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On a Casselman-Shalika type formula for unramified Speh representations (2407.07774v4)
Published 10 Jul 2024 in math.RT and math.NT
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.