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 $p\ne 2$, let $G={\mathrm{GL}}_n(F)$ and let $H$ be an orthogonal subgroup of $G$. For $\pi$ a complex smooth supercuspidal representation of $G$, we give a full characterization for the distinguished space ${\mathrm{Hom}}_H(\pi ,1)$ 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 $\pi$ 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 $\pi$ evaluating at $-1$ is $1$.

arXiv commentPublished version in JNT with minor changes

References in corpus (4)

• A `relative' local Langlands correspondence
• Galois self-dual cuspidal types and Asai local factors
• Comparison of Bushnell-Kutzko and Yu's constructions of supercuspidal representations
• Supercuspidal representations of ${\mathrm{GL}}_n(F)$ distinguished by a unitary involution