Cuspidal representations of quaternionic $GL_n(D)$ with symplectic periods
Vincent Sécherre, Shaun Stevens
TL;DR
The paper resolves Prasad's conjecture in the cuspidal case for inner forms $G={ m GL}_n(D)$ of ${ m GL}_{2n}(F)$ with $D$ non-split and odd residue characteristic, showing that a cuspidal representation is ${ m Sp}_n(D)$-distinguished iff its Jacquet--Langlands transfer to ${ m GL}_{2n}(F)$ is non-cuspidal. The proof blends endo-class and simple-character techniques from Bushnell--Kutzko with a detailed involution analysis to produce a σ-stable type whose κ-part is distinguished, reducing the problem to unitary-distinction questions for finite fields and a global-to-local argument via Verma-type globalization. The work also provides a depth-zero description of the base-change map $oldsymbol{ m b}_{D/F}$ and clarifies how depth and transfer interact with distinction, connecting local period problems to non-cuspidal JL transfers through a refined type theory. These results illuminate Prasad's conjecture in the cuspidal regime and emphasize the role of endo-classes and simple characters in period problems for $p$-adic groups.
Abstract
We prove a conjecture of Prasad predicting that a cuspidal representation of $GL_n(D)$, for an integer $n > 1$ and a non-split quaternion algebra $D$ over a non-Archimedean locally compact field $F$ of odd residue characteristic, has a symplectic period if and only if its Jacquet--Langlands transfer to $GL_{2n}(F)$ is non-cuspidal.