Unramified Godement-Jacquet theory for the spin similitude group

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 $\mathrm{Clif}(V)$.

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