Computations of orbital integrals and Shalika germs

2022/07/22 by Cheng-Chiang Tsai, Tsai, Cheng-Chiang
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2207.11296

openalex publication_date 2022/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a reductive group G over a non-archimedean local field, with some assumptions on (residue) characteristic we give an method to compute certain orbital integrals using a method close to that of Goresky-Kottiwitz-MacPherson but in a different language. These orbital integrals allow us to compute the Shalika germs at some ``very elliptic'' elements in terms of number of rational points on some quasi-finite covers of the Hessenberg varieties of GKM, which are subvarieties of (partial) flag varieties. Such values of Shalika germs determine the Harish-Chandra local character expansions of the so-called very supercuspidal representations.

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