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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Gelfand–Zeitlin systems
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Gelfand–Zeitlin systems
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Gelfand–Zeitlin systems
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).
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Cluster structure of the Gelfand–Zeitlin system
The cluster variety GLc,cn /AdH has an automorphism τ called theQ-system/Dehn twist/discrete factorization dynamics: mutate atall green vertices:
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Cluster structure of the Gelfand–Zeitlin system
The quiver for Uq(gln) is obtained by gluing quivers forgl1, gl2, . . . , gln−1 Toda spaces:
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Cluster structure of the Gelfand–Zeitlin system
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Cluster structure of the Gelfand–Zeitlin system
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 ...)
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Coulomb branches of 3d N = 4 gauge theories
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Coulomb branches of 3d N = 4 gauge theories
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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 .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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 .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Minuscule monopole operators
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〉.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
Minuscule monopole operators
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 .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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).
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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 .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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).
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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. . . . . .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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 .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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 .
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform
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.
Gus Schrader Gelfand–Zeitlin systems under Whittaker transform