Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution

Abstract

Let F be a non-archimedean locally compact field of residue characteristic p = 2, let G = GLn(F ) and let H be an orthogonal subgroup of G. For π a complex smooth supercuspidal representation of G, we give a full characterization for the distinguished space HomH(π, 1) being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of π in the space of smooth functions on the set of symmetric matrices in G, as a complex vector space, is non-zero and of dimension four, if and only if the central character of π evaluating at −1 is 1.