Gelfand Zeitlin systems under Whittaker transformmath.columbia.edu/~schrader/montreal.pdf · gl nbreaks up into 1-dimensional weight spaces for GZ , where each weightspace has multiplicity

Gelfand–Zeitlin systems under Whittaker transform

Gus Schrader

Columbia University

Faces of Integrability, CRM Montreal

May 8, 2019

Slides available atwww.math.columbia.edu/∼ schrader

Gus Schrader Gelfand–Zeitlin systems under Whittaker transform

This talk is based on work in progess with A. and M. Shapiro, andwith P. Di Francesco, R. Kedem and A.Shapiro.


Gelfand–Zeitlin systems

In the 50’s, Gelfand and Zeitlin introduced the following approachto the representation theory of the Lie algebra gln.

Let h ⊂ gln be the Cartan subalgebra, and let Zn be the center ofthe enveloping algebra U(gln).

Then we have Harish–Chandra’s isomorphism:

Zn ' C[h]Sn

' polynomial ring in n variables.



The chain of embeddings

gl1 ⊂ gl2 ⊂ · · · ⊂ gln−1 ⊂ gln,

where glk is embedded as matrices zero outside of the top-leftk × k square, induces embeddings

U(gl1) ⊂ U(gl2) ⊂ · · · ⊂ U(gln).

Idea: (Gelfand, Zeitlin) If

Zk = center of U(glk),

then [Zj ,Zk ] = 0 for all 1 ≤ j , k ≤ n.



The subalgebraGZn ⊂ U(gln)

generated by Z1, . . . ,Zn is a commutative subalgebra in

1 + 2 + . . . n =n(n + 1)

2variables,

called the Gelfand-Zeitlin subalgebra.



GZ showed any finite dimensional irreducible representation V ofgln breaks up into 1-dimensional weight spaces for GZn, whereeach weightspace has multiplicity ≤ 1.

=⇒ we get a canonical basis in V .

An identical construction works if we replace U(gln) by thequantum group Uq(gln).


Semiclassical limit

There is also a semiclassical limit of the GZ construction:(Guillemin-Sternberg 80’s)

U(g) degenerates to the Poisson algebra C[gl∗n] of functions on gl∗nwith the Kirillov-Kostant-Souriau bracket,

Uq(gln) degenerates to the Poisson algebra C[GL∗n] where GL∗n isthe Poisson-Lie dual of the standard Poisson-Lie structure on GLn.


Integrability of Gelfand–Zeitlin systems

Generic symplectic leaves in the varieties gl∗n and GL∗n are ofdimension n2 − n, and correspond to fixing the spectrum of amatrix.

The restriction of the semiclassical limit of GZn to these leaves is aPoisson commutative algebra generated by

n(n + 1)

2− n =

n2 − n

2

independent functions,

i.e. an integrable system.


Cluster structure of the Gelfand–Zeitlin system

In fact, the Gelfand-Zeitlin system for Uq(gln) (or, semiclassically,GL∗n) can be added to the list of integrable systems whose phasespace is a cluster Poisson variety.

The basic building block for describing its cluster structure is thequiver for the cluster structure on relativistic Toda spaceGLc,cn /AdH where c is a Coxeter element of the Weyl group Sn.



The cluster variety GLc,cn /AdH has an automorphism τ called theQ-system/Dehn twist/discrete factorization dynamics: mutate atall green vertices:



The quiver for Uq(gln) is obtained by gluing quivers forgl1, gl2, . . . , gln−1 Toda spaces:



The blue variables X1,1, . . . ,Xn−1,1 are frozen ones.

In the natural functional representation of the quantum clusteralgebra on the Laurent ring in n(n − 1)/2 variables

C[X±1ij ]1≤j≤i≤n−1

they act as multiplication operators by the correspondingcoordinate functions.

The Gelfand-Zeitlin Hamiltonians are given by the collection of allgl1, gl2, . . . , gln−1 Toda Hamiltonians

Hglik , 1 ≤ k ≤ i ≤ n − 1.



