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 diag(g, I(k-1)c), where g ∈ GLc(F) 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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