On relative cuspidality

Abstract

Let $(\mathbb{G},\mathbb{H})$ be a symmetric pair of reductive groups over a $p$-adic field with $p\ne 2$, attached to the involution $\hat{I}¸$. Under the assumption that there exists a maximally $\hat{I}¸$-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.

Main theorem wrong. (GL4n,Sp4n) provides a counter-example due to Lapid-Offen's classification of relative discrete series: none is strongly relatively cuspidal but there are G^- elliptic tori. Mistake in Section 7: μcannot be chosen uniform with respect to all parabolics. The first 6 sections might help prove existence of relatively cuspidal representations, but we did not find how