The quantum group generators are embedded into the uppercluster algebra as follows:

Ei 7→i+1∑k=0

Hgli+1

k τk(Xi ,1),

where Hglik is the k–th gli Toda Hamiltonian, and

τk(Xi ,1) is the k–th power of the Dehn twist applied to the frozenvariable Xi ,1.

(with similar formulas for the Fi ...)


Coulomb branches of 3d N = 4 gauge theories

A unifying perspective: the phase spaces of these clusterintegrable systems arise K -theoretic Coulomb branches of 3dN = 4 gauge theories.

Braverman-Finkelberg-Nakajima ’16 : let G be a complexreductive group, V a representation of G .

LetG (O) = G [[z ]], G (K) = G ((z)),

so thatGrG = G (K)/G (O)

is the affine Grassmannian.



BFN define a space

RG ,V ⊂ GrG × V ((z))

RG ,V ={

[g ], s∣∣ [g ] ∈ GrG , s ∈ V [[z ]] ∩ g · V [[z ]]

},

which roughly plays the role that the Steinberg variety does inSpringer theory.

RG ,V carries a natural action of C∗~ n G (O) where C∗~ acts by looprotation, and fits into a convolution diagram.



BFN show that this convolution diagram defines an associativealgebra structure on

HC∗nG(O)• (RG ,V ), K

C∗nG(O)• (RG ,V ),

and that the ~ = 0 fibers HG(O)• (RG ,V ), K

G(O)• (RG ,V ) are

commutative (and thus Poisson) algebras.


Coulomb branches and integrable systems

The K-theoretic Coulomb branch MC = Spec KG(O)• (RG ,V ) is a

normal, affine Poisson variety of dimension 2 rank(G ).

The subalgebra K •G (pt) is Poisson commutative withrank(G )-many independent generators, so gives an integrablesystem on MC .


Coulomb branches of quiver gauge theories

Important special case: Suppose Γ is a quiver with vertices Γ0,edges Γ1, and that we label each vertex v ∈ Γ0 with an naturalnumber nv .

If Γ′0 ⊂ Γ0 is a subset of nodes whose complement consists only ofleaves, then for we have a reductive group

GΓ =∏v∈Γ′0

GL(nv ),

and a representation

VΓ =⊕e∈Γ1

Hom(Cns(e) ,Cnt(e))

of GΓ.


Example

E.g. Γ orientation of type A Dynkin diagram

1 2 3 4 5

C C2 C3 C4 C5

with Γ0 = {1, 2, 3, 4}, so that

GΓ = GL(1)× GL(2)× GL(3)× GL(4),

andVΓ = Mat1×2 ⊕Mat2×3 ⊕Mat3×4 ⊕Mat4×5.


Minuscule monopole operators

When the pair (G ,V ) comes from a quiver, the BFN algebra

KG(O)• (RG ,V ) can be made explicit: it is generated by minuscule

monopole operators (Weekes ’19): classes of the structuresheaves

[ORωi ,k], [ORω∗

i,k

]

of pre-images Rωi ,k ,Rω∗i ,k of the closed G (O)-orbits in GrG underthe projection RG ,V → GrG .



Better yet, equivariant localization with respect to the maximal

torus T ⊂ GΓ gives an embedding of KG(O)• (RG ,V ) into an algebra

of rational q-difference operators

KT• (RT

G ,V ) ' Aq(Γ) =⊗i∈Γ′0

Aq(ni ),

where

Aq(ni ) =C〈Dij ,wij〉nij=1

〈Dijwrs = qδir δjswrsDij〉.



The minuscule monopole operators are represented by theq–difference operators

[ORω1,i] 7→

ni∑r=1

∏arrowsj←i

nj∏s=1

(1 + qwirw−1js ) Di ,r

where

Di ,r =

ni∏s 6=r

(1− wisw−1ir )−1Dir .


Example: the quantum group

To recover Uq(gln), we consider the oriented Dynkin quiver of typeAn−1.

