Smooth representations and Hecke algebras of p-adic GL_n(D)

2023/06/22 by Mondal, Amiya Kumar, Pattanayak, Basudev
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2306.12938

Abstract

The main question we are going to address in this paper is: How much does the representation theory of the p-adic group GL_n(D) depend on the p-adic division algebra D? Let D be a central division algebra defined over some locally compact non-archimedean local field. Using Bushnell-Kutzko theory of types and Sécherre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category R ( GL_n(D) ) of smooth complex representations of GL_n(D) do not depend on the p-adic division algebra D. In particular, when n=1 or 2, the category R ( GL_n(D) ) does not depend on the p-adic division algebra D.

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