2012/07/26 by Yumiko Hironaka, Hironaka, Yumiko, Yasushi Komori +1
Mathematics · #11F70 #11F85 (Primary) 11E95 #22E50 #33D52 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories #math.NT #msc:11E95 #msc:11F70 #msc:11F85 #msc:22E50 #msc:33D52
paper · pdf · doi:10.48550/arxiv.1207.6189
to appear in International Journal of Number Theory
openalex publication_date 2012/07/26 · arxiv created 2013/09/09 · arxiv updated 2013/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the space X of unitary hermitian matrices over \frp-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e., 2∉ \frp. In the latter half we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall-Littlewood symmetric polynomials of type Cn appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show the Schwartz space \SKX is a free Hecke algebra \hec-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on \SKX.