Whittaker functions and Demazure characters

TL;DR

This paper links Iwahori--Whittaker functions with Demazure characters, providing explicit formulas and generalizing key formulas in p-adic representation theory under specific combinatorial conditions.

Contribution

It introduces a new explicit formula connecting Iwahori--Whittaker functions and Demazure characters, extending the Casselman--Shalika formula and generalizing Bump--Nakasuji's results.

Findings

Derived an explicit formula for Iwahori--Whittaker functions using Demazure characters.

Generalized the Casselman--Shalika formula under certain combinatorial conditions.

Computed the transition matrix between bases for Iwahori fixed vectors in induced representations.

Abstract

In this paper, we consider how to express an Iwahori--Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman--Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a p-adic group; this generalizes a result of Bump--Nakasuji.