Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series

Abstract

In this paper we prove a conjecture of Kudla and Rallis. Let $\chi$ be a unitary character, $s\in \mathbb{C}$ and $W$ a symplectic vector space over a non-archimedean field with symmetry group $G(W)$. Denote by $I(\chi ,s)$ the degenerate principal series representation of $G(W\oplus W)$. Pulling back $I(\chi ,s)$ along the natural embedding $G(W)\times G(W)\hookrightarrow G(W\oplus W)$ gives a representation $I_{W,W}(\chi ,s)$ of $G(W)\times G(W)$. Let $\pi$ be an irreducible smooth complex representation of $G(W)$. We then prove \[\dim _\mathbb{C}\mathrm{Hom}_{G(W)\times G(W)}(I_{W,W}(χ,s),π\otimes π^\lor)=1.\] We also give analogous statements for $W$ orthogonal or unitary. This gives in particular a new proof of the conservation relation of the local Theta correspondence for symplectic-orthogonal and unitary dual pairs.

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References in corpus (1)

• Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field