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The nonarchimedean theta correspondence for

Authors:	Brooks Roberts

Institution:	Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada

Abstract:	In this paper we consider the theta correspondence between the sets $\operatorname{Irr} (\operatorname{GSp} (2,k))$ and $\operatorname{Irr} (\operatorname{GO} (X))$ when is a nonarchimedean local field and $\dim _{k} X =4$ . Our main theorem determines all the elements of $\operatorname{Irr} (\operatorname{GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname{Irr} (\operatorname{O} (X))$ that occur in the theta correspondence between $\operatorname{Irr} (\operatorname{Sp} (2,k))$ and $\operatorname{Irr} (\operatorname{O} (X))$ . We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

Keywords:	Nonarchimedean theta correspondence {$\GSp (2)$ $\GO (4)$}

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