· zl1 zl2 0 0 is equal to

l " l #! l X 2 2 2 Y 2 2 2 Y 2 2 (zj1 + zj2) t (t + ai ) + t (t + ai ) (7.40) j=1 i=1, i6=j i=1 Now, we evaluate the determinant in (7.39) by expanding by cofactors along the 2l + 1 column to get 121

t det(A) where A is the matrix in (7.36). We know that by our above work this is equal to (7.40). Canceling t then gives us the desired result. Q.E.D.

Remark 7.4.1. We note that the polynomial in (7.38) is invariant under sign changes aj → −aj. So that the above computations and the ones that follow are independent of the Pfaffian of the 2l × 2l cutoff of the matrix in (7.36).

Now, we want to see that we can choose the zij in the matrix in (7.36) so that that it is conjugate in so(2l + 1) to an element of the form:   0 b1    −b 0   1     0 b2       −b2 0    (7.41)  ..   .     0 b   l     −b 0   l  0 with bi ∈ C, bi 6= 0, for all i and bi 6= (+/−)bj and satisfying the disjointness condition that ai 6= (+/−)bj. We argue in a similar manner as we did in the case of extending Type

Bl to Type Dl+1. We denote the polynomial in equation (7.38) by f(t) and the one in equation (7.37) by g(t). Then we note that since both polynomials are monic and both the above matrix and the matrix in (7.36) have trace 0, the polynomial h(t) = f(t) − g(t) is of degree at most 2l − 1. Now, we want to find conditions on the zij such that h(t) ≡ 0. As in the previous case, if we can choose the zij such that there exist 2l distinct complex numbers x1, ··· , x2l such that f(xi) = g(xi) for 1 ≤ i ≤ 2l, then, since f − g is of degree at most 2l − 1, we would be forced to have f − g ≡ 0 ⇒ f = g. 122

We choose for our xi the 2l distinct eigenvalues of the 2l × 2l cutoff of the matrix in (7.36). i.e. We take x1,1 = a1ı, x1,2 = −a1ı, ··· , xl,1 = alı, xl,2 = −alı. Note that these are distinct precisely because of the regularity condition on the

2l × 2l cutoff of (7.36) and because no ai = 0, by the eigenvalue disjointness condition. We have that

Ql 2 2 2 2 i=1(bi − aj ) f(xj,i) = g(xj,i) ⇔ (z + z ) = (7.42) j1 j2 Ql 2 2 i=1, i6=j(ai − aj ) for i = 1, 2. The RHS of the (7.42) is defined precisely because the 2l × 2l cutoff of the matrix in (7.36) is regular and it is non-zero because of the disjointness condition on the ai and bj. If we let v u Ql 2 2 u i=1(bi − aj ) dj = t (7.43) Ql 2 2 i=1, i6=j(ai − aj ) then we have that f(t) = g(t) if and only if

2 2 2 zj1 + zj2 = dj for 1 ≤ j ≤ l and with dj 6= 0. Again, we note that dj depends only on the values of the aj and bi. Thus, X ∈ so(2l + 1) satisfying (7.42) is conjugate in SO(2l + 1) to an element of the form (7.41) by Proposition 7.3.2.

As in the case of extending from Type Bl to Type Dl+1, we see that the set of extensions form an irreducible, smooth affine variety of dimension l that is algebraically isomorphic to SO(2)l. We refer to these solutions as Z. Now, in this case the centralizer in SO(2l) of the 2l × 2l cutoff of the matrix in (7.36) consists of matrices of the form   SO(2)    SO(2)     .  (7.44)  ..    SO(2) 123

l Thus, G2l ' SO(2) as algebraic groups. G2l acts on the solution variety Z by the adjoint action. The same reasoning as used in the previous case shows that this action is nothing more than l copies of the left regular representation of SO(2) acting along the last column of the matrix in (7.36). Thus, the same analysis in the case of extending from Type Bl to Type Dl+1 allows us to arrive at the following theorem.

