Whittaker Functions. by Daniel Bump. and Quantum Groups. Supported in part by NSF grant

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1 Whittaker Functions and Quantum Groups by Daniel Bump A B C A B C C B A C B A upported in part by NF grant

2 Whittaker Functions Let F =, C, Q p or F q. Let G=GL(n,F) or more generally a split reductive group. Let U be the maximal unipotent subgroup. Let ψ be a nondegenerate character of F. If G=GL(3) we may use ψ 1 x y 1 z =ψ 0 (x+z) 1 where ψ 0 :F C is an additive character, e.g. ψ 0 (x)=e 2πix if F =. heorem 1. (Gelfand-Graev, Piatetski-hapiro, halika, odier) If (π, V) is an irreducible representation of G there admits at most one linear functional Ω:V C such that Ω(π(u)v)=ψ(u)Ω(v) for all v V and u U. If v V then W(g) = Ω(π(g)v) is called a Whittaker function. Usually we are interested in particular v, e.g. the unique (up to scalar) K-fixed vector where K= O(n) if F = U(n) if F = C. GL(n, Z p ) if F = Q p For this choice, W is called the spherical Whittaker function. It may be defined by an integral hence has a natural normalization. 2

3 he Archimedean Case If F =, Kazhdan and Kostant observed that the differential equations satisfied by the spherical Whittaker function are observables (Hamiltonians) for the Quantum oda Lattice. his has led to many developments of which we mention the work of Gerasimov, Lebedev and Oblezin (GLO). he Nonarchimedean Case his talk will mainly be about the nonarchimedean case. Interpolation One intriguing aspect of GLO is that they are able to interpolate between the archimedean and nonarchimedean cases. his is perhaps similar to the case with Macdonald polynomials which interpolate between the spherical functions for the archimedean case ( Zonal functions ) and the nonarchimedean case ( Hall-Littlewood polynomials ). his alk Without further ado we turn to the question of how nonarchimedean Whittaker functions relate to quantm groups. 3

4 he Weyl Character Formula Let Ĝ(C) be a complex reductive Lie group, realized as an affine algebraic group. Let ˆ be a maximal split torus, and z ˆ(C). Let P =X (ˆ) be the weight lattice. Let λ be a dominant weight and let χ λ be the irreducible character of Ĝ(C) with highest weight λ. By the Weyl character formula (1z α )χ λ (z)= ( ) (1) l(w) z w(λ+ρ)+ρ ρ= 1 2 α Φ w W he Casselman-halika Formula α Φ + α Let F be a nonarchimedean local field and o its ring of integers. Let G be the split reductive group in duality with Ĝ. his means if is the maximal split torus of G then the weight lattice P of Ĝ may be identified with the cocharacter group X () of G, and with (F)/(o). If λ P let t λ be a representative in (F). he Casselman-halika formula shows that for the spherical Whittaker function W W(t λ )= { ( ) α Φ (1q1 z α )χ λ (z) if λ is dominant, 0 otherwise. Here q is the residue field size. he unimportant constant ( ) is a power of q. he expression is a deformation of the Weyl character formula. 4

5 Example We will soon enter territory where each Cartan type must be handled individually, and although results are available for other Cartan types we will restrict ourselves to ype A, that is, GL(n). Here is the Casselman-halika formula for GL(n). (It was proved earlier by hintani for this case.) If G=GL(n) then the Langlands dual Ĝ= GL(n) also. We may identify the weight lattice P with Z n. If λ=(λ 1,,λ n ) P then λ is dominant if and only if λ 1 λ 2 λ n. t λ = λ1 λn he Casselman-halika formula asserts, = a generator of the maximal ideal p of o. W(t λ )=δ 1/2 (t λ ) i<j (1q 1 z i z j 1 )s λ (z 1,,z n ) where z= z 1 ˆ(C) z n is the atake or Langlands parameter. he character χ λ (z)=s λ (z 1,,z n ) is a chur polynomial. 5

