Unramified Godement-Jacquet theory for the spin similitude group

TL;DR

This paper suggests an analogous approach to the classical Godement-Jacquet theory for $ ext{SO}(V)$ groups, replacing matrices with Clifford algebras to facilitate local unramified $L$-function calculations.

Contribution

It introduces an approximate Godement-Jacquet framework for $ ext{SO}(V)$ using Schwartz-Bruhat functions on Clifford algebras, extending the classical theory beyond $ ext{GL}_n$.

Findings

Simplifies local unramified $L$-function calculations for $ ext{SO}(V)$

Proposes a new method replacing matrices with Clifford algebras

Provides evidence for an analogous theory to Godement-Jacquet for special orthogonal groups.

Abstract

Suppose $F$ is a non-archimedean local field. The classical Godement-Jacquet theory is that one can use Schwartz-Bruhat functions on $n \times n$ matrices $M_n(F)$ to define the local standard $L$-functions on $\mathrm{GL}_n$. 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 $\mathrm{SO}(V)$: One replaces $\mathrm{GL}_n(F)$ with $\mathrm{GSpin}(V)(F)$ and $M_n(F)$ with $\mathrm{Clif}(V)(F)$, the Clifford algebra of $V$. More precisely, we explain how a few different local unramified calculations for standard $L$-functions on $\mathrm{SO}(V)$ can be done easily using Schwartz-Bruhat functions on $\mathrm{Clif}(V)(F)$. We do not attempt any of the ramified or global theory of $L$-functions on $\mathrm{SO}(V)$ using Schwartz-Bruhat functions on $\mathrm{Clif}(V)$.