A Casselman-Shalika formula for the Shalika model of GL_n

TL;DR

This paper extends the Casselman-Shalika formula to the Shalika model of GL_n, providing explicit formulas for unramified representations and simplifying calculations related to lifts from orthogonal groups.

Contribution

It develops a new variant of the Casselman-Shalika method tailored for the Shalika model, enabling more straightforward computations of periods for unramified representations.

Findings

Derived explicit formulas for the Shalika model of GL_n.

Reduced complex calculations to straightforward group-based computations.

Enhanced the Casselman-Shalika method with a new variant.

Abstract

The Casselman-Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of p-adic groups that are associated to unique models (i.e. multiplicity-free induced representations). We apply this method to the case of the Shalika model of GL_n, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y.Hironaka, and in effect reduce many such problems to straightforward calculations on the group.