Non-Archimedean Whittaker functions as characters: a probabilistic approach to the Shintani-Casselman-Shalika formula

TL;DR

This paper introduces a probabilistic method to derive the Shintani-Casselman-Shalika formula for non-Archimedean groups with minuscule cocharacters, connecting Whittaker functions to characters of the Langlands dual group.

Contribution

It develops a novel probabilistic approach using minuscule random walks and reflection principles to prove the formula, extending previous Archimedean results.

Findings

Poisson kernel formula for Whittaker functions

Probability of random walk staying in Weyl chamber

Connection to Weyl character formula

Abstract

For a reductive group $G$ over a non-Archimedean local field (e.g $GL_n( \mathbb{Q}_p )$ ), Jacquet's Whittaker function is essentially proportional to a character of an irreducible representation of the Langlands dual group $G^\vee( \mathbb{C} )$ ( a Schur function if $G = GL_n( \mathbb{Q}_p )$). We propose a probabilistic approach to this claim, known as the Shintani-Casselman-Shalika formula, when the group $G$ has at least one minuscule cocharacter in the coweight lattice. Our presentation goes along the following lines. Thanks to a minuscule random walk $W^{(z)}$ on the coweight lattice and a related random walk on the Borel subgroup, we establish a Poisson kernel formula for the non-Archimedean Whittaker function. The expression and its ingredients are similar to the one previously obtained by the author in the Archimedean case. A simple manipulation reduces the problem to evaluating the probability of $W^{(z)}$ never exiting the Weyl chamber. Then, an implementation of the reflection principle forces the appearance of the Weyl character formula and therefore retrieves characters of $G^\vee\left( \mathbb{C} \right)$. The construction of the random walk on the Borel subgroup requires some care. It is extracted from a spherical random walk whose increments have a distribution that can be understood as elements from the spherical Hecke algebra.