Gelfand-Tsetlin theory

Gelfand-Tsetlin theory

Vyacheslav Futorny

University of Sao Paulo

Israel Moiseevich Gelfand (1913-2009)

Michael L’vovich Tsetlin (1924-1966)

Example

Assume the set {1, a, b, c , . . .} ⊂ C is linearly independent over Z

1 a + 1 a b 0a b − 1 b a + 1

c c + 1 c

a a− 1

a + 1

An element v ∈ Cn(n+1)

2

1 0 a + 1 a ba + 1 a b b − 1

c + 1 c c

a a− 1

a + 1

An element in normal form in Sµv

a + 1 a 1 0 ba a b − 1 b − 1

c c c

a− 1 a− 1

a− 1

A seed in (Zµ0 #Sµ)v

Fix a seed v

Sv is the stabilizer of v in G = S1 × . . .× Sn

Above: Sv = S1 × S2 × S3 × (S2 × S2)

D(v) = {z ∈ Zn(n−1)

2 |v + z in normal form}

If z ∈ D(v) then (Sv )z is the stabilizer of z in Sv

Szv the set of minimal length representatives of Sv/(Sv )z

Denote I (v) the set of intervals corresponding to the blocks of theseed v

In the example above: I (v) = {(1)1, (12)2, (123)3, (12)4, (34)4}

For z ∈ D(v) denote I (v , z) the refinement of I (v) correspondingto the blocks of v + zThen

{Dσ(v + z) | z ∈ D(v), σ ∈ Szv }

is a C-basis (of derivative tableaux) of the universal moduleV (T (v)) = U(gln)χ

For σ ∈ Szv denote D<σ(v + z) an arbitrary linear combination of

tableaux Dτ (v + z) with τ ∈ Szv strictly smaller than σ in the

induced Bruhat order

I = [a, b]k ⊂ I (v) with k < n set

eI =

k+1∏j=1

(xk,a − xk+1,j)∏(k,j)/∈I

(xk,a − xk,j); fI =

k−1∏j=1

(xk,b − xk−1,j)∏(k,j)/∈I

(xk,b − xk,j)

TheoremThe action of the canonical generators of gl(n,C) on V (T (v)) isgiven by the formulas

Ek,k+1Dσ(v + z) =

= −∑

I∈I(v ,z)[k]

∑τ≤σα(I )

Dv+zτ,σα(I )(eI )Dτ (v + z + δk,a(I )),

Ek+1,kDσ(v + z) =

=∑

I∈I(v ,z)[k]

∑τ≤σβ(I )

Dv+zτ,σβ(I )(fI )Dτ (v + z − δk,b(I )),

EkkDσ(v + z) = hk(v + z)Dσ(v + z),

where hk = xk,1 + · · ·+ xk,k − (xk−1,1 + · · ·+ xk−1,k−1) + k − 1,α(I ) = (b, b − 1, . . . , a), β(I ) = (a, a + 1, . . . , b) and τ,σ are thePostnikov-Stanley operators

Fix a complex vector space V , Λ = S(V ∗), L the fraction field of Λ

Φ a finite root system with base Σ, W = W (Φ) the correspondingreflection group with minimal generating set S . Then W acts on Λand L, set Γ = ΛW

For s ∈W ,

∇s =1

αs(1− s) ∈ L#W ,

a twisted derivation of L: for f , g ∈ L,

∇s(fg) = ∇s(f )g + s(f )∇s(g)

Example

Let V = C2 and {x , y} ⊂ (C2)∗ the dual basis to the canonicalbasis. Let s the reflection given by s(z1, z2) = (z2, z1), soαs = x − y . Then

∇s(f )(x , y) =f (x , y)− f (y , x)

x − y

For σ ∈W take a reduced decomposition σ = s1 · · · s` and set

∂σ = ∇s1 ◦ · · · ◦ ∇s`

∇s(Λ) ⊂ Λ for any s ∈ S

Set ∆(Φ) =∏α∈Φ+ α, for each σ ∈W set

Sσ =1

|W |∂σ−1ω0

∆(Φ)

{Sσ | σ ∈W } are Schubert polynomials, they form a basis of Λ asa Γ-module, degSσ = `(σ)

SσSτ =∑ρ∈W

cρσ,τSρ mod IW ,

where cρσ,τ are the generalized Littlewood-Richardson coefficients,IW the ideal of Λ generated by the elements of Γ of positive degree

