generalized gelfand-zeitlin systems - cornell universitypi.math.cornell.edu/~bsh68/geneve.pdf ·...

Generalized Gelfand-Zeitlin systems

Benjamin Hoffman

Department of MathematicsCornell University

Groupes de Lie et espaces des modules 28 April 2020

Joint with Jeremy Lane (McMaster/Fields)

Benjamin Hoffman (Cornell University) Generalized Gelfand-Zeitlin systemsGroupes de Lie et espaces des modules 28 April 2020 Joint with Jeremy Lane (McMaster/Fields) 1

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Outline

Gelfand-Zeitlin systemsCanonical bases and their parametrizationsGeneralized Gelfand-Zeitlin systemsProof of the main result


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Linear Poisson manifolds

For a Lie group K , let (k, [·, ·]) be the Lie algebra of K , and k∗ its lineardual.

Define a Poisson bracket on C∞(k∗):

{X ,Y }(ξ) = 〈ξ, [X ,Y ]〉

for X ,Y ∈ k ∼= (k∗)∗ ⊂ C∞(k∗) and ξ ∈ k∗. Then (k∗, {·, ·}) is a linearPoisson manifold.

Symplectic leaves of k∗ are the coadjoint orbits.

If (M, ω, µ) is a Hamiltonian K -manifold, then the moment mapµ : M → k∗ is a Poisson map.


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Gelfand-Zeitlin functions on u(n)∗

Consider the unitary group U(n).Identify u(n)∗ with H(n), the set of n × n Hermitian matrices.

For 1 ≤ i ≤ j ≤ n, let

µij : H(n)→ R

where µij(M) is the ith largest eigenvalue of the j × j submatrix M[1,j],[1,j].

The functions µin are Casimirs of the Poisson structure on H(n).


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Some properties of the Gelfand-Zeitlin system

Collectively:µ : H(n)→ Rn(n+1)/2.

It is smooth on an open dense subset.

The vector fields {µij , ·} and {µi′j ′ , ·} commute, for all i , i ′, j , j ′. The

flows of these vector fields descend to an effective torus action(S1)n(n−1)/2 on the smooth locus of µ.

? Completely integrable, on a dense subset. ?

The image of µ is the polyhedral cone in Rn(n+1)/2 cut out by theinequalities µij+1 ≥ µij ≥ µ

i+1j+1

This is the Gelfand-Zeitlin system


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Generalizations of the Gelfand-Zeitlin system

Guillemin-Sternberg (83): Original construction for u(n) and o(n).The construction uses a chain of subalgebras, eg:

o(1) ⊂ o(2) ⊂ · · · ⊂ o(n − 1) ⊂ o(n).

Harada (04): Completely integrable system on sp(2n)∗.

Harada-Kaveh (15): Completely integrable systems on integralcoadjoint orbits, for arbitrary compact k.

Goal: give a generalization of the GZ system on k∗ for all compact K .


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Outline



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Lie Theory Notation

Throughout what follows:

K is a compact Lie group, and G is its complex form.

B,B− ⊂ G is choice of opposite Borels. N ⊂ B,N− ⊂ B− aremaximal unipotent subgroups.

H = B ∩ B− is Cartan subgroup, and T = H ∩ K is maximal torus.Fraktur letters denote Lie algebras.

P = Hom(H,C×) ↪→ h∗ is the set of weights of G .P+ is the set of dominant weights of g, and t

∗+ is the positive Weyl

chamber.


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Canonical bases

Theorem (Lusztig): There is a canonical basis B of U(n−).

If λ ∈ P+ is a dominant integral weight of G , letV (λ) be the irreducible G module with high weight λ

vλ ∈ V (λ) be a fixed highest weight vectorThen Bλ = B · vλ is a weight basis of V (λ). The dual basis B∗λ is a weightbasis of V (λ)∗.


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Parametrizations of canonical bases

Let λ ∈ P+. A polyhedral parametrization of Bλ (or B∗λ) is a bijection

Bλ ↔4λ

where 4λ ⊂ Zm is the set of lattice points of a rational convex polytope inRm.

A polyhedral parametrization of B∗ :=∐λ∈P+ B

∗λ is

C =∐λ∈P+

{λ} ×4λ ⊂ P × Zm.

We require that C is the set of lattice points of a rational convexpolyhedral cone in Rm+r .


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Gelfand-Zeitlin cone

Let G = GLnC. Then P ∼= Zn and λ ∈ P+ can be written:

λ =(λ1, λ2, . . . , λn) ∈ Zn

λ1 ≥ λ2 ≥ · · · ≥ λn

The Gelfand-Zeitlin pattern 4GZλ associated with λ is the set of λij ∈ Z,1 ≤ i ≤ j ≤ n, so that

λij+1 ≥ λij ≥ λi+1j+1There exists a bijection Bλ ↔4GZλ .

