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, see [12, Conjecture V.3.2]. Let χ be a unitary character, $s\in C$ and W a symplectic vector space over a non-archimedean field with symmetry group $G\left(W\right)$. Denote by $I(\chi ,s)$ the degenerate principal series representation of $G(W\oplus W)$. Pulling back $I(\chi ,s)$ along the natural embedding $G\left(W\right)\times G\left(W\right)\hookrightarrow G(W\oplus W)$ gives a representation $I_{W,W}(\chi ,s)$ of $G\left(W\right)\times G\left(W\right)$. Let π be an irreducible smooth complex representation of $G\left(W\right)$. We then prove${\dim }_C{\mathrm{Hom}}_{G\left(W\right)\times G\left(W\right)}\left(I_{W,W}\right(\chi ,s),\pi \otimes {\pi }^{\vee })=1.$ 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.