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arXiv · 2407.07774

On a Casselman-Shalika type formula for unramified Speh representations

Elad Zelingher
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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.

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Elad Zelingher. 2024-07-10. On a Casselman-Shalika type formula for unramified Speh representations. https://doi.org/10.1090/proc%2F17185

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