On some relatively cuspidal representations of $mathrm{GL}_{n}$ over $p$-adic fields (Representation theory and various problems in algebra, analysis, and geometry)

Abstract

This work provides certain examples of relatively cuspidal representations, attached to symmetric spaces over p-adic fields. We study three concrete examples of symmetric spaces: ・GL_{n}(E)/GL_{n}(F), Galois involution; ・GL_{2m}(F)/GL_{m}(E), inner involution (I); ・GL_{n}(F)/(GL_{n-r}(F)cross GL_{r}(F)), inner involution (II), where E/F is a quadratic extension of p-adic fields. Examples are given by inductions from particular kinds of parabolic subgroups stable under the involution.