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Title: Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials
Abstract We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters $$g_i$$ 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 $$g_i$$ 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 $$g_i$$ 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 $$g_i$$ 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.  more » « less
Award ID(s):
2001537 1623501
PAR ID:
10259996
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Selecta Mathematica
Volume:
27
Issue:
3
ISSN:
1022-1824
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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