K-orbits on the flag variety and

the Gelfand-Zeitlin integrable

system

Mark Colarusso, University of Wisconsin

Milwaukee

May, 20 2014

Contents

The talk is divided into three sections:

I: Motivation

II: Lie-Poisson Structure and the Gelfand-

Zeitlin system

III: K-orbits on the flag variety and the

Gelfand-Zeitlin system

IV: Applications to Geometric Representa-

tion Theory

1

I: Motivation

Ritz Values

Let g = gl(n,C) = n× n complex matrices.

Given x ∈ g, xi = upper i × i left hand cornerof x.

Call xi the i× i cutoff of x.

Let σ(xi) = {λi1, . . . , λii} be the eigenvalues ofxi.

Define: R := {σ(x1), . . . , σ(xn)}.

R = Ritz values of x.

2

Ritz Values are important in numerical linear

algebra.

(See Parlett and Strang: “Matrices with Pre-

scribed Ritz Values”, Linear alg and its app.

(2008).)

Rayleigh-Ritz:

If x ∈ g is Hermitian, then σ(xm) for m < napproximates σ(x).

3

Let R be a specified set of Ritz values.

Questions:

1. Does there exist a Hermitian matrix x with

Ritz Values R?

2. Does there exist a general n × n complexmatrix x with Ritz Values R?

4

Answers: (1) Answer due to Cauchy:

(Real) eigenvalues for xj must interlace with

those for xj+1.

If σ(xj) = {µ1, . . . , µj} and

σ(xj+1) = {λ1, . . . , λj+1}, then

λ1 ≤ µ1 ≤ λ2 ≤ . . . ≤ µj ≤ λj+1.

Can construct Herimitian x with interlacng Ritz

values R.

5

(2) Answered by Kostant and Wallach using

Gelfand-Zeitlin theory.

For any complex Ritz values R construct an xwhose Ritz values are given by R.

Let gR := {x ∈ g : such that x has Ritz values R}.

Questions of Parlett and Strang concerning

the family gR can be answered using GZ theoryand the GZ system.

6

The family gR and the Kostant-Wallach map

How to describe family gR geometrically?

x, y ∈ gR if and only if σ(xi) = σ(yi) for all i,1 ≤ i ≤ n.

Let pi,j(x) be coefficient of tj−1 in the charac-

teristic polynomial of xi.

Define the Kostant-Wallach map

Φ : gl(n,C)→ Cn(n+1)

2 .

Φ(x) = (p1,1(x), . . . , pi,j(x), . . . , pn,n(x)).

Then

gR = Φ−1(c).

for some fixed c ∈ Cn(n+1)

2 depending on R.7

Hessenberg Matrices:

Hessenberg Matrices Hess: x ∈ g is said to bean (upper) Hessenberg matrix if :

x =

a11 a12 · · · a1n−1 a1n1 a22 · · · a2n−1 a2n0 1 · · · a3n−1 a3n... ... . . . ... ...0 0 · · · 1 ann

Φ : Hess→ Cn(n+1)

2 is an isomorphism.

This answers a question of Parlett and Strang.

Hessenberg matrices provide a canonical form

for the family gR.

8

It turns out that the functions

{pi,j : 1 ≤ i ≤ n,1 ≤ j ≤ i}

can be used to define an integrable system.

Can study family gR using Lie-Poisson theory.

9

II: The Lie-Poisson structure and the GZ

integrable system:

Integrable systems were developed in 19th cen-

tury as a method for studying classical me-

chanical systems.

In modern times, integrable systems play a

large role in geometric representation theory.

i.e. Can use integrable systems to construct

geometrically representations of Lie groups/algebras.

This is the main focus of our work.

11

Integrable Systems:

Let (M,ω) be a symplectic manifold with dimM =

2n.

The functions on M form a Poisson algebra

with bracket {·, ·}.

An integrable system on M is a collection of

n independent functions F1, . . . , Fn such that

{Fi, Fj} = 0 for all i, j.

12

Moment map and Lagrangian submanifolds:

F1, . . . , Fn an integrable system on symplectic

(M,ω), dimM = 2n.

Moment Map: F : M → Cn,

F(m) = (F1(m), . . . , Fn(m)).

Definition: A submanifold N ⊂M is Lagrangianif TpN = TpN⊥ for all p ∈ N .

Fact: A regular level set of F is a Lagrangiansubmanifold of M

Lagrangian submanifolds play crucial role in

geometric construction of representations via

quantization.

