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

TL;DR

This paper proves that the restricted degenerate principal series $I_{W,W}(χ,s)$, obtained by pulling back the degenerate principal series $I(χ,s)$ along the embedding $G(W)×G(W)↪G(W⊕W)$, has a unique (up to scalar) copy mapping into $π⊗χπ^{ extlor}$ for every irreducible smooth representation $π$ of $G(W)$, i.e. $ ext{dim Hom}_{G(W)×G(W)}(I_{W,W}(χ,s),π⊗χπ^{ extlor})=1$. The authors achieve this through a threefold strategy: (i) a standard filtration of $I_{W,W}(χ,s)$ by $G(W)×G(W)$-orbits on the Lagrangian Grassmannian, with explicit subquotients; (ii) a detailed boundary/Jacquet-module analysis giving explicit realizations of the subquotients and a filtration on the Jacquet module; and (iii) a multiplicity-one input from Minguez for GL-r representations together with MVW-duality to control duals. A Rankin-type zeta functional is constructed to show the existence of a nonzero map, while the filtration together with Minguez’ and MVW tools yields the desired uniqueness; the result also yields a new proof of the local conservation relation for symplectic–orthogonal and unitary dual pairs. The work clarifies the role of boundary components in the degenerate principal series and provides a robust framework potentially adaptable to broader dual-pair settings, subject to metaplectic considerations.

Abstract

In this paper we prove a conjecture of Kudla and Rallis. Let $χ$ 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 $I(χ,s)$ the degenerate principal series representation of $G(W\oplus W)$. Pulling back $I(χ,s)$ along the natural embedding $G(W)\times G(W)\hookrightarrow G(W\oplus W)$ gives a representation $I_{W,W}(χ,s)$ of $G(W)\times G(W)$. Let $π$ 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.

Theorems & Definitions (49)

• Theorem 1: Howe Duality Conjecture for Type I, Waldspurger, GanTakeda, Gan2017
• Theorem 2: Conservation Relation, sun
• Theorem 3
• Theorem 4
• Theorem 5: Minguez
• Proposition 6
• Lemma 2.1.1
• proof
• Proposition 2.2.1: Bernstein2 Proposition 1.14
• Proposition 2.2.2: Bernstein2 1.16