Theorem 7.4.2. Given a matrix of the form (7.36) i.e.   0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22     .. .   . .     0 a z   l l1     −a 0 z   l l2  −z11 −z12 −z21 −z22 · · · −zl1 −zl2 0 with the 2l × 2l cutoff regular semi-simple, we can choose the zij such that this matrix has characteristic polynomial given by

" l # Y 2 2 t (t + bi ) i=1 with bi 6= 0 for all i, bi 6= (+/−)bj, and with the aj and the bi satisfying the disjointness condition that bi 6= (+/−)aj and aj 6= 0 for all j. By Remark (7.4.1), this computation is independent of the sign of the Pfaffian of the 2l × 2l cutoff. Thus, we can extend from any regular semi-simple orbit in so(2l) to any regular semi-simple orbit in so(2l + 1) satisfying the disjointness condition. Moreover, each of these extensions is isomorphic as a variety to the torus SO(2)l and the l centralizer G2l = SO(2) acts simply transtively on the extensions. 124

7.5 The Γn construction for generic elements of so(n)

The construction proceeds in a similar manner to the case of M(n) discussed in detail in Chapter 4. In this case, however, we only define an algebraic action of d/2 d/2+rk n SO(2) on each fibre so(n)c where c ∈ C satisfies the eigenvalues disjoint- ness condition. (Recall d is half the dimension of a generic adjoint orbit of SO(n) d/2+rk n in so(n)). Now for each such c ∈ C we write c = (c1, c2, ··· , ci, ··· , cn) with rk i c ∈ C where rk i is the rank of so(i). Then each ci represents a regular semi- simple element in the standard Cartan hi ⊂ so(i) up to the action of the Weyl group

Wi and the elements represented by ci and ci+1 have no eigenvalues in common. Re- rk i call that we identify C with hi/Wi via the map Φi(X) = (fi,1(X), ··· , fi,rk i(X)) which takes X ∈ hi to its values on the fundamental Weyl group invariants. (Re- call also that for i = 2l + 1 the standard Cartan hi ⊂ so(i) is given by matrices of the form (7.13) and for i = 2l hi is given by matrices of the form (7.18).) We consider the principal subset of hi/Wi given by the quotient by the action of Wi on the principal open subset of hi defined by the non-vanishing of the polynomial 2 Q 2 2 Qrk i π with π = 1≤j

D = {z ∈ hreg|α(Rez) ≥ 0, if α(Rez) = 0, α(Imz) > 0 for all α ∈ Φ+} (7.45)

Consider the case g = M(i), h the i × i diagonal matrices, and W = Si, the symmetric group on i letters. If we take the standard positive roots for g, then it is easy to see that fundamental domain we have just described is nothing other than the set reg {(z1, ··· , zi) ∈ h |z1 > z2 > ··· > zi} where “>” denotes the lexicographical ordering on C that we introduced in the beginning of Chapter 3 and used to write down elements of the solution varieties Ξi ⊂ M(i + 1). ci, ci+1 Now, given c ∈ Cd/2+rk n satisfying the eigenvalue disjointness condition, rk i we write c = (c1, c2, ··· , ci, ··· , cn) with c ∈ C as above. We want to define a regular isomorphism c d/2 Γn : SO(2) → so(n)c We define Ξi ⊂ so(i + 1) the orthogonal solution variety at level i as follows. ci, ci+1 0 Recall that each ci represents a regular semi-simple element in (hi) /Wi. We choose 0 0 a representative of (hi) /Wi and (hi+1) /Wi+1 using the fundamental domain given equation (7.45). This gives us a canonical way of writing down elements of the standard Cartan subalgebras hi ⊂ so(i) and hi+1 ⊂ so(i + 1) that satisfy the eigenvalue disjointness condition whose values on the fundamental Weyl group in- rk i rk i+1 variants are as prescribed by ci ∈ C and ci+1 ∈ C . Once the representatives for c and c have been chosen Ξi is defined to be the set of solutions to the i i+1 ci, ci+1 extension problem given in Theorem 7.4.1 if i is odd and if i is even then Ξi ci, ci+1 is given by Theorem 7.4.2. Let l = rk (so(i)). First, we need to see that we can find an element g(z) ∈ SO(i+1) with the property that g(z) conjugates the element z ∈ Ξi to ci, ci+1 the element in the standard Cartan in so(i+1) determined by ci+1 and lying in the fundamental domain Di+1 ⊂ hi+1 in equation (7.45) and with the property that the function z → g(z) is regular on Ξi . If i is odd, then g(z) would conjugate ci, ci+1 126 the element   0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22     .. .   . .       0 al zl1     −a 0 z   l l2     0 zl+1    −z11 −z12 −z21 −z22 · · · −zl1 −zl2 −zl+1 0 into the element   0 b1    −b1 0       0 b2       −b2 0  ∈ Di+1    ...       0 b   l+1  −bl+1 0

