On a Casselman-Shalika type formula for unramified Speh representations

2024/07/10 by Elad Zelingher, Zelingher, Elad
Mathematics · #11F66 #22E50 #33D52 #Advanced Algebra and Geometry #FOS: Mathematics #Geology #Holomorphic and Operator Theory #Mathematical functions and polynomials #Mathematics #Number Theory (math.NT) #Primary: 11F70. Secondary: 05E05 #Pure mathematics #Representation Theory (math.RT) #Type (biology)

paper · pdf · doi:10.48550/arxiv.2407.07774

openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 ∈ GL_c(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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