E.g. n = 5

1 2 3 4 5

C C2 C3 C4 C5

with Γ0 = {1, 2, 3, 4}, so that

GΓ = GL(1)× GL(2)× GL(3)× GL(4).


Example: the quantum group

The minuscule monopole operators are

[ORω1,i] 7→

i∑r=1

ni+1∏s=1

(1 + qwirw−1i+1,s)Di ,r

=i∑

r=1

ni+1∑k=0

qkek(w−1i+1,•)w

ki ,rDi ,r

=i+1∑k=0

ek(w−1i+1,•)Di [k],

where we set

Di [k] =i∑

r=1

qkwki ,rDi ,r .


Difference operators

We recognize the Di [k] as the first generalized nil-Macdonaldoperator (DiFrancesco-Kedem ’15) in the variables Xi ,1, . . . ,Xi ,ni .

Di [k] =i∑

r=1

qkwki ,rDi ,r

=i∑

r=1

qkwki ,r

ni∏s 6=r

(1− wisw−1ir )−1Dir


Reminder on Whittaker transform

Recall from Sasha’s talk that the q–Whittaker transform for theToda chain intertwines the Toda Hamiltonians with operators ofmultiplication by elementary symmetric functions:

W ◦ HTodak = ek ◦W,

It also intertwines the action of the nil-Macdonald operators D[k]with Dehn twists of the multiplication operators Xi ,1

corresponding to the frozen variables:

W ◦ τk(X1) = D[k] ◦W.


Whittaker transform of the monopole operators

So if apply Whittaker transform

Wgl1 �Wgl2 � · · ·�Wgln−1

on each tensor factor of the representation of Aq by q-differenceoperators, we recover our formulas for the cluster realization ofUq(gln):

[ORω1,i] =

i+1∑k=0

ek(w−1i+1,•)Di [k]

7→i+1∑k=0

Hgli+1

k τk(Xi ,1).


Example: Big Toda space

Another example is the ‘big Toda’ system from Sasha’s talk, whichcorresponds to the quiver gauge theory for Γ given by

. . . . . .

Cn−2 Cn Cn−2. . . . . .


Cluster realizations of K -theoretic Coulomb branches forquiver theories

In fact, a cluster structure exists for more general quiver gaugetheories.

Theorem (A. Shapiro-M. Shapiro-S.)

For each quiver without cycles, the K -theoretic Coulomb branchalgebra can be embedded into a quantum upper cluster algebra,such that each minuscule monopole operator is a cluster monomial.

We conjecture that this embedding is an isomorphism.

In particular, we get cluster realizations of all generalized affineGrassmannian slices Wλ

µ(g) for g of type ADE .


Jordan quiver

It is also interesting to consider quivers with cycles or self-loops.

E.g. Jordan quiver GΓ = gln, and the representation VΓ = gln isthe adjoint

n

The minuscule monopole operators are now the Macdonaldoperators

Mk =∑

J⊂[1,...,n]|J|=k

∏j∈Jr 6∈J

twr − wj

wr − wj

∏j∈J

Dj .


Cluster realization of spherical DAHA

Theorem (Di Francesco-Kedem-S.-A.Shapiro)

The K -theoretic Coulomb branch construction for the Jordanquiver gives an embedding of the spherical double affine Heckealgebra into a quantum upper cluster algebra, and the modulargroup SL(2,Z) acts by cluster transformations.


Cluster realization of spherical DAHA

In this case an interesting cluster-theoretic phenomenon occurs:these quivers are not ’locally-acyclic’, and their cluster algebras donot coincide with their upper cluster algebras.

e.g. n = 2, Markov quiver

Indeed, the Macdonald operators in DAHA are contained in theupper cluster algebra, but cannot be expressed as polynomials incluster variables.


Documents

Gelfand Zeitlin systems under Whittaker transformmath.columbia.edu/~schrader/montreal.pdf · gl nbreaks up into 1-dimensional weight spaces for GZ , where each weightspace has multiplicity