If i is even, g(z) would conjugate the element in z ∈ Ξi given by ci, ci+1   0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22     .. .   . .     0 a z   l l1     −a 0 z   l l2  −z11 −z12 −z21 −z22 · · · −zl1 −zl2 0 into the element 127

  0 b1    −b 0   1     0 b2       −b2 0    ∈ Di+1  ..   .     0 b   l     −b 0   l  0 This can be done non-explicitly as follows. By Theorems 7.4.1 and 7.4.2, we know that Ξi ' SO(2)l and that G = SO(2)l acts simply transitively on ci ci+1 i Ξi . Thus, we have a natural notion of an identity element in Ξi under this ci ci+1 ci ci+1 identification. More explicitly, in the case where i is even, we can take the identity element to be the following matrix   0 a1 d1    −a 0 0   1     0 a2 d2       −a2 0 0    (7.46)  .. .   . .     0 a d   l l     −a 0 0   l  −d1 0 −d2 0 · · · −dl 0 0 where dj is given by equation (7.43). We can do a similar construction in the odd case. 128

In the case where i is even, we can act by the element

  (1/d1)z11 −(1/d1)z12    (1/d1)z12 (1/d1)z11       (1/d2)z21 −(1/d2)z22       (1/d2)z22 (1/d2)z21   .   ..       (1/d )z −(1/d )z   l l1 l l2  (1/dl)zl2 (1/dl)zl1 (7.47) in Gi to obtain the arbitrary element of the solution variety   0 a1 z11    −a 0 z   1 12     0 a2 z21       −a2 0 z22     .. .   . .     0 a z   l l1     −a 0 z   l l2  −z11 −z12 −z21 −z22 · · · −zl1 −zl2 0 The strategy is as follows. We concentrate on the case where i = 2l is even. The case of i odd is similar. Given an element z ∈ Ξi we write z = Ad(k(z))·X ci, ci+1 for a unique k(z) ∈ G where X ∈ Ξi is the matrix given in (7.46). We then i ci, ci+1 fix a choice of matrix h ∈ SO(i + 1) such that Ad(h) · X is in the fundamental domain Di+1 in hi+1 ⊂ so(i + 1). Notice that such a choice is independent of the coordinates z on Ξi . We then observe that the matrix g(z) = h k(z)−1 is such ij ci, ci+1 −1 that Ad(g(z)) · z is in the fundamental domain Di+1 ⊂ hi+1. The matrix k(z) is the inverse of the matrix in equation (7.47), which is clearly a regular function of z ∈ Ξi . Thus the function z → g(z) is regular on Ξi . ci, ci+1 ci, ci+1 c d/2+rk n Now, we are ready to define the mapping Γn for c ∈ C satisfying the eigenvalues disjointness condition. Again, we write c = (c1, ··· , ci, ··· , cn) with c ∈ rk i. We identify the solution variety at level i,Ξi with SO(2)rk i via i C ci, ci+1 129

Theorems (7.4.1) and (7.4.2). For z ∈ Ξi , we let g (z) ∈ SO(i + 1) be as ci, ci+1 i,i+1 defined in the last paragraph. Then we can define the following regular map

c rk i rk (n−1) Γn : SO(2) × · · · × SO(2) · · · × SO(2) → so(n)c

given by: (7.48)

c −1 −1 −1 Γn(z2, ··· , zn−1) = Ad(g2,3(z2) g2,3(z2) ··· gn−2,n−1(zn−2))zn−1 where z ∈ Ξi ' SO(2)rk i for 2 ≤ i ≤ n − 1. Recall also that in this definition i ci, ci+1 we are thinking of SO(j) ,→ SO(n) via the embedding

