A Yang-Baxter Equation for Metaplectic Ice
We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic -fold cover of , where is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra , modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter is specialized to the inverse of the residue field cardinality.)
For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted -matrix of the quantum group . This is a piece of the twisted -matrix for , mentioned above.
2010 Mathematics Subject Classification:Primary 16T25; Secondary 22E50
The formula of Casselman and Shalika  expresses values of the spherical Whittaker function for a principal series representation of a reductive algebraic group over a -adic field in terms of the characters of irreducible finite-dimensional representations of the Langlands dual group. Their proof relies on knowing the effect of the intertwining integrals on the normalized Whittaker functional. Since the Whittaker functional is unique, the intertwining integral just multiplies it by a constant, which they computed.
In contrast with this algebraic case, Whittaker models of principal series representations of metaplectic groups are generally not unique. The effect of the intertwining operators on the Whittaker models was computed by Kazhdan and Patterson . Specifically, they computed the scattering matrix of the intertwining operator corresponding to a simple reflection on the finite-dimensional vector space of Whittaker functionals for the -fold metaplectic cover of , where is a -adic field. Some terms in this matrix are simple rational functions of the Langlands parameters, while others involve -th order Gauss sums. Though complicated in appearance, this scattering matrix was a key ingredient in their study of generalized theta series, and also in the later development of a metaplectic Casselman-Shalika formula by Chinta and Offen  and McNamara .
One of the two main results of this paper is that this scattering matrix computed by Kazhdan and Patterson is the -matrix of a quantum group, quantum affine , modified by Drinfeld twisting to introduce Gauss sums. This appears to be a new connection between the representation theory of -adic groups and quantum groups, which should allow one to use techniques from the theory of quantum groups to study metaplectic Whittaker functions.
Although we can now prove this directly, we were led to this result by studying lattice models whose partition functions give values of Whittaker functions on a metaplectic cover of . In , it was predicted that a solvable such model should exist; i.e., one for which a solution to the Yang-Baxter equation exists. Such a solvable model has important applications in number theory: it gives easy proofs (in the style of Kuperberg’s proof of the alternating sign matrix conjecture) of several facts about Weyl group multiple Dirichlet series . The other main result of this paper is the discovery of a solvable lattice model whose partition function is a metaplectic Whittaker function. Moreover, we relate this solution to an -matrix for the quantum affine superalgebra . The relation between the two main results follows from the inclusion of (quantum affine) into .
We now explain these results in more detail. Let denote an -fold metaplectic cover of where the non-archimedean local field contains the -th roots of unity. Given a partition of length , we will exhibit a system whose partition function equals the value of one particular spherical Whittaker function at , where is a standard section.
The systems proposed in  were generalizations of the six-vertex model. The six-vertex model with field-free boundary conditions was solved by Lieb , Sutherland  and Baxter  and were motivating examples that led to the discovery of quantum groups (cf. [29, 23, 17]). In Baxter’s work, the solvability of the models is dictated by the Yang-Baxter equation where the relevant quantum group is . In the special case (so when we are working with non-metaplectic ), the systems proposed in  are six-vertex models that coincide with those discussed in Brubaker, Bump and Friedberg [9, 11] and there is a Yang-Baxter equation available. However even in this case these models differ from those considered by Lieb, Sutherland and Baxter since they are not field-free. Based on the results of this paper, we now understand that the relevant quantum group for the lattice models in [9, 11] is , as we will make clear in subsequent sections.
It was explained in  that a Yang-Baxter equation for metaplectic ice would give new proofs of two important results in the theory of metaplectic Whittaker functions. The first is a set of local functional equations corresponding to the permutation of the Langlands-Satake parameters. The second is an equivalence of two explicit formulas for the Whittaker function, leading to analytic continuation and functional equations for associated Weyl group multiple Dirichlet series. The proof of this latter statement occupies the majority of .
However, no Yang-Baxter equation for the metaplectic ice in  could be found. In this paper we will make a small but crucial modification of the Boltzmann weights for the model in . This change does not affect the partition function, but it makes possible a Yang-Baxter equation. This is Theorem 3.1 in Section 3. The solutions to the Yang-Baxter equation may be encoded in a matrix commonly referred to as an -matrix.
We further prove that the resulting -matrix has two important properties:
It is a Drinfeld twist of the -matrix obtained from the defining representation of quantum affine , a Lie superalgebra.
It contains the -matrix of a Drinfeld twist of which, as we have already explained, we will identify with the scattering matrix of intertwining operators on Whittaker models for metaplectic principal series.
Consider the quantized enveloping algebra of the untwisted affine Lie algebra , i.e. the central extension of the loop algebra of . We denote the quantized enveloping algebra as instead of the usual because in our application the deformation parameter will be , where is the cardinality of the residue field of . If and are vector spaces, let denote the flip operator . The Hopf algebra is almost quasitriangular; given any two modules and , there is an -matrix such that is a module homomorphism (though it will not always be an isomorphism). The -matrices for acting on a tensor product of two evaluation modules were found by Jimbo  (see also Frenkel and Reshetikhin , Remark 4.1.); they satisfy a parametrized Yang-Baxter equation.