Moreover,CGZ =

∐λ∈P+

{λ} ×4GZλ ⊂ Zn × Zn(n−1)/2

is the set of integral points of a convex polyhedral cone.


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String parametrizations

Let G be a reductive complex Lie group. Let i = (i1, i2, . . . , im) be areduced expression for the longest element w0 of the Weyl group W of G .

There is a polyhedral parametrization 4iλ of B∗λ which depends only on thechoice of i. Each 4iλ is a string polytope (Berenstein-Zelevinsky).

Moreover,Ci =

∐λ∈P+

{λ} ×4iλ ⊂ P × Zm

is the set of integral points of a convex polyhedral cone (Littelmann).


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String parametrizations, in detail

Fix λ ∈ P+, and consider b∗ ∈ B∗λ ⊂ V (λ)∗. Define

vi(b∗) = (v1, . . . , vN) ∈ 4iλ ⊂ ZN

by:

v1 = max{v | F vi1b∗ 6= 0}

v2 = max{v | F vi2Fv1i1b∗ 6= 0}

...

vN = max{v | F viNFvN−1iN−1

· · ·F v1i1 b∗ 6= 0}

Then vi is a bijection B∗λ ↔4iλ.


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Example

Let G = SL3C. Let i = (1, 2, 1). Generators of Ci:

F : N6 → Ci ⊂ P+ × N3

(a1, a2, a3, a12, a13, a23) 7→

∑i

ai ,∑ij

aij , a2 + a23, a3 + a13 + a23, a3

Then:

F (0, 1, 0, 0, 1, 0) = F (1, 0, 0, 0, 0, 1).

C[Ci] =C[z1, z2, z3, z12, z13, z23]

(z2z13 − z1z23)


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Outline



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Problem statement

The moment map image µ(u(n)∗) of the Gelfand-Zeitlin system isR≥0 · CGZ .

The integral points CGZ parametrize∐λ∈P+ B

∗λ, the dual canonical bases

of GLn(C) modules.


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Problem statement

Let k be the Lie algebra of a compact Lie group K , and let C ⊂ ZN be apolyhedral parametrization of of

∐λ∈P+ B

∗λ. Find a proper continuous map

µ : k∗ → RN which satisfies the following:1 For each open face σ of t∗+, the map µ is smooth on an open dense

subset of K · σ.2 On its smooth locus, µ : K · σ → RN generates a completely

integrable torus action.

3 µ(k∗) = R≥0 · C.

Theorem (H.-Lane)

For all compact Lie groups K and all string parametrizations Ci, such amap µ : k∗ → RN exists.


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Outline



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Proof outline

1 Pass to a stratified space ET ∗K which is symplectic realization of k∗.2 Identify ET ∗K with the affine variety G � N.3 Take a toric degeneration of G � N to a toric variety XCi .4 Use gradient-Hamiltonian flow to build a map G � N → XCi .5 Compose with a moment map XCi → RN , and divide by a small torus

action, to getk∗ → RN

satisfying the desired properties.


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Symplectic implosion

Step 1: Pass to a space ET ∗K which is a symplectic realization of k∗.

The symplectic implosion ET ∗K of T ∗K is a stratified space constructedby Guillemin-Jeffrey-Sjamaar.

As a set: ET ∗K =∐σ≤t∗+

(K/[Kσ,Kσ])× σ.

It is a topological space. Each piece (K/[Kσ,Kσ])× σ is a smoothsymplectic manifold.

There is a proper map ET ∗K → k∗. It is piecewise Poisson.


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Ambient affine space

Step 2: Identify ET ∗K with the affine variety G � N.

Fix a string parametrization Ci of∐λ∈P+ B

∗λ. It is a semigroup.

Let Π ⊂ P+ be a finite subset so that∐λ∈Π{λ} ×4iλ generates Ci as a

semigroup.

Let E = ⊕λ∈ΠV (λ). It is a G -module.

Example: G = SL3(C), Π = {ω1, ω2}, E = V (ω1)⊕ V (ω2).


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Base affine space

C[G ] ∼= ⊕λ∈P+V (λ)∗ ⊗ V (λ) as a G × G module.

C[G ]1×N ∼= ⊕λ∈P+V (λ)∗ ⊗ Cvλ ∼= ⊕λ∈P+V (λ)∗.Then B∗ =

∐λ∈P+ B

∗λ is a basis for C[G ]N .

Let G � N = SpecC[G ]N .

⊕λ∈ΠV (λ)∗ generates C[G ]N , so

G � N ↪→ ⊕λ∈ΠV (λ) = E .


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Example

Let G = SL3C, Π = {ω1, ω2}. Then

C[G ]N ∼=C[z1, z2, z3, z12, z13, z23](z2z13 − z1z23 − z3z12)

Since

C[V (ω1)] ∼= C[z1, z2, z3]C[V (ω2)] ∼= C[z12, z13, z23]

this embedsG � N ↪→ V (ω1)⊕ V (ω2).


