On some relatively cuspidal representations: Cases of Galois and inner involutions on ${\rm GL}_n$

Abstract

Relatively cuspidal representations attached to a $p$-adic symmetric space $G/H$ are thought of as the building blocks for all the irreducible $H$-distinguished representations of $G$. This work provides certain new examples of relatively cuspidal representations. We study three examples of symmetric spaces; ${\rm GL}_n(E)/{\rm GL}_n(F)$, ${\rm GL}_{2m}(F)/{\rm GL}_m(E)$, and ${\rm GL}_n(F)/\bigl({\rm GL}_{n-r}(F)\times{\rm GL}_r(F)\bigr)$ where $E/F$ is a quadratic extension of $p$-adic fields. Those representations are given by induction from cuspidal distinguished representations of particular kinds of parabolic subgroups stable under the involution.