Abstract
We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters g_i, and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters g_i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters g_i. It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the g_i, and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.
Highlights
This paper contains two main results concerning a somewhat mysterious action of the Weyl group of a reductive Lie group on the algebra of rational functions on its torus. This action was first introduced in type A by Kazhdan and Patterson [24], and in full generality by Chinta and Gunnells [15,16], who used it to obtain formulas for the local parts ( p-parts) of Weyl group multiple Dirichlet series
The key role in the proof is played by a certain representation of the affine Hecke algebra that we construct in Theorem 3.7 below, and which we refer to as the metaplectic representation
Let us conclude with remarking that the localization procedure we use in this paper is instrumental in Cherednik’s construction of quantum affine Knizhnik–Zamolodchikov equations attached to affine Hecke algebra modules
Summary
This paper contains two main results concerning a somewhat mysterious action of the Weyl group of a reductive Lie group on the algebra of rational functions on its torus. There is a striking analogy between the Chinta–Gunnells setting and the theory of Macdonald polynomials [12,29,31] The latter are a family of orthogonal polynomials on the torus that depend on two or three “root-length” parameters, and which generalize many important polynomials in representation theory and algebraic combinatorics, including spherical functions for real and p-adic groups. In the present paper we introduce, without proofs, the metaplectic polynomials in type A, where many of the essential ideas already appear This is the setting of [24] and of Macdonald’s book on symmetric functions [30], which is of considerable independent interest in algebraic combinatorics. This will be presented in a forthcoming paper [36], which will include the detailed proofs
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