13

Lie-Poisson Structure:

Consider g = gl(n,C) ∼= gl(n,C)∗ via the traceform < x, y >= tr(xy).

Then g is a Poisson manifold equipped with

the Lie-Poisson structure.

i.e. Holomorphic functions on g are equipped

with a Poisson bracket {·, ·}.

Poisson bracket of linear functions:

fx(z) = tr(xz), fy(z) = tr(yz), x, y ∈ g.

{fx, fy}(z) = tr(z[x, y]) for z ∈ g.

14

Adjoint Orbits

Poisson manifolds are foliated by symplectic

submanifolds called symplectic leaves.

Fact: Adjoint orbits Ad(G) · x are symplecticleaves of the Lie-Poisson structure on g .

Where Ad(G) ·x has canonical Kostant-Kirillovsymplectic structure.

We call an adjoint orbit Ad(G) · x regular if

dim Ad(G) · x = n2 − n is maximal.

15

Define GZ functions:

JGZ = {pi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ i}.

Facts: [Kostant-Wallach]

1. The functions pi,j are algebraically indepen-

dent.

2. Gelfand-Zeitlin theory⇒ {pi,j, pk,l} = 0 withrespect to Lie-Poisson structure.

3. Ad(G) · x regular adjoint orbit

|JGZ|Ad(G)·x| =n(n−1)

2 =12 dim(Ad(G) · x)

Conclusion: JGZ forms integrable system on

regular adjoint orbits.

16

The moment map for the GZ system is the

Kostant-Wallach map:

Φ : gl(n,C)→ Cn(n+1)

2 .

Φ(x) = (p1,1(x), . . . , pi,j(x), . . . , pn,n(x)).

Regular level sets of Φ are Lagrangian subva-

rieties of regular Ad(G) · x.

Philosophy of quantization: Lagrangian sub-

manifolds of Ad(G) · x↔ Irreps of g.

17

Compact Gelfand-Zeitlin system

U(n,C) = n× n unitary matrices.

Guillemin and Sternberg in 1980’s constructed

analogous GZ system on

u(n,C)∗ = n× n Hermitian matrices.

18

They showed:

Geometric quantization of integral coadjoint

orbit Ad∗(U(n,C)) · λ using GZ Lagrangian fo-liation

⇒ classical GZ basis of highest weight U(n,C)-module V λ.

(Their proof uses eigenvalue interlacing rule for

Hermitian matrices.)

19

Lagrangian Fibres of KW map

Definition: Call x ∈ g strongly regular if thedifferentials

dpi,j(x) are linearly independent for all i, j.

gsreg=set of strongly regular elements.

Fact: For any c ∈ Cn(n+1)

2 ,

Φ−1(c)sreg := gsreg ∩Φ−1(c) 6= ∅.

20

NOTE Φ−1(c)sreg ⊂ Ad(G) · x is Lagrangian.

Today: Use orbits of a symmetric sub-

group of GL(n,C) on flag variety to studygsreg and Φ−1(c)sreg.

21

Quantum Version of GZ system for gl(n,C)

Category of (g,Γ)-modules (also called GZ mod-

ules)

Γ = GZ subalgebra of enveloping algebra of g.

Γ ∼= Z(gl(1,C))⊗C · · · ⊗C Z(gl(n,C)),

where Z(gl(i,C)) is centre of U(gl(i,C)) ⊂ U(gl(n,C)).

A (g,Γ)-module V is an infinite dimensional rep

of g on which Γ acts locally finitely.

22

(g,Γ)-modules have been studied by Drozd,

Futorny, and Ovsienko.

Goal: Develop program to produce (g,Γ) mod-

ules using the geometry of the GZ integrable

system.

(g,Γ)-modules closely related to generalized Harish-

Chandra modules studied by Zuckerman, Penkov,

Serganova.

23

Harish-Chandra Modules

Example: g = gl(n,C),

K = GL(n − 1,C) × GL(1,C) block diagonalmatrices.

Classical HC-module = (g,K)-module.

(i.e. Infinite dimensional g reps which are sum

of finite dimensional K-reps.)

(g,K)-modules are used to study rep theory of

real Lie groups (in the this case U(n− 1,1).)

24

Geometric Construction of HC modules:

Uses flag variety B of g.

B = variety of Borel subalgebras of g.

B ∼= GL(n,C)/B where B ⊂ G is any Borelsubgroup.

Beilinson-Bernstein give geometric construc-

tion of (g,K)-modules using algebraic geome-

try and theory of differential operators on flag

variety B of g.

