# 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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