Measuring the Space of Metaplectic Whittaker Functions

TL;DR

This paper investigates the dimension of the space of metaplectic Whittaker functions, providing explicit formulas and exploring their connection to quantum group modules, revealing when certain homomorphisms are well-defined.

Contribution

It derives two explicit formulas for the dimension of Whittaker function spaces on metaplectic covers and analyzes their relation to quantum group modules.

Findings

Two precise formulas for the dimension of Whittaker function spaces.

Conditions under which the homomorphism to quantum group modules is well-defined.

Abstract

Whittaker functions are special functions that arise in $p$-adic number theory and representation theory. They may be defined on representations of reductive groups as well as their metaplectic covering groups: fascinatingly, many of their number theoretic applications survive the transition between the reductive and metaplectic cases. However, one notable difference is that the space of Whittaker functions on a reductive group over a nonarchimedean local field $F$ is one-dimensional, whereas this is no longer true in the metaplectic case. In a previous paper, the second author showed that the dimension of the space of Whittaker functions on an arbitrary $n$-fold metaplectic cover of $GL_r(F)$ can be counted in terms of the number of solutions to a particular system of linear Diophantine equations in terms of $n$ and $r$. In this paper, we calculate two precise formulae for $\dim(\mathfrak{W})$, one inspired by viewing this system as a homogenous specialization of an inhomogenous system and the other by the structure of the coroot lattice of $GL_r(F)$. Then we use these formulae to investigate a homomorphism between $\mathfrak{W}$ and a particular quantum group module, built by the second author in a previous paper, and show precisely when this map is well-defined for any choice of basis for $\mathfrak{W}$.