Journal of the Ramanujan Mathematical Society

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Unramified Godement-Jacquet Theory for the Spin Similitude Group


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1 Department of Mathematics, Institute for Advanced Study, Princeton, NJ 08540
     

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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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  • Unramified Godement-Jacquet Theory for the Spin Similitude Group

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Authors

Aaron Pollack
Department of Mathematics, Institute for Advanced Study, Princeton, NJ 08540

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 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 ).

References