6 okuyama s deformation of the WCF he Weyl character formula has a deformation (okuyama, 1988) that may be exactly matched with the Casselman-halika formula. It produces, with t a parameter α Φ + (1tz α )χ λ (z). okuyama expressed this as a sum over strict Gelfand-setlin patterns with shape λ + ρ. here are different ways of expressing his result. As a sum over the Kashiwara crystal B λ+ρ. As the partition function of a solvable lattice model. At first glance it seems a stretch to lay too much significance on this as a formula for the spherical Whittaker function. However, we affirm that this is important on the following evidence: the identity of okuyama s formula with the Casselman-halika formula may be extended to a formula for the far more subtle Whittaker functions on metaplectic groups. 6

7 okuyama s Formula: Crystal Version okuyama s formula may be expressed as a sum over Kashiwara s crystal with highest weight λ+ρ: (1tz α )χ λ (z)= G(v)z wt(v)w0ρ. α Φ + v B λ+ρ he data defining G(v) are the lengths of segments in a path through the crystal from v to the highest weight vector For the marked vertex the segments have lengths 2,2, hese are the lengths of the paths in the red-blue-red directions

8 okuyama s Formula: Crystal Version okuyama s formula may be expressed as a sum over Kashiwara s crystal with highest weight λ+ρ: (1tz α )χ λ (z)= G(v)z wt(v)w0ρ. α Φ + v B λ+ρ G(v)= g(b i ) sometimes h(b i ) ditto q b i 0 where b i runs through the path lengths (0,2,2 for the marked v) and: g(b)=t 1b h(b)=(t 1 1)t 1b In this case: G(v)=1 g(2) h(2) Change g and h to Gauss sums to get metaplectic Whittakers. 8

9 okuyama s Formula: ix Vertex Model here is another way of writing okuyama s formula that was first found by Hamel and King. his represents α Φ + (1tz α )χ λ (z) as the partition function of a statistical mechanical system. his uses the six-vertex model, an example that was studied extensively prior to the discovery of quantum groups. Begin with a grid, usually (but not always) rectangular: Each exterior edge is assigned a fixed spin + or. he inner edges are also assigned spins but these will vary. 9

10 ix Vertex Model A state of the model is an assignment of spins to the inner edges. (he outer edges have preassigned spins. Every vertex is assigned a set of Boltzmann weights. hese depend on the spins of the four adjacent edges. For the six-vertex model there are only six nonzero Boltzman weights: a 1 a 2 b b 2 c 1 c 2 he Boltzmann weight of the state is the product of the weights at the vertices. he partition function is the sum over the states of the system. 10

11 he Partition Function he Boltzmann weights are called field-free if a 1 =a 2, b 1 =b 2 and c 1 =c 2. hey are called free-fermionic if a 1 a 2 +b 1 b 2 c 1 c 2 =0. heorem 2. (Lieb, utherland, Baxter, Korepin-Izergin) If every vertex has the same field-free Boltzmann weights then the partition function can be evaluated. Kuperberg used this to prove the AM conjecture. One method of proof is Baxter s and uses the Yang-Baxter equation. After Onsager s work on Ising model this gave a second example of a solvable lattice model. heorem 3. (Hamel and King s reformulation of okuyama) For certain freefermionic Boltzmann weights the partition function can be evaluated and equals α Φ + (1tz α )χ λ (z). he set of states injects into the B λ+ρ crystal using Gelfand-setlin patterns. Hamel and King gave a novel proof using jeu de taquin. Brubaker, Bump and Friedberg used instead the Yang-Baxter equation. he version of the Yang-Baxter equation they use was first found by Korepin. Brubaker, Bump, Chinta, Friedberg and Gunnells gave a variant that produces metaplectic Whittaker functions. 11

12 Ice Consider the following Boltzmann weights: (z): + a 1 a b 1 b z t z z(t + 1) 1 c c 2 his is in the free-fermionic regime: a 1 a 2 +b 1 b 2 =c 1 c 2. Consider a system: (z 1 ) (z 1 ) (z 1 ) (z 1 ) (z 2 ) (z 2 ) (z 2 ) (z 2 ) he boundary conditions: + on left and bottom edges, on right and signs on top edge are determined by the partition λ by some rule. Use (z i ) in i-th row: z i eigenvalues of z. + (z 3 ) (z 3 ) (z 3 ) (z 3 ) okuyama, Hamel-King and Brubaker-Bump-Friedberg proved the partition function equals α Φ + (1tz α )χ λ (z). 12