" # g 0 g ,→ 0 Idn−j A few observations are in order about the map defined in (7.48). The c existence of the map Γn, which is based on the existence of the solution varieties Ξi for 2 ≤ i ≤ n, gives that the fibre so(n) is non-empty for c ∈ d/2+rk n ci, ci+1 c C c satisfying the eigenvalue disjointness condition. It is easy to see that Γn is surjective 2 onto the fibre so(n)c. For given an X ∈ so(n)c, we have that X3 ∈ Ξc2, c3 . This defines an element z2 ∈ SO(2). Then we conjugate by g2,3(z2) ∈ SO(3) to get a 3 new element Y . The 4 × 4 cutoff of Y is then in the solution variety Ξc3, c4 . This defines for us a z4. We then conjugate Y by g3,4(z4) ∈ SO(4). Proceeding in this c fashion we obtain that X = Γn(z2, ··· , zn−1). This argument is totally analogous to the one we used to characterize the map Γn defined for M(n) in Chapter 4 (see Remark 4.2.1 in that chapter). We can also use an analogous argument to the one c used in the case of M(n) to see that the map Γn is injective using the fact that SO(i)rk i is always acting freely on the solution variety Ξi . ci,ci+1 c In this way, we have also defined an inverse to the map Γn. We want to also d/2 observe that this map is regular as a map from so(n)c → SO(2) . The argument is analogous to the one we used in Chapter 4 to see that the inverse to the map

a1,··· ,an−1 Γn is continuous. We proceed by induction. The point is that on the fibre 2 the eigenvalues of each of the cutoffs are fixed. Given X ∈ so(n)c,X3 ∈ Ξc2 c3 . 130

2 So that z2 = (x13, x23) ∈ Ξc2, c3 ' SO(2). From our above discussion, we know that the element g2,3(z2) ∈ SO(3) which conjugates X3 into the standard Cartan in so(3) and g2,3 depends regularly on z2 and thus on X. Thus, the map from so(n)c → so(n) given by

X → Ad(g2,3(z2)) · X is regular. This starts off the induction. We now follow the following steps. (1) We assume inductively that the map:

Φ0 : so(n)c → so(n)

given by

X → Ad(gj,j−1(zj−1)) ··· Ad(g2,3(z2)) · X is regular and z ∈ Ξl for 2 ≤ l ≤ j − 1 have already been computed as regular l cl, cl+1 functions of X. (2) It then follows that the map so(n) → Ξj given by c cj , cj+1

Φ1(X) = [Ad(gj,j−1(zj−1)) ··· Ad(g2,3(z2)) · X]j+1 is regular. Thus, we have obtained zj+1 as a regular function of X.

(3) Now, we have a regular map Ξj → SO(j + 1, ) given by taking z → cj , cj+1 C j gj,j+1(zj) (where gj,j+1(zj) is the element of SO(j + 1) that conjugates zj into the standard Cartan in so(j + 1)). (4) Composing the map from (2) with the map from (3), we have a regular map:

Φ2 : so(n)c → SO(j + 1)

given by

X → gj,j+1(zj)

(5) Now, we note that we have a natural regular map on the product defined as 131 follows: Φ3 : SO(j + 1) × so(n) → so(n)

given by

(g, Z) → Ad(g) · Z This is nothing more than the adjoint action of the subgroup SO(j + 1) ,→ SO(n) on so(n). (Here we have used the embedding of SO(j + 1) mentioned above).

(6) Using the universal property of the product in the category of varieties we get a regular map

Φ4 : so(n)c → SO(j + 1) × so(n)

given by

X → (Φ2(X), Φ0(X))

(7) Lastly, we note that the compositions of the regular maps Φ3 and Φ4 is the map

Ψ(X) = Φ3 ◦ Φ4 = Ad(gj,j+1(zj)) ··· Ad(g2,3(z2)) · X is regular. This completes the inductive step. Thus by, induction we can find all of the z ∈ Ξj as regular functions of X and hence we can define a regular j cj , cj+1 c inverse to Γn. We have now proven the following theorem.

Theorem 7.5.1. Let c ∈ Cd/2+rk n satisfy the orthogonal eigenvalue disjointness condition. Then the fibre so(n)c is non- empty. Moreover, as an affine variety we have that c d/2 Γn SO(2) ' so(n)c

c (Where Γn is the map defined in equation (7.48).) Thus, so(n)c is a smooth, affine variety of dimension d/2. (Where d is the dimension of a generic adjoint orbit of SO(n) in so(n)). 132

Remark 7.5.1. Note that as an algebraic group, SO(2)d/2 ' (C×)d/2. Thus, Theo- rem 7.5.1 is the orthogonal analogue of Theorem 3.23 in [KW, pg 45].

