Hecke modules from metaplectic ice

Abstract

Abstract. We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of padic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.An important tool in studying representations of p-adic groups is the affine Hecke algebra. Ion [28], Brubaker, Bump and Licata [8], Brubaker, Bump and Friedberg [7,9] and other papers considered representations of Hecke algebras related to models of representations of p-adic groups. These works mainly focused on unique functionals such as the Whittaker, spherical and Bessel functionals, but, as we shall show here, the ideas are also applicable to certain non-unique functionals on central extensions of p-adic groups. The goal is to show that these functionals act as Hecke algebra module maps to various target spaces. This results in recursion formulas for Iwahori fixed vectors in the model using Demazure-like operators based on the Hecke action. It allows for effective computation, provides links to the geometry of associated Bott-Samelson varieties, and proves that these functions match important classes of symmetric functions, such as various specializations of non-symmetric Macdonald polynomials.In a separate direction, Brubaker, Bump and Friedberg [6] showed that the values of the spherical Whittaker function for unramified principal series of GL r (F ), where F is a p-adic field, may be realized as partition functions of a solvable lattice model. A similar model for the spherical Whittaker functions for covers of GL r (F ) -a situation in which the Whittaker functional is no longer unique -was given in Brubaker, Bump, Chinta, Friedberg and Gunnells [5]. However, except when the cover is trivial, this model was not solvable in the sense of Baxter [3]. The Boltzmann weights were altered to obtain a solvable model in Brubaker, Buciumas and Bump [4]. This is the "metaplectic ice" of the title. Moreover, [4] provided a new algebraic framework, relating the models to the R-matrices of a Drinfeld twist of the quantum affine Lie superalgebra U q ( gl(n|1)). As a biproduct of this discovery, a relationship between the scattering matrices of the intertwining integrals on Whittaker coinvariants and R-matrices of a Drinfeld twist of U q ( gl(n)) came to light.In the present paper, we will unify these directions by providing simultaneous foundations for both. The search for new foundations is motivated by the following consideration. In [7], we took the point of view of the universal principal series. This means that the induced character becomes C[Λ]...

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

“…Closely related to it is the role of the localization procedure for type A in the context of integrable vertex models with U q ( sl n )-symmetry, in the special cases that the associated braid group action descends to an affine Hecke algebra action, in which case the localization procedure is often referred to as Baxterization (see, e.g., [12,37] and references therein). This is exactly the context in which the metaplectic Whittaker function can be realized as a partition function, the corresponding integrable model being "metaplectic ice", see [1][2][3]. It is an intriguing open question whether there is a conceptual connection with the current interpretation of the Chinta-Gunnells action through localization.…”

A Yang–Baxter equation for metaplectic ice

“…In Reshetikhin [39] Section 3, Drinfeld twisting is used to obtain multiparameter deformations of U q (sl(n)). We explain in [5], Section 4 (at least for the gl(n) part) how the Drinfeld twist on the quantum group produces the desired change to the R-matrix that we present below. Notice that in Figure 3, if we have a nonzero weight for the vertex of form Proof.…”

“…Here g(a − b) is an n-th order Gauss sum. These are not present in the out-of-the-box U √ v ( gl(n)) R-matrix, but may be introduced by Drinfeld twisting that will be discussed in Section 4 (see also Section 4 of [5]). This procedure does not affect the validity of the Yang-Baxter equations, but is needed for comparison with the R-matrix for the partition functions of metaplectic ice giving rise to Whittaker functions.…”

“…We conclude by reviewing some recent papers which are sequels to this one. The paper by Brubaker, Buciumas, Bump and Friedberg [5] 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 p-adic groups or from certain Schur-Weyl dualities for quantum affine algebras.…”

Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

“…It was clarified by Nakasuji and Naruse [19] that the basis µ w is essentially the "Yang-Baxter basis" of Lascoux, Leclerc and Thibon [17], and the consistency of the definition follows from the Yang-Baxter equation. The appearance of the Yang-Baxter equation in the context of p-adic intertwining operators is then related to the viewpoint in Brubaker, Buciumas, Bump and Friedberg [4]. Suppose that s = s α is a simple reflection.…”