Hecke Modules from Metaplectic Ice

TL;DR

This paper introduces a new algebraic framework connecting affine Hecke modules, p-adic group representations, and quantum group R-matrices, providing fresh proofs for properties of metaplectic Whittaker functions.

Contribution

It develops a broad class of affine Hecke algebra modules linked to p-adic and quantum group representations, offering new algebraic proofs for metaplectic Whittaker function results.

Findings

New algebraic framework for affine Hecke modules

Connections between p-adic representations and quantum R-matrices

Algebraic proofs of properties of metaplectic Whittaker functions

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 $p$-adic 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.