On relative cuspidality

TL;DR

This paper extends existing proofs to show that certain reductive p-adic groups have strongly relatively cuspidal representations, confirming prior conjectures in the field.

Contribution

It generalizes Beuzart-Plessis' proof to a broader class of symmetric pairs, establishing the existence of strongly relatively cuspidal representations.

Findings

Proves existence of strongly relatively cuspidal representations for specified symmetric pairs.

Confirms conjectures of Kato and Takano.

Extends methods to cases with a maximally θ-split torus that is anisotropic modulo its split component.

Abstract

Let $(\mathbb{G},\mathbb{H})$ be a symmetric pair of reductive groups over a $p$-adic field with $p\neq 2$, attached to the involution $\theta$. Under the assumption that there exists a maximally $\theta$-split torus in $\mathbb{G}$, which is anisotropic modulo its intersection with the split component of $\mathbb{G}$, we extend Beuzart-Plessis' proof of existence of cuspidal representations, and prove that $\mathbb{G}(F)$ admits strongly relatively cuspidal representations. This confirms expectations of Kato and Takano.