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The conservation relation for discrete series representations of metaplectic groups

Let $F$ denote a non-archimedean local field of characteristic zero with odd residual characteristic and let $\widetilde{; ; ; Sp(n)}; ; ; $ denote the rank $n$ metaplectic group over $F$. If $r^{; ; ; \pm}; ; ; (\sigma)$ denotes the first occurrence index of the irreducible genuine representation $\sigma$ of $\widetilde{; ; ; Sp(n)}; ; ; $ in the theta correspondence for the dual pair $(\widetilde{; ; ; Sp(n)}; ; ; , O(V^{; ; ; \pm}; ; ; ))$, the conservation relation, conjectured by Kudla and Rallis, states that $r^{; ; ; +}; ; ; (\sigma) + r^{; ; ; -}; ; ; (\sigma) = 2n$. A purpose of this paper is to prove this conjecture for discrete series which appear as subquotients of generalized principle series where the representation on the metaplectic part is strongly positive. Assuming the basic assumption, we also prove the conservation relation for general discrete series of metaplectic groups by explicitly determining the first occurrence indices.

discrete series; metaplectic groups; theta correspondence

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