Supercuspidal representations of GL_n(F) distinguished by an orthogonal involution

2020/11/14 by Jiandi Zou, Zou, Jiandi
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2011.07349

openalex publication_date 2020/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a non-archimedean locally compact field of residue characteristic p≠2, let G=GL_n(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 Hom_H(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.

Citations

• A `relative' local Langlands correspondence
• Supercuspidal representations of GL_n(F) distinguished by a unitary involution

Related

• Supercuspidal representations of GL_n(F) distinguished by a unitary involution
• Tame Supercuspidal Representations of \GL_n Distinguished by Orthogonal Involutions
• Distinguished Tame Supercuspidal Representations and Odd Orthogonal Periods
• Representations lisses de GL(m,D), IV : representations supercuspidales
• The supercuspidal representations of p-adic classical groups