13 Baxter In the field-free case let be Boltzmann weights for some vertex with a=a 1 =a 2 and b= b 1 =b 2 and c=c 1 =c 2. Following Lieb and Baxter let = a2 +b 2 c 2. 2ac Given one row of ice with Boltzmann weights at each vertex: δ 1 δ 2 δ 2 δ 2 µ µ ǫ 1 ǫ 2 ǫ 3 ǫ 4 oroidal boundary conditions: Equal spins at left and right edges, so sum over µ=+ and. Effectively µ is an interior edge. Non-toroidal BC can be handled by a variant of this method. Let δ=(δ 1,δ 2, ) and ε=(ε 1,ε 2, ) be the states of the top and bottom rows. he partition function then is a row transfer matrix Θ (δ, ε). he partition function with several rows is the product of the row transfer matrices. heorem 4. (Baxter) If = then Θ and Θ commute. hough we will not explain this point, the commutativity of transfer matrices yields, among other things, the solvability of the model. he proof uses the Yang-Baxter equation and the ideas here are key for us. 13

14 he Yang-Baxter equation heorem 5. (Baxter) Let and be vertices with field-free Boltzmann weights. If = then there exists a third with = = such that the two systems have the same partition function for any spins δ 1,δ 2,δ 3,ε 1,ε 2,ε 3. ǫ 3 ǫ 3 ǫ 2 δ 1 ǫ 2 δ 1 ǫ 1 δ 2 ǫ 1 δ 2 δ 3 δ 3 14

15 Commutativity of ransfer Matrices his is used to prove the commutativity of the transfer matrices as follows. Consider the following system, whose partition function is the product of transfer matrices Θ Θ (φ,δ)= ε Θ (φ,ε)θ (ε,δ): φ 1 φ 2 φ 3 φ 4 µ 1 µ 1 µ 2 µ 2 δ 1 δ 2 δ 3 δ 4 oroidal Boundary Conditions: µ 1,µ 2 are interior edges and so we sum over µ 1,µ 2. Insert and another vertex 1 that undoes its effect: 15

16 φ 1 φ 2 φ 3 φ 4 µ 1 1 µ 1 µ 2 µ 2 δ 1 δ 2 δ 3 δ 4 Now use YBE repeatedly: φ 1 φ 2 φ 3 φ 4 µ 1 1 µ 1 µ 2 µ2 δ 1 δ 2 δ 3 δ 4 Due to toroidal BC now, 1 are adjacent again and they cancel. he transfer matrices have been shown to commute. 16

17 Algebraic Formulation he investigation of the Yang-Baxter equation by mathematical physicists in ussia and Japan (descended from Faddeev and ato) in the 1980s led in an algebraic direction culminating in the discovery of quantum groups (quasitriangular or co-quasitriangular Hopf algebras) by Drinfeld and Jimbo. Now we understand things as follows. he Yang-Baxter equation, which we considered previously as the identity of the partition functions of ǫ 3 ǫ 3 ǫ 2 δ 1 ǫ 2 δ 1 ǫ 1 δ 2 ǫ 1 δ 2 δ 3 δ 3 can be interpreted as follows. Let V be a two-dimensional free vector space on the spins + and. hen, and may be interpreted as elements of End(V V) and the Yang-Baxter equation is an identity in End(V V V). It is written = where if A End(V V) then A ij is A I with A acting on the i, j components of a tensor in V V V and I acting on the third component. 17

18 ǫ 2 ǫ 1 µ ν ǫ 3 δ 3 ρ δ 1 δ 2 emember that we sum over interior edge spins. Let us label these µ,ν,ρ for the purpose of the following discussion. V U U V -Matrices We imagine the spins ε 1,ε 2 and µ,ν as specifying vectors each in a two-dimensional vector space. It will help us later if we two vector spaces U and V with ε 1 U, ε 2 V, µ U and ν V. he four spins together specify a vector in U V U V = End(U V). his endomorphism of U V is called an -matrix. 18

19 Braid Picture Deform the picture above. U V W U V W W V U W V U We interpret as being an endomorphism of U V, where U and V are vector spaces. imilarly End(U W) and End(V W). he Yang-Baxter equation is still written = It is not important whether U =V =W. ince and the cancelling 1 are distinct, we draw the vertex as an over-and-under crossing. his will help us keep track of things and also introduces the Artin braid group. 1 19