Key Point: K-acts on B by conjugation withfinitely many orbits.

25

Problem: For (g,Γ)-modules, Γ does not in-

tegrate to a group which acts on flag variety

with finitely many orbits.

So a Beilinson-Bernstein type construction is

not clear.

Solution: Describe components of Φ−1(c)sregusing theory of K-orbits on B.

Refs:

[CE1] Colarusso, Mark and Evens, Sam: “K-

orbits on the flag variety and strongly regular

nilpotent matrices” in Selecta Mathematica,

2012.

[CE2] Colarusso, Mark and Evens, Sam: “K-

orbits on the flag variety and eigenvalue coin-

cidences”, arxiv 1303.6661, 17 pages.

26

Strategy:

Study “Partial KW” maps for each i = 1, . . . , n.

Φi : gl(i,C)→ Ci−1 × C1,

Φi(xi) = (pi−1,1(xi), . . . , pi,i(xi))

Note: Fibres of Φi are stable under conjuga-

tion by Ki = GL(i−1,C)×GL(1,C) ⊂ GL(i,C).

(1) Construct components of the fibres of Φiusing Ki-orbits on the flag variety of gl(i,C).

(2) “Glue” together components of partial KW

maps Φi, i = 2, . . . , n to construct Lagrangian

components of fibres of Φ.

27

We now discuss (1):

Let i = n, g = gl(n,C),

K = GL(n− 1,C)×GL(1,C).

Partial KW map:

Φn(x) = (pn−1,1(x), . . . , pn,n(x)).

Note x, y in same fibre of Φn ⇔ σ(xi) = σ(yi)for i = n− 1, n.

28

Eigenvalue coincidence varieties

So we study geometry of eigenvalue coinci-

dence varieties:

g(l) := {x ∈ gl(n,C) : |σ(xn−1) ∩ σ(x)| = l},

for l = 0, . . . , n− 1.

(The intersection in spectra is computed count-

ing repetition.)

Any fibre Φ−1n (c) ⊂ g(l) where l depends uponc.

29

Notation:

Let Q = K · bQ be K-orbit of bQ on B.

Let YQ := KbQK−1 = set of all matrices in

gl(n,C) which are K-conjugate to elements ofbQ.

30

Theorem 1. [CE2]

The irreducible component decomposition of

g(l) is:

g(l) =⋃

codim(Q)=lYQ ∩ g(l).

To prove theorem use Grothendieck’s simulta-

neous resolution and algebro-geometric prop-

erties of partial KW map Φn.

31

Interesting Corollary:

Consider

g(n− 1) = {x ∈ g : |σ(x) ∩ σ(xn−1)| = n− 1}

(i.e. maximal number of coincidences.)

Corollary 1. The irreducible component de-

composition of g(n− 1) is:

g(n− 1) =⋃

Q closedYQ

Corollary relates most complicated eigenvalue

coincidences to simplest K-orbits on B.

32

Algebro-Geometric Properties of Φn.

x ∈ gl(n,C),

Φn(x) = (pn−1,1(x), . . . , pn,n(x)).

Theorem 2. [CE2] The morphism Φn is flat

and is a GIT quotient for the action of K =

GL(n−1,C)×GL(1,C) on gl(n,C) by conjuga-tion.

33

Nilfibre of Φn:

We prove Theorem 2 using symplectic geom-

etry.

Φn is flat ⇔ fibres of Φn are equidimensional.

By homogeneity of Φn enough to show nilfibre

Φ−1n (0) is equidimensional.

Φ−1n (0) = {x ∈ g : x, xn−1 are nilpotent }.

To study: Φ−1n (0), we introduce a variant ofthe Steinberg variety.

34

B := flag variety of gl(n,C),

Bn−1 := flag variety of gl(n− 1,C),

Generalized Steinberg Variety:

Z = {(x, b, b′) : b ∈ B , b′ ∈ Bn−1 and

x ∈ [b, b], xn−1 ∈ [b′, b′]}

We have a diagram:

Z↙ µ ↘ π

Φ−1n (0) B × Bn−1

where µ and π are projections to the appropri-

ate factors.

35

Fact: K = GL(n − 1,C) × GL(1,C) acts onB ×Bn−1 diagonally with finitely many oribts.

We prove:

Z is a union of conormal bundles to K-diagonal

orbits ⇒:

Z ⊂ T ∗(B)×T ∗(Bn−1) is Lagrangian and there-fore Z equidimensional.