Remark 7.5.2. In the case of orthogonal Lie algebras, it is not automatic that the d/2+rk n fibre soc(n) is non-empty for c ∈ C satisfying the eigenvalue disjointness condition, unlike in the case of M(n). In the case of M(n), we know that all fibres (n+1) Mc(n) are non-empty for c ∈ C 2 because the map

n(n+1) Φn : M(n) → C 2

given by

Φn(X) = (p1,1(X1), ··· , pi,j(Xi), ··· , pn,n(Xn))

(Where pi,j is the elementary symmetric polynomial of degree i − j + 1 in the eigenvalues of Xi) is surjective by Theorem 2.3 in [KW, pg 16].

7.6 The action of B on generic fibres so(n)c

c We want to use the isomorphism Γn to define an algebraic action of the d/2 d/2+rk n group SO(2) on the fibre so(n)c for c ∈ C satisfying the eigenvalue disjointness condition. We define the action as follows. Let X ∈ so(n)c. If c −1 (Γn) (X) = (z2, ··· , zn−1) then

c 0 0 g · X = Γn(z2z2, ··· , zn−1zn−1) (7.49)

d/2 rk i for g = (z2, ··· , zn−1) ∈ SO(2) with zi ∈ SO(2) . (Here are identifying the action of G = SO(2)rk i on Ξi ' SO(2)rk i via conjugation with action of i ci, ci+1 SO(2)rk i on itself by left translation, as in the previous sections).

d/2 Remark 7.6.1. We note that the above action of SO(2) on so(n)c is a simply, transitive group action on so(n)c. We also note that this is an algebraic action d/2 of SO(2) on so(n)c. We can see this as follows. We want to see that the map d/2 d/2 SO(2) × so(n)c → so(n)c defining the action (z,X) → z · X with z ∈ SO(2) is regular. We can realize this map as the composition of the following regular 133

rk i maps: (We write for z = (z2, ··· , zn−1) with zi ∈ SO(2) .)

d/2 d/2 d/2 Φ1 : SO(2) × so(n)c → SO(2) × SO(2)

given by

c −1 ((z2, ··· , zn−1),X) → ((z2, ··· , zn−1), (Γn) (X))

d/2 d/2 d/2 Φ2 : SO(2) × SO(2) → SO(2)

given by

0 0 0 0 ((z2, ··· , zn−1), (z2, ··· , zn−1)) → (z2z2, ··· , zn−1zn−1)

c Φ3 = Γn

then the action map is given by the composition

Φ3 ◦ Φ2 ◦ Φ1

Now, we can easliy describe this action by noticing that

0 0 −1 g(zjzj) = g(zj)(zj) (7.50)

−1 This follows from our definition of g(zj) as g(zj) = h k(zj) where h ∈ SO(j + 1) conjugates the “identity element” given in equation (7.46) into the standard Cartan and k(zj) is given in equation (7.47). Equation (7.50) follows from the identity 0 0 k(zjzj) = zj k(zj). This identity follows from the fact that the map in (7.34) was 134 an SO(2) equivariant isomorphism. Using (7.50), we can write the action as

0 0 (z2, ··· , zn−1) · X

is given by

0 −1 0 −1 −1 0 Ad(z2g2,3(z2)z3g3,4(z3) ··· gn−2,n−1(zn−2)zn−1 ··· gn−2,n−1(zn−2) ··· g2,3(z2)) · X (7.51) c For X ∈ so(n)c,X = Γn(z2, ··· , zn−1). Note the similarities with the Type A case.

The B action on the generic fibres so(n)c is now given by the following theorem.

Theorem 7.6.1. For c ∈ Cd/2+rk n satisfying the eigenvalue disjointness condition, we have soc(n) is exactly one B orbit of maximal dimension d/2. Thus, elements of Ωn are strongly regular. On the fibre soc(n) the action of B has the same orbits as a simply transitive algebraic action of the torus SO(2)d/2 ' (C×)d/2.