20 Monoidal Categories A monoidal or tensor category is a category with an associative composition law with natural isomorphisms A (B C) (A B) C. It is assumed that there is a unit I with natural A I isomorphisms A and A I A. Coherence: Any two ways of getting from one parenthesization of A 1 A 2 to another give the same result. (Maclane) (A B) (C D) A (B (C D)) ((A B) C) D A ((B C) D) (A (B C)) D 20

21 ymmetric vs Braided Monoidal Categories A symmetric monoidal category (Maclane) adds natural isomorphisms τ A,B : A B B A. Coherence: any two ways of going from one permutation of A 1 A 2 to another give the same result. A B C τ A B,C C A B 1 A τ B,C τ A,C 1 B Important generalization! Maclane assumed that τ A,B :A B and τ B,A :B A are inverses. But... A C B Joyal and treet proposed omitting this assumption. his leads to the important notion of a braided monoidal category. 21

22 Coherence in a Braided Monoidal Category In a symmetric monoidal category τ A,B B and τ 1 :A B,A B are the same but in :A a braided monoidal category they may not be. We may distinguish them by using crossings: A B τ A,B B A A B B A τ 1 B,A he top row is A B he bottom row is B A Coherence: Any two ways of going from A 1 A 2 to itself gives the same identity provided the two ways are the same in the Artin braid group. For example: A B C A B C C B A C B A hese diagrams describe two morphisms A B C C B A. 22

23 Quantum Groups Hopf algebras are convenient substitutes essentially generalizations of the notion of a group. he category of modules or comodules of a Hopf algebra is a monoidal category. (A module over H has a multiplication H V V so dually a comodule has a comultiplication V H V.) he group G has morphisms µ:g G G and :G G G, namely the multiplication and diagonal map. hese become the multiplication and the comultiplication in the Hopf algebra. he modules over a group form a symmetric monoidal category. here are two types of Hopf algebras with an analogous property. In a cocommutative Hopf algebra, the modules form a symmetric monoidal category. In a commutative Hopf algebra, the comodules form a symmetric monoidal category. oughly a quantum group is a Hopf algebra whose modules or comodules form a braided monoidal category. 23

24 Groups as Hopf Algebras. If G is a Lie group, its enveloping algebra U(g) is the convolution ring of distributions at the identity. Modules of G correspond to modules of U(g). hey form a symmetric monoidal category. Dually, consider the coordinate ring O(G) of an affine algebraic group. ince the functor X O(X) from affine schemes to commutative rings is contravariant, the multiplication in G corresponds to the comultiplication in O(G). he comodules of O(G) correspond to modules of G and they form a symmetric monoidal category. Quantum Groups If g is a complex Lie algebra, Drinfeld and Jimbo defined a quantized enveloping algebra U q (g). he modules form a braided monoidal category. he group O(G) may also be deformed. he comodules form a braided monoidal category. 24

25 Parametrized Yang-Baxter Equation Let us assume that we have a collection of possible vertices,,,, each representing a set of Boltzmann weights. We call them -matrices. Assume for every pair and of vertices that there exists a vertex such that the Yang-Baxter equation is true in the sense that the following two partition functions are equal: U V W U V W W V U W V U It is to be imagined that U,V,W are all copies of the same vector space which will eventually become modules in some category. We may think of (, ) as a kind of multiplication on the set Γ of -matrices. Given, the condition on is overdetermine so (if it exists) is undoubtedly unique up to scalar multiple. he composition tends to be associative. If U = V = W = this may give a group or monoid structure on a subset of P(End(V V)) 25

26 Associativity of composition of -matrices Consider the following setup. U V W X U V W X () () X W V U We ll prove these are equal. X W V U 26

27 Here goes: U V W X U V W X () () X W V U Use YBE on the right hand side. X W V U 27

28 U V W X U V W X () () X W V U X W V U YBE on the upper left side. 28

29 U V W X U V W X () () X W V U X W V U Next on the lower left. Done. 29

30 imilarly... U V W X U V W X () () X W V U X W V U hese two partition functions are also equal. (imilar proof.) 30

31 Comparison We ve exhibited two endomorphisms of U W X (represented as greenies below) such that for either endomorphism the two partition functions are equal: U V W X U V W X?? X W V U X W V U his is an overdetermined system of equations on the endomorphism of U W X. 31

