The twisted Satake isomorphism and Casselman-Shalika formula

TL;DR

This paper establishes a new perspective on the unramified Whittaker space for split adjoint groups, leading to a derivation of the Casselman-Shalika formula through the identification with skew-invariant functions.

Contribution

It introduces a novel approach to relate the Whittaker space to skew-invariant functions, providing a new proof of the Casselman-Shalika formula for split adjoint groups.

Findings

Unramified Whittaker space identified with skew-invariant functions

Derived Casselman-Shalika formula from this identification

Provides a new conceptual framework for understanding these structures

Abstract

For an arbitrary split adjoint group we identify the unramified Whittaker space with the space of skew-invariant functions on the lattice of coweights and deduce from it the Casselman-Shalika formula.