Combinatorics of Iwahori Whittaker Functions

TL;DR

This paper provides a combinatorial evaluation of Iwahori Whittaker functions on metaplectic covers of GL(n), introducing new colored combinatorial data and an explicit Iwahori decomposition, extending previous work on spherical Whittaker functions.

Contribution

It introduces new colored combinatorial data and an explicit Iwahori decomposition, extending the evaluation of Whittaker functions to metaplectic covers.

Findings

Equivalence of colored Lusztig data, Gelfand-Tsetlin patterns, and lattice models.

Explicit Iwahori decomposition for the maximal unipotent subgroup.

Extension of McNamara's evaluation to metaplectic covers.

Abstract

We give a combinatorial evaluation of Iwahori Whittaker functions for unramified genuine principal series representations on metaplectic covers of the general linear group over a non-archimedean local field. To describe the combinatorics, we introduce new combinatorial data that we call colored data: colored Lusztig data, colored Gelfand-Tsetlin patterns, and colored lattice models. We show that all three are equivalent. To achieve the result, we give an explicit Iwahori decomposition for the maximal unipotent subgroup of a split reductive group which gives the parametrization of the generalized Mirkovi\'c-Vilonen cycles in the affine flag varieties and is of interest in itself. Our result is based and naturally extends Peter McNamara's evaluation of the metaplectic spherical Whittaker function in terms of Lusztig data.