32 Overdetermined system should have only one solution... so the two different endomorphisms are equal. hus U W X U W X () equals () X W U X W U hat means that () has the property characterizing () so ()=(). 32

33 Parametrized Yang-Baxter Equation Fix a vector space. uppose End(V V) is defined (as above) for all and in some collection of endomorphisms of V V. his means = satisfies = , or in pictures V V V V V V V V V V V V Beginning with Baxter s field-free example, one often draws,, from a collection of matrices parametrized by a group Γ. More generally Γ can be a monoid. 33

34 Make a Category We may think of having one copy V γ of a fixed vector space for every γ Γ and the equation becomes where now (ξ.η) End(V ξ V η ). 12 (ξ/η) 13 (ξ/ζ) 23 (η/ζ)= 23 (η/ζ) 13 (ξ/ζ) 12 (ξ/η) V ξ V η V ζ V ξ V η V ζ (ξ/η) (η/ζ) (ξ/ζ) (ξ/ζ) (η/ζ) (ξ/η) V ζ V η V ξ V ζ V η V ξ We might adjoin kernels and cokernels (Karoubi completion) to get an abelian category out of the V γ and hope it s a braided monoidal category. We might then hope it s the category of modules over a QHA (we ll explain later) 34

35 Example: parametrized YBE: Baxter First, in Baxter s field free case, we recall that = a2 +b 2 c 2 2ab or a 2 +b 2 c 1 c 2 2ab It is OK to allow c 1 c 2 as long as a 1 =a 2 and b 1 =b 2. Find q such that = 1 2 (q+q1 ). he parameter group Γ will be C. olve: a=zq 2, b=q(z1), c 1 =(1q 2 ), c 2 =z(1q 2 ). he -matrix is (for suitable basis a (z)= b c 2 c 1 b a = q 2 z q(1z) z(q 2 1) (q 2 1) q(1z) q 2 z in End(V ξ V η ) where z=ξ/η. Or the parameter monoid Γ can be C: if z=0 this gives us a solution q 2 q q 2 q q 2 35

36 Example: parametrized YBE: Eight-vertex model Baxter realized that the eight vertex model led to parametrized YBE with parameter group an elliptic curve. In the field free case the -matrices look like: a d b c c b. d a Example: parametrized YBE: Free-Fermionic Case he set of Boltzmann weights with a 1 a 2 + b 1 b 2 = c 1 c 2 forms a parametrized YBE with -matrices a 1 b 1 c 2 c 1 b 2. a 2 hus if =0 we can drop the field free hypothesis. he subset of this monoid with c 1 c 2 0 forms a group parametrized YBE with nonabelian parameter group Γ=GL 2 GL 1. 36

37 Quasitriangular Hopf algebras Drinfeld defined the notion of a quasitriangular Hopf algebra (QHA). If H is a QHA then its category of modules forms a braided monoidal category. he exact definition is not important for our discussion and we omit it. imilarly there are dual QHA s with the property that its comodules form a braided monoidal category. riangular = ymmetric If the QHA or dual QHA is triangular the category of modules is a symmetric monoidal category meaning the commutativity constraint θ A,B B A satisfies θ A,B = θ 1 B,A. his is less interesting. For example, the applications of quantum :A B groups to knot theory depend on being able to distinguish A B τ A,B B A A B B A τ 1 B,A 37

38 annakian problem aavedra-ivano proved given a symmetric monoidal category satisfying certain axioms (rigidity, fiber functor) there exists a commutative Hopf algebra having the category as comodule category. Given a suitable braided monoidal category, can we find a QHA (or dual QHA) having an equivalent category of modules (or comodules)? (Joyal and treet, Majid.) More modestly, given a solution to YBE or parametrized YBE, construct a QHA (or dual QHA) having that vector space or family of vector spaces as a module (or comodule). wo main methods: Drinfeld double glues two dual Hopf algebra to make a QHA. Faddeev, eshetikhin and akhtajan (F). his method first produces a dual QHA, and a QHA may be obtained by duality. F method was extended to parametrized case by Cotta-amusino, Lambe and inaldi. Buciumas reconsiders the F method in the parametrized case more functorially with an eye on the Free-Fermionic YBE. 38