The quantum group has an -dimensional evaluation module for every complex parameter value . We will label a basis of the module where runs through the integers modulo . The parameter will be called a positive decorated spin (to be supplemented later by another one, denoted ). We may think of the decoration (mod ) as roughly corresponding to the sheets of the metaplectic cover of degree .
The resulting -matrix in is the matrix whose entries are indexed by positive decorated spins and such that
These values are given by the following table:
Here is an -th order Gauss sum. These are not present in the out-of-the-box -matrix, but may be introduced by Drinfeld twisting that will be discussed in Section 4 (see also Section 4 of ). This procedure does not affect the validity of the Yang-Baxter equations, but is needed for comparison with the -matrix for the partition functions of metaplectic ice giving rise to Whittaker functions.
To obtain the full -matrix used in the Yang-Baxter equation for metaplectic ice, we must enlarge the set of positive decorated spins to include one more, labelled . Thus there are decorated spins altogether, the positive ones and one more. The -dimensional vector space is enlarged to an “super” vector space . The positive decorated spins are a basis for the odd part , and the even part is one-dimensional, spanned by . In Section 3, we present an -matrix that gives a solution of the Yang-Baxter equation for the metaplectic ice model. In Section 4, we show that the solution of the Yang-Baxter equation is equivalent to the -matrix corresponding to the defining representation of the quantum affine Lie superalgebra modified by a Drinfeld twist.
Finally, we explain the connection between the -matrix of Theorem 3.1 and the structure constants alluded to in item (2) above. The local functional equations for metaplectic Whittaker functions mentioned earlier may be understood as arising from intertwining operators. Let be the diagonal torus in , the Langlands dual group of . Each diagonal matrix
indexes a principal series representation of . Let be the finite-dimensional vector space of spherical Whittaker functions for . If , is one-dimensional, but not in general since if the representation does not have unique Whittaker models. If is a simple reflection in the Weyl group , then let denote the standard intertwining integral (see (5.5) for the precise definition). This induces a map . If then has an interesting scattering matrix on the Whittaker model that was computed by Kazhdan and Patterson (Lemma I.3.3 of ). This calculation underlies their work on generalized theta series, and was used by Chinta and Offen  and generalized by McNamara  to study the analog of the Casselman-Shalika formula for the spherical Whittaker functions.
Let be the module of Whittaker coinvariants of the representation . By definition this is the quotient of the underlying space of characterized by the fact that a linear functional is a Whittaker functional if and only if it factors through . Thus is the dual space of the space of Whittaker functionals on . Its dimension is . In Section 5, we will prove that the scattering matrix of the intertwining integrals on the Whittaker coinvariants is essentially , where is the -matrix for a Drinfeld twist of .
There is an isomorphism of the space of Whittaker coinvariants to the vector space that takes the vectors into the basis of dual to the basis of given in [26, 16, 35] (see Section 5). Then the following diagram commutes:
where denotes the map induced by the normalized intertwining operator defined in (5.6).
The notation means that the operator is applied to the tensor components, while we take the identity map on the remaining components.
This offers a new and seemingly fundamental connection between the representation theory of quantum groups and -adic metaplectic groups. It also suggests several immediate questions.
First, one may ask for generalizations to other Cartan types. For symplectic groups, Yang-Baxter equations based on those found here are given in Gray . A categorical framework for some of these operations would be desirable. Even for central extensions of there are open questions. We required the -th roots of unity to be in the ground field , in order to twist the Matsumoto cocycle defining the metaplectic central extension of by a cocycle of the form as in (5.1). We may ask whether other choices of cocycle admit a similar story; in particular, some choices result in a strictly smaller dimensional space of Whittaker models, so wouldn’t biject with basis elements in the tensor product of vector spaces appearing in Theorem 1.1.
One may also ask for connections with other literature such as Weissman . It seems particularly important to understand the relation between our work and the the quantum geometric Langlands program initiated by Lurie and Gaitsgory in , and more specifically the relation to the work of Lysenko  and Gaitsgory and Lysenko .
While we have given an interpretation of the decorated spins which are the possible states of the horizontal edges as basis vectors for an evaluation module of , the edges of vertical type have no known similar interpretation. One may ask whether has a two-dimensional module such that the Boltzmann weights in Figure 2 are interpreted as the -matrix for the pair , . We know no reason for such an to exist, except that if it does not, then Theorem 3.1 is an example of a parametrized Yang-Baxter equation that is not predicted by quasitriangularity.
We conclude by reviewing some recent papers which are sequels to this one.