Proof: We claim that the action of the torus SO(2)d/2 defined in (7.49) on the fibre so(n)c has the same orbit structure as the B action on the fibre. To see this, we differentiate the action defined in (7.49) and compute the tangent space to so(n)c. We note that since the action of SO(2) is algebraic it is in particular holomorphic on so(n)c, which, as a smooth affine subvariety of so(n) naturally has the structure of an embedded complex submanifold of so(n). To differentiate the action in (7.49), it is easier to use the description of the action given in equation (7.51). We note that differentiating the orbit map given in equation (7.51) at the identity gives us a description of TX (so(n)c), for X ∈ soc(n), since the orbit map of a holomorphic group action is a submersion. In this case, it is in fact a diffeomorhisphm since the action is free (see [Lee, pg 230]). To differentiate this action of SO(2), we want to have a simple coordinate system on SO(2) in a neighbourhood of the identity. The coordinate system we 135 use is the following one. Let g ∈ SO(2), then g has the form " # u −v v u for u, v ∈ C with u2 + v2 = 1. We claim that for g in a neighbourhood of the identity that g can be written as " # cos(z) − sin(z) sin(z) cos(z) for a unique z ∈ C. We can see this as follows. Consider the map Ψ : C → SO(2) given by " # cos(z) − sin(z) Ψ(z) = sin(z) cos(z) Now, Ψ(0) = Id and if we differentiate this map at z = 0 we see that we get " # 0 −1 Ψ0(0) = 1 0

So that the map is non-singular. Now, applying the inverse function theorem, we see that Ψ is a diffeomorphism from a neighbourhood of 0 ∈ C onto a neighbour- hood of the identity in SO(2). Since we are only interested in differentiating the orbit map in (7.51) at the identity, we can use this coordinate representation. Thus, we can use as a basis ∂ for Lie(SO(2)) ∂z |z=0 . For the convenience of the reader, we restate the expression of the orbit map in (7.51) below.

0 0 (z2, ··· , zn−1) · X

is given by

0 −1 0 −1 −1 0 Ad(z2g2,3(z2)z3g3,4(z3) ··· gn−2,n−1(zn−2)zn−1 ··· gn−2,n−1(zn−2) ··· g2,3(z2)) · X 136

0 rk i Using the above coordinates, we represent zi ∈ SO(2) as follows. Let l = rk i (recall rk i represents the rank of so(i)). For i odd we have   cos(zi1) − sin(zi1)    sin(z ) cos(z )   i1 i1     cos(zi2) − sin(zi2)       sin(zi2) cos(zi2)     ..   .     cos(z ) − sin(z )   il il     sin(z ) cos(z )   il il  1

And for i even we have:   cos(zi1) − sin(zi1)    sin(zi1) cos(zi1)       cos(zi2) − sin(zi2)       sin(zi2) cos(zi2)   .   ..       cos(z ) − sin(z )   il il  sin(zil) cos(zil)

(Recall that we are thinking of SO(i, C) ,→ SO(n, C) as embedded in the top left hand corner). To get the differential of the orbit map in coordinates we compute:

∂ 0 −1 −1 0 |zij =0Ad(z2g2,3(z2) ··· gn−2,n−1(zn−2)zn−1 ··· gn−2,n−1(zn−2) ··· g2,3(z2)) · X ∂zij For 2 ≤ i ≤ n − 1, 1 ≤ j ≤ l. For the case of i = 2l even, we get

−1 −1 ad(g2,3(z2) ··· gi−1,i(zi−1)Aijgi−1,i(zi−1) ··· g2,3(z2)) · X (7.52) 137

where Aij ∈ so(i) ,→ so(n) is the matrix given by   0 0    0 0     .   ..       0 −1  Aij =      1 0   .   ..       0 0    0 0

Similarly, for the case where i = 2l + 1 is odd. Now, in the case where i = 2l is even the element hi = gi−1,i(zi−1) ··· g3,4(z3)g2,3(z2) ∈ SO(i) conjugates the i × i cutoff of X, Xi into the standard Cartan in so(i). Namely, it conjugates it into a matrix of the form   0 b1    b1 0       0 b2       −b2 0     ...       0 b   l  −bl 0

Recall that we are assuming that each cutoff Xi is regular, so that the centralizer of such a matrix in so(2l) is given by:   so(2)    so(2)     .   ..    so(2) which has basis given by matrices of the form Aij for 1 ≤ j ≤ l. Thus, we have that equation (7.52) gives us tangent vectors of the form

−1 [hi Aij hi,X] ∂X (7.53) 138

−1 Now, the Aij, 1 ≤ j ≤ l form a basis for the centralizer of the matrix hi X hi , so −1 that the elements hi Aijhi, 1 ≤ j ≤ l form a basis of the centralizer in so(i) of Xi. Performing this argument for all 2 ≤ i ≤ n − 1, we see that the tangent space at

X ∈ so(n)c is given by

[so(i)Xi ,X] TX (so(n)c) = span{∂X }

Xi (where so(i) denotes the centralizer of Xi in so(i)).