Since µ(Z) = Φ−1n (0), it follows from elemen-tary considerations that:

Φ−1n (0) is equidimensional.

36

Applications to representation theory:

Consider generalized Harish-Chandra modules

for the pair (g,Γn).

Γn = partial GZ algebra, i.e.

Γn ∼= Z(gl(n− 1,C))⊗C Z(gl(n,C))

Z(gl(i,C)) = centre of U(gl(i,C)), i = n− 1, n.

37

Given (g,Γn)-module M its associated variety:

Ass(M) ⊂ Φ−1n (0).

Using Theorems 1 and 2, can show

Irreducible components of Φ−1n (0)↔ closed K-orbits on B × Bn−1.

38

Idea: Study (g,Γn)-modules by studying K-

equivariant D-modules on B × Bn−1 and thegeometry of Φ−1n (0).

Key Point: K acts diagonally on B×Bn−1 withfinitely many orbits.

Therefore a Beilinson-Bernstein construction

type of (g,Γn)-modules will be possible.

39

Iteration:

Question: What about the (g,Γ)-modules

studied by Futorny et al?

Recall “Full” Kostant-Wallach map:

Φ(x) = (p1,1(x), . . . , pi,j(x), . . . , pn,n(xn)).

Describe geometry of strongly regular (i.e. La-

grangian) fibres Φ−1(c)sreg using geometry ofg(l):

So far only for Φ−1(0)sreg [CE1].

40

Nilfibre:

Φ−1(0) = {x ∈ gl(n,C) : xi is nilpotent for all i}.

i = 1, . . . , n.

Note: If M is a (g,Γ)-module, then Ass(M) ⊂Φ−1(0).

41

Components of Nilfibre:

Note:

Φ−1(0) ⊂ g(n− 1) =⋃

Q closedYQ.

It turns out:

Irreducible components of strongly regular fi-

bre Φ−1(0)sreg given by 2n−1 Borel subalgebrasof gl(n,C) constructed using certain closed K-orbits.

42

For i = 1, . . . , n, Ki = GL(i− 1,C)×GL(1,C).

Let Q±,i be Ki-orbit of i× i upper (resp lower)triangular matrices in the flag variety of gl(i,C).

Note: Q±,i are Ki-orbits related via Beilinson-Bernstein to holomorhic/anitholomorphic dis-

crete series for the real Lie group U(i− 1,1).

Let bQ1,...,Qn := {b ∈ B : bi ∈ Q±,i}.

43

Theorem 3. (1) bQ1,...,Qn is a single Borel sub-

algebra of gl(n,C).

(2) Let nregQ1,...,Qnbe the regular nilpotent ele-

ments in bQ1,...,Qn. Then:

Φ−1(0)sreg =∐

nregQ1,...,Qn

is the irreducible com-

ponent decomposition of Φ−1(0)sreg.

Remark: This result precedes the results of

[CE2]. Uses completely different techniques.

44

Case g = gl(3)

There are 4 such Borel subalgebras.

b−,− =

h1 0 0a1 h2 0a2 a3 h3

b+,+ = h1 a1 a20 h2 a3

0 0 h3

b+,− =

h1 a1 00 h2 0a2 a3 h3

b−,+ = h1 0 a1a2 h2 a3

0 0 h3

45

In Future:

(1) Geometrically construct (g,Γn)-modules us-

ing geometry developed in [CE2] and theory of

differential operators on flag varieties.

(2) Use iteration method of nilpotent case [CE1]

and methods of [CE2] to describe all fibres

Φ−1(c)sreg geometrically using K-orbits on B.

46

(3) Use (1) and (2) to describe (g,Γ)-modules

geometrically.

Theorem 3 suggests that certain (g,Γ)-modules

are related to holomorphic/antiholomorphic dis-

crete series for the real Lie groups U(i− 1,1),i = 1, . . . , n.

47

Other Projects and Directions:

1. Nonlinear GZ systems for Poisson Lie groups

with applications of the structure and rep

theory of quantum groups and Yangians.

(with S. Evens).

2. Development of Lie-Poisson theory for in-

finite dimensional Lie algebras and applica-

tions to rep theory (with M.Lau).

49

Conclusion:

Ritz Values R −→ Kostant-Wallach Map Φ −→

Gelfand-Zeitlin integrable system −→

Further study of Φ−1(c) = gR −→

K-orbits on B −→ GZ-modules and generalizedHarish-Chandra modules −→

K-oribts and Eigenvalue coincidences (i.e. gR)

So we have come full circle.

50