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Example

C[G ]N ∼= C[z1,z2,z3,z12,z13,z23](z2z13−z1z23−z3z12)

The dual canonical basis B∗ of C[G ]N is the set of monomials in

z1, z2, z3, z12, z13, z23

which are not divisible by z2z13.


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Kähler structure

Give E a K -invariant Hermitian form 〈·, ·〉.Then E is Kähler, with Kähler form ωE = −=〈·, ·〉.

Theorem (Guillemin-Jeffrey-Sjamaar, Hilgert-Manon-Martens): ET ∗K isisomorphic to G � N, equipped with the restriction of ωE .


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High weight map

E = ⊕λ∈ΠV (λ) is a Hamiltonian K × T space.

T moment map:

hw : E → t∗+ ⊂ t∗∑λ∈Π

uλ 7→ π∑λ∈Π||uλ||2λ, uλ ∈ V (λ)

Key fact: hw is proper.

Recall ET ∗K =∐σ≤t∗+

K[Kσ ,Kσ]

× σ. Stratification:

K

[Kσ,Kσ]× σ ∼= G � N ∩ hw−1(σ)


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Toric degenerations

Step 3: Take a toric degeneration of G � N to a toric variety XCi .

Let XCi = SpecC[Ci].

Theorem (Caldero): There is a flat family π : X → C whereπ−1(C×) ∼= G � N × C× and π−1(0) ∼= XCi

Reformulation, due to Kaveh: The string parametrization can be extendedto a valuation

vi : C[G � N]→ P × ZN

with one dimensional leaves.


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Toric degenerations

Embed it into E × C:X E × C

C

←↩ →←

→π

←→ pr2

Technical point: Let T be the compact torus of XCi . Then the embeddingX ↪→ E × C can be chosen so that T acts on E via unitarytransformations (and hence is a Hamiltonian action).


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Example

Let G = SL3C, i = (1, 2, 1) as before. Then

C[Ci] =C[z1, z2, z3, z12, z13, z23]

(z2z13 − z1z23)

Take the algebra:

R = C[z1, z2, z3, z12, z13, z23, t](z2z13 − z1z23 − tz3z12)

R is a C[t] module. X = SpecR → SpecC[t] = C.R[t−1] ∼= C[G ]N ⊗ C[t±1]R/(t) ∼= C[Ci].


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Stratified gradient-Hamiltonian flow

Step 4: Use gradient-Hamiltonian flow to build a map G � N → XCi .

For each σ ≤ t∗+, restrict attention to X ∩ (hw−1(σ)× C). Smooth awayfrom π−1(0). Let

V σ = − ∇


LetV =

∐σ≤t∗+

V σ : X\π−1(0)→ T (E × C).

Technical fact: V is continuous.(Note this doesn’t always happen for stratified gradient flows).


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Fact: π∗V = − ddt .

Let ϕt be the time t flow ofV (as long as it is defined). Then

ϕt : π−1(1)→ π−1(1− t). (∗)

for t ∈ [0, 1).

Technical facts: This flow (∗)existsand is unique for t ∈ [0, 1);is continuous;

preserves the symplecticform (stratum-wise).


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Technical fact (partly following Harada-Kaveh): ϕt can be extended to acontinuous map

ϕ1 : G � N ∼= π−1(1)→ π−1(0) ∼= XCi

which, when restricted to

G � N ∩ hw−1(σ)

is a smooth symplectic isomorphism on an open dense subset.


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Step 5

ET ∗K G � N XCi R≥0 · Ci RN

←→step 2 ←→ϕ1 ←→ ←↩ →


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Step 5

ET ∗K G � N XCi R≥0 · Ci RN

k∗←→step 2

←→

←→ϕ1 ←→ ←↩ →←

→µ


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Collective Hamiltonians

Let (M, ω, µ) be a compact connected Hamiltonian K -manifold.

M is multiplicity free if µ−1(Oλ)/K is 0-dimensional, for all λ ∈ t∗+.

An integrable system on M is collective if its moment map is of the form

Mµ−→ k∗ → RN .

Question: When does (M, ω, µ) have a collective integrable torus action?


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Collective Hamiltonians

Theorem (H.-Lane)

Let (M, ω, µ) be any compact connected Hamiltonian K -manifold. Then,

the composition MΦ−→ k∗ µ−→ RN is continuous, proper, smooth on an open

dense subset of M, and generates a Hamiltonian (S1)N action on itssmooth locus.

In particular, if M is multiplicity free, then this is a completely integrabletorus action on an open dense subset of M.

Proof:

M → Msc = (EM × ET ∗K ) � T → (EM × XCi) � T → RN


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