The paper by Brubaker, Buciumas, Bump and Friedberg  was written after the first draft of this one was already posted to the arxiv, and depends on this one. In it we give a very general method of constructing representations of the affine Hecke algebra and show that examples of such representations can come either from the theory of Whittaker functionals on metaplectic -adic groups or from certain Schur-Weyl dualities for quantum affine algebras. Theorem 1 in the present paper is used to prove the two representations mentioned are in fact the same. The paper also contains a more formal discussion of the Drinfeld twisting, an important supplement to the brief treatment we give below in Section 4.
The paper by Brubaker, Buciumas, Bump and Gray  was also written after this one. It uses the Yang-Baxter equations from this paper, and supplementary ones from Gray , to reprove the main result of , which may be expressed as the equality of the partition functions of two different ice models. One of the two ice models is described below in Section 2. The other one is similar but has different weights. The equality of the two partition functions is reminiscent of dualities for physical systems, similar for example to the Kramers-Wannier duality that relates the partition functions of the low-temperature and high temperature Ising models.
In the paper Brubaker, Buciumas, Bump and Gustafsson  it is shown (extending the earlier paper  in the case) that the row transfer matrices for metaplectic ice can be interpreted as operators on the Fermionic Fock space of Kashiwara, Miwa and Stern , after Drinfeld twisting. This is a module for (twisted) . To achieve this one modifies the boundary conditions so that the grid has infinitely many columns. Then a sequence of spins in a row of vertically oriented edges may be interpreted as a basis vector in , and the main theorem is that the row transfer matrices have expressions resembling vertex operators. In particular they are -module homomorphisms. This partially addresses the lack of an interpretation of the vertical edges as -modules noted in Remark 1.2.
Acknowledgements: This work was supported by NSF grants DMS-1406238 (Brubaker) and DMS-1001079, DMS-1601026 (Bump and Buciumas). We thank Gautam Chinta, Solomon Friedberg and Paul Gunnells for their support and encouragement, and David Kazhdan, Daniel Orr and the referee for helpful comments.
2. The partition function
In statistical mechanics, the partition function of a model is a generating function. This means that through its dependence on global parameters of the system (such as temperature) it carries information about properties of the system such as entropy and free energy. Here we are concerned with two-dimensional lattice models that represent metaplectic Whittaker functions, and the global parameters on which it depends are the Langlands parameters.
Consider a finite two-dimensional rectangular grid of fixed size, composed of interior edges connecting to vertices of the grid and boundary edges adjacent to a single vertex in the grid. Every edge will be assigned a spin, which has value or . The spins along the boundary edges will be fixed as part of the data specifying the system; the spins on the interior edges will be allowed to vary. Thus with the spins on the boundary fixed, a state of the system will be an assignment of spins to the interior edges.
We associate a system to any integer partition as follows. The size of the grid will have rows and columns, where may be any integer greater than or equal to . The boundary edge spins are set to be at all left and bottom boundary edges, and on all right edges. The boundary edges along the top of the grid depend on the strict partition with . The spins along the top edge will be in the columns numbered for all and on all remaining columns. See Figure 1 for an example of a state in the system for and .111Strictly speaking, our systems correspond to an integer partition and the choice of sufficiently large integer specifying the number of columns. However, the partition function is unchanged if we increase , and we suppress from the notation.
Define the charge at each horizontal edge in the configuration to be the number of spins at or to the right of the edge, along the same row. (This notion was introduced in .) We also will speak of the charge at a vertex, defined to be the charge on the edge to the right of the vertex. The charges are labeled in Figure 1 as decorations above each vertex.
The state will be called admissible if the four spins on adjacent edges of any vertex are in one of the six configurations in Figure 2 (top). It will be called -admissible if it is admissible and if furthermore every horizontal edge with a spin has charge modulo .
An example of an admissible state is shown in Figure 1. (The appearance of labels on the vertices in the figure will be explained momentarily.) The illustrated state is -admissible only if or , since it has a horizontal edge with charge .
The Boltzmann weight of a state is obtained as a product of weights attached to each vertex in the model. The weight attached to any vertex makes use of a pair of functions and defined on the integers satisfying certain properties which we will now explain.
Let be a fixed positive integer and a fixed parameter. Let be a function of the integer which is periodic modulo , and such that , while if does not divide . Let
Choose nonzero complex numbers and associate one to each row, as indicated in Figure 1. The rows are labeled down to in descending order and is associated with the -th row as in Figure 1. Given a vertex in the -th row, its Boltzmann weight is given in Figure 2 (top). Note that this weight depends on the spins and the charges on adjacent edges, and the row in which it appears. Then the Boltzmann weight of the state is the product of the Boltzmann weights over all vertices in the grid. We often omit the or the in the notation for , as the weights may be stated uniformly for all such choices.
In  and , the functions and are defined using -th order Gauss sums, with , and shown to satisfy the above properties. We will use this specific choice later in Theorem 2.5 and in Section 5 to connect the partition function to metaplectic Whittaker functions. However, only the above properties are required for their study using the Yang-Baxter equation. The function is defined in (5.11) below, and we already defined by (2.1).