Consider Θ : Λ→ DerC(Λ), Θ(xi ) = ∂∂xi

Let (−,−)Θ : Λ× Λ→ C the bilinear form given by(f , g) = Θ(f )(g)(0)

Pσ the unique element such that (Pσ,Sτ ) = δσ,τ for all σ, τ ∈W ,and (Pσ, IW )Θ = 0

{Pσ | σ ∈W } Postnikov-Stanley polynomials

For σ, τ ∈W with τ ≤ σ in the Bruhat order set Dσ = Θ(Pσ).Then

Dτ,σ =∑ρ∈W

cστ,ρDρ

GT the category of all Gelfand-Tsetlin gln-modules

Fix ζ ∈ Cn(n+1)

2 /(Zn(n−1)

2 #G ), v ∈ ζ

GTζ the full subcategory of GT of modules with support in ζ, i.e.V ∈ GTζ if

V = ⊕z∈Z

n(n−1)2

M[v + z ]

GT =⊕

ζ∈Cn(n+1)

2 /(Zn(n−1)

2 #G)

GTζ

The F.-Ovsienko inequality:

dimM[v + z ] ≤ |Sv ||(Sv )z |

Strong FO Conjecture:The set of all z such that equality above holds (thecolorred essential support) is nonempty

hence the FO inequality gives a sharp bound in each subcategoryGTζ

Theorem (V.F., Grantcharov, Ramirez, Zadunaisky, 2018)

Let v be a seed in Cn(n+1)

2 and ζ = ζv . Then

(i) The module V (T (v)) has a simple socle Vsoc

(ii) The Strong F.-Ovsienko Conjecture holds for Vsoc

(iii) For any v + z in the essential support of Vsoc, the moduleVsoc is the unique simple Gelfand-Tsetlin module having v + zin its support

(v) For any parabolic subgroup G ⊂ Sv ,|Sv ||G | appears as a

Gelfand-Tsetlin multiplicity in module Vsoc

Recent developments

Following Webster:Consider a smash product F = L#W and K = LW

The standard flag order is the subalgebra

FΛ = {x ∈ F|x(Λ) ⊂ Λ}

A subalgebra F ⊂ FΛ is a principal flag order if KF = FΛ andW ⊂ F

Then F is a Galois order over Λ (G = {1}, M =MnW )

Let e = 1|W |∑

w∈W w ∈ FΛ. Then K ' eFe and KΓ ' eFΛe

� For any flag order F , the centralizer algebra U = eFe is aprincipal Galois order, and the category of U-modules is a quotientof the category of F -module

� Any principal Galois order U has form eFDe for someΛ#W ⊂ D ⊂ L#W , where

FD = De ⊗Γ U ⊗ ΓeD

Coulomb branches recently defined by Braverman, Finkelberg,Nakajima

A Coulomb branch is attached to each connected reductivecomplex group G and representation N

Let G [t] be the Taylor series points of the group G , G ((t)) itsLaurent series points and

Y = (G ((t))× N[t])/G [t]

with a natural map π : Y → N[t]

Let H = NGL(N)(G ) the connected component of the identity inthe normalizer of G , TH maximal torus, Q ⊂ H the subgroupgenerated by G and TH

The Coulomb branch is the convolution algebra

A = HQ×C∗∗ (π−1(N[t]))

There is a module structure on the Q × C∗-equivariant homologyof any G [t]-invariant subvariety in N[t]

Let V = t∗H⊕ where tH is the Cartan Lie algebra of H and M thecocharacter lattice of TG , acting by the h-scaled translations, Wthe Weyl subgroup of G , MnW the extended affine Weyl group

SetΛ = H∗TH×C∗(∗) = S(tH)[h]

Γ = H∗Q×C∗(∗) = S(tH)W [h]

� A ⊂ KΓ is a principal Galois order

Gelfand-Tsetlin theory has strong connection with categorificationtheory and Khovanov-Lauda-Rouquier-Webster algebras

(Kamnitzer, Tingley, Webster, Weekes, Yacobi, 2018)

Invariants in symmetric algebra

g a simple Lie algebra over C with basis Y1, . . . ,Yl s.t.

[Yi ,Yj ] =∑k

c kij Yk

The adjoint action of g on itself extends to the symmetric algebraS(g) by

Y (X1 . . .Xk) =k∑

i=1

X1 . . . [Y ,Xi ] . . .Xk

The subalgebra of invariants is

S(g)g = {P ∈ S(g)|Y (P) = 0 for allY ∈ g}

Let n = rankg.

