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Journal of the Ramanujan Mathematical Society

Volume 33, Issue 3, September 2018  pp. 249–282.

Unramified Godement-Jacquet theory for the spin similitude group

Authors:  Aaron Pollack
Author institution:Department of Mathematics, Institute for Advanced Study, Princeton, NJ 08540

Summary:  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 Mn(F) to define the local standard L-functions on~GLn. 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 GLn(F) with GSpin(V)(F) and Mn(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).

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