On a Casselman-Shalika type formula for unramified Speh representations

TL;DR

This paper derives a new explicit formula for unramified Speh representations' spherical functions, linking them to Hall--Littlewood polynomials and simplifying previous complex proofs.

Contribution

It provides a simple, combinatorial Casselman-Shalika type formula for Speh representations, connecting representation theory with Hall--Littlewood polynomials.

Findings

Formula expresses Speh representation values via Hall--Littlewood polynomials.

Proof uses Macdonald's formula and Ginzburg--Kaplan integral computations.

Addresses a question posed by Lapid-Mao.

Abstract

We give a Casselman-Shalika type formula for unramified Speh representations. Our formula computes values of the normalized spherical element of the $(k,c)$ model of a Speh representation at elements of the form $\operatorname{diag}\left(g, I_{(k-1)c}\right)$, where $g \in \mathrm{GL}_c\left(F\right)$ for a non-archimedean local field $F$. The formula expresses these values in terms of modified Hall--Littlewood polynomials evaluated at the Satake parameter attached to the representation. Our proof is combinatorial and very simple. It utilizes Macdonald's formula and the unramified computation of the Ginzburg--Kaplan integral. This addresses a question of Lapid-Mao.