Unramified Godement-Jacquet theory for the spin similitude group

2017/04/19 by Aaron Pollack, Pollack, Aaron
Chemistry · Mathematics · #Advanced Algebra and Geometry #Advanced NMR Techniques and Applications #FOS: Mathematics #Molecular spectroscopy and chirality #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1704.05897

openalex publication_date 2017/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose F is a non-archimedean local field. The classical Godement-Jacquet theory is that one can use Schwartz-Bruhat functions on n × n matrices M_n(F) to define the local standard L-functions on GL_n. The purpose of this partly expository note is to give evidence that there is an analogous and useful "approximate" Godement-Jacquet theory for the standard L-functions on the special orthogonal groups SO(V): One replaces GL_n(F) with GSpin(V)(F) and M_n(F) with Clif(V)(F), the Clifford algebra of V. More precisely, we explain how a few different local unramified calculations for standard L-functions on SO(V) can be done easily using Schwartz-Bruhat functions on Clif(V)(F). We do not attempt any of the ramified or global theory of L-functions on SO(V) using Schwartz-Bruhat functions on Clif(V).

Citations

• Normalized intertwining operators and nilpotent elements in the Langlands dual group

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