Chevalley: S(g)g = C[P1, . . . ,Pn] for certain algebraicallyindependent invariants P1, . . . ,Pn of certain degrees d1, . . . , dndepending on g.

For g = glN set E = (Eij)and write

det(u + E ) = uN + C1uN−1 + ....+ CN

Then S(glN)glN = C[C1, ...,CN ]

Also Tk = trE k ∈ S(glN)glN for all k > 0 and

S(glN)glN = C[T1, . . . ,TN ]

S(g) has the Lie–Poisson bracket {Yi ,Yj} =∑l

k=1 c kij Yk

S(g)g is Poisson commutative (Poisson centre of S(g))

Mischenko-Fomenko subalgebras

Problem: Extend S(g)g to a maximal Poisson commutativesubalgebra of S(g)

Let µ ∈ g∗. Mischenko-Fomenko subalgebra Aµ is generated bythe µ-shifts of elements in S(g)g, that is, by all the derivativesD jµ(p) for p ∈ S(g)g and j ∈ {0, . . . , deg p − 1}, where

D jµ(p)(x) =

dj

dtp(x + tµ)|t=0, x ∈ g∗.

Mischenko-Fomenko, 1978: Aµ is a Poisson-commutativesubalgebra of S(g).

Identify g ' g∗ via the Killing form. Let Y1, . . . ,Yl be a basis of g

Take P = P(Y1, . . . ,Yl) ∈ S(g) of degree d , µ ∈ g∗.Substitute Yi 7→ Yi + z µ(Yi ) and expand:P(Y1 + z µ(Y1), . . . ,Yl + z µ(Yl)

)= P(0) + P(1)z + · · ·+ P(d)zd

Then Aµ is generated by all elements P(i) associated with allg-invariants P ∈ S(g)g

Example

det(u + µ+ Ez−1)∑

0≤i≤k≤NC ikz−k+iuN−k

The elements C ik with k = 1, ...,N and i = 0, 1, ..., k − 1 are

algebraically independent generators of Aµ for regular µ

µ ∈ g∗ ∼= g is regular, if the centralizer gµ of µ in g has minimalpossible dimension = the rank of g.

TheoremIf µ ∈ g∗ is regular, then

i) Aµ is maximal Poisson commutative(A.Tarasov, 2002, for regular semisimple µ; Panyushev,Yakimova, 2008);

ii) P(i)k with k = 1, . . . , n and i = 0, 1, . . . , dk − 1, are

algebraically independent generators of Aµ(colorblue Mischenko-Fomenko for regular semisimple µ;Feigin, E.Frenkel and Toledano Laredo, 2010)

Vinberg’s problem

The universal enveloping algebra U(g) has a canonical filtrationand

grU(g) ' S(g).

Vinberg, 1990: find a commutative subalgebra Aµ ∈ U(g) which“quantizes” Aµ, i.e. grAµ = Aµ.

Explicit free generators of Aµ for g =glN :

- A. Tarasov, 2000

- Chervov and Talalaev, 2006, 2009

Solution of Vinberg’s problem for any g:

- Rybnikov, 2006, regular semi-simple µ,

- Feigin, Frenkel and Toledano Laredo, 2010, any regular µ

Explicit generators

Let g = glN , Eij , i , j ∈ {1, . . . ,N} the standard basis of glN . LetE = (Eij) and µ = (µij).

Write

cdet(−∂z + µ+ Ez−1) =∑

0≤i≤k≤Nφ

(i)k z−k+i∂N−kz

andtr(−∂z + µ+ Ez−1)k =

∑0≤i≤k≤N

ψ(i)k z−k+i∂N−kz

TheoremFor any µ elements φ

(i)k and ψ

(i)k are two families of generators of a

commutative subalgebra Aµ of U(glN). If µ is regular, then theelements of each of these families with k = 1, ...,N andi = 0, 1, ..., k − 1 are algebraically independent

Algebraically independent generators of the algebra Aµ for regularµ: for gl2 : trE , trµE , trE 2

for gl3 : trE , trµE , trµ2E , trE 2, trµE 2, trE 3

for gl4 : trE , trµE , trµ2E , trµ3E , trE 2, trµE 2,

2 trµ2E 2 + tr (µE )2, trE 3, trµE 3, trE 4