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

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 $${\dim }_{\mathbb{C}}{\mathrm{H}\mathrm{o}\mathrm{m}}_{G(W)\times G(W)}(I_{W,W}(\chi ,s),\pi \otimes {\pi }^{\vee })=1.$$ 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.