39 Duality (classical case, i.e. q = 1) Let G be an affine algebraic group over C. Its coordinate ring O(G) is a Hopf algebra: the comultiplication corresponds to the multiplication in G. Let g be its Lie algebra. he universal enveloping algebra U(g) is a Hopf algebra: the multiplication in g corresponds to convolution. hat is, U(g) may be identified with the convolution ring of distributions on G concentrated at the identity. Applying a distribution to a function gives a dual pairing U(g) O(G) C. We have ξ,f 1 f 2 = ξ,f 1 f 2, ξ η, f = ξη,f where is comultiplication. If V is a module of g then V is a comodule of G. he modules/comodules form a symmetric monoidal category. Deformation Let q be a parameter. Let U q (g) be the quantized enveloping algebra and O q (G) the quantum G. Both are Hopf algebras, in duality. U q (g) is a QHA and O q (G) is a dual QHA. he above case is q=1. he modules/comodules form a braided monoidal category. 39

40 Faddeev, eshetikhin and akhtajan (F) Problem: beginning with a solution of (unparametrized) YBE End(V V) where V is a vector space, produce a bialgebra with V as a comodule. hen τ should be a comodule endomorphism. (τ(x y)= y x.) Categorically, it corresponds to the commutativity constraint. olution: Let be a matrix of noncommuting variables equal in number to dim(v) 2. hese variables are subject to the condition 1 2 = 2 1 ( 1 = I, 2 =I ). (1) hen A is the algebra generated by these variables. Perspective 1: For F, the motivation is from inverse scattering theory. Here plays the role of the Lax operator which satisfy the Fundamental Commutation elation (1). Perspective 2: Assuming that V is a comodule for an algebra A, the comultiplication V A V may be expressed as v i j t ij v j. Now consider the algebraic consequences of the requirement that τ: V V V V is a coalgebra homomorphism, where τ(x y)=(y x). his requirement leads to the identity (1). Cotta-amusino, Lambe, inaldi, Bucimuas: his works in the parametrized case with one set of variables (z) for each z Γ. 40

41 Dual Affinization If g is a complex semisimple Lie algebra let ĝ be the derived Lie algebra of the Kac- Moody affinization. It is 0 C c ĝ g C[t,t 1 ] 0. Dual to the enlargement U q (g) to U q (ĝ): pecial case: Γ=C. uppose we already have a Hopf algebra A with V as a comodule. he O(Γ)=C[t,t 1 ] is a Laurent polynomial ring. his is a Hopf algebra. is a Hopf algebra. A Γ = Hom(O(Γ),A) Proposition 6. A Γ has a comodule V z for every z Γ. Proof. If z Γ, then comultiplication z :V A Γ V may be defined making V an A Γ comodule V z. If : V A V is the comultiplication, v = a i v i then z v = φ i v i where φ i A Γ sends f O(Γ) to f(z)a i. uppose Γ = C inside the multiplicative monoid C. (Call this algebraic monoid Γˆ.) he algebra A can be obtained by taking the -matrix at 0, which satisfies then applying the F construction =

42 riangularity In order for a braided monoidal category to be symmetric we need the commutativity constraint θ A,B B A satisfy θ :A B B,A =θ 1 A,B. Given a parametrized YBE :Γ End(V V) We have one copy V z for each z Γ and θ Vz,V w =τ(z,w 1 ). his means we need τ(z)τ(z 1 )=I. In the Baxter case we may check that this is true, after adjusting (z) by a scalar: a ( ) qzq 1 (z)= b c 2 c 1 b = 1 z1 z(qq zq 1 1 ) q qq 1 z1 a qzq 1 in End(V ξ V η ) where z = ξ/η. Hence this Yang-Baxter equation is triangular and the resulting monoidal category is symmetric. 42

43 Limiting Case is not triangular he parameter group in this example can be enlarged to the multiplicative monoid C which contains two idempotents, z = 1 (the unit) and z = 0. aking z = 0 gives = (0) satisfying = and the F construction gives GL q (2) which is quasitriangular but not triangular since ττ=i is not true. What about the Free-Fermionic Case? A Hopf algebra exists with two-dimensional comodules in bijection with the parameter group Γ=GL(2) GL(1). Is it Hom(O(Γ), L 1 (2)) with dual QHA structure? Is it dual to some enlargement of U 1 (sl 2 )? What about limiting cases? What about the eight vertex model? What about the metaplectic case? 43