Now, for X ∈ Ωn ⊂ so(n) this is none other than the subspace VX . Recall that we had in equation (7.2)

[(dfi,j )(Xi),X] VX = span{(∂ )X |2 ≤ i ≤ n − 1, 1 ≤ j ≤ rk i} where fi,j are fundamental adjoint invariants for so(i).

Now if Xi is regular, then we have that (dfi,j)(Xi), 2 ≤ j ≤ rk i form a basis for the centralizer of Xi (see [K2, pg 382]). Thus, we have

[so(i)Xi ,X] TX (so(n)c) = span{∂X } = VX (7.54)

d/2 Now, we also clearly have that dim TX (so(n)c) = d/2 = dim SO(2) . Thus, for sreg sreg X ∈ so(n)c, dim VX = d/2. So that so(n)c ⊂ so(n) and so(n) is non-empty.

We know from Proposition 7.3.1 that for any X ∈ so(n)c, we have B · X ⊂ so(n)c.

Thus, we can write so(n)c as a disjoint union of B orbits, a so(n)c = B · X(i) i∈I with X(i) ∈ so(n)c. Now, the dimension of each B orbit is exactly d/2, since

dim B · X(i) = dim VX(i) = dim TX(i)(so(n)c) = d/2

Hence, each B orbit B · X(i) is an open submanifold of so(n)c. Now, we argue as we did in the case of M(n). We note that B · X(i), i ∈ I is given by so(n)c \ ` ` ( j6=i, j∈I B · X(j)), since the union is disjoint. But j6=i, j∈I B · X(j) is open, so that B · X(i) is closed. Thus B · X(i) is both open and closed in the connected manifold so(n)c. Hence, since B · X(i) 6= ∅, we must have

so(n)c = B · X(i) 139 is exactly one B orbit. This completes the proof of the theorem. Q.E.D.

Remark 7.6.2. Theorem 7.6.1 generalizes Theorems 3.23 and 3.28 in [KW, pgs 45, 48] for the analogous set of regular semi-simple matrices in M(n) to the orthogonal Lie algebras so(n).

We have a corollary to Theorem 7.6.1.

Corollary 7.6.1. The functions {fi,j| 2 ≤ i ≤ n−1, 1 ≤ j ≤ rk i} are algebraically independent.

Proof: sreg sreg We now know that so (n) is non-empty from Theoem 7.6.1, since Ωn ⊂ so(n) . Then, we can apply Theorem 2.7 in [KW, pg 19], which carries over to the orthog- onal case by Theorem 7.2.2, to see that there exist a non-empty Zariski open set of elements for which the differentials (dfi,j)x for 2 ≤ i ≤ n, 1 ≤ j ≤ rk i are linearly independent. The algebraic independence follows immediately. Q.E.D.

Remark 7.6.3. Using this corollary, we see that the Gelfand-Zeitlin Algebra GZ(g) of g = so(n) defined in equation (1.22) and the algebra J(g) = P (so(2))SO(2) ⊗ · · · ⊗ P (so(n))SO(n) are both commutative algebras which are free on d/2 + rk n generators and hence they are abstractly isomorphic as we promised in section 1.6.

As another corollary to Theorem 7.6.1, we see that we can polarize open submanifolds of adjoint orbits of elements in Ωn ⊂ so(n).

Corollary 7.6.2. Let X in Ωn ⊂ so(n) and let OX be its adjoint orbit. Then we sreg sreg have that OX = OX ∩ so(n) is non-empty Zariski open subset of OX . The B sreg orbits of dimension d/2 in OX form the leaves of a polarization of OX .

Proof: sreg From Theorem 7.6.1, we know that OX is a non-empty Zariski open subset of sreg OX , since X ∈ so(n) . One argues in exactly the same manner as Theorem 3.36 140 in [KW, pg 86] using Proposition 1.5.2 to see that the B orbits of dimension d/2 sreg in OX are Langrangian submanifolds of OX . Bibliography

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