On a Casselman-Shalika type formula for unramified Speh representations

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 $\mathrm{diag}\left(g,I_{(k-1)c}\right)$, 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.

13 pages. Comments are welcome! v4: This version contains all changes that were made for submission in Proceedings of the AMS