2018/05/10 by U. K. Anandavardhanan, Anandavardhanan, U. K., Nadir Matringe +1
Computer Science · Mathematics · #20C33 #22E50 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1805.04047
openalex publication_date 2018/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E/F be a quadratic extension of finite fields. By a result of Gow, an irreducible representation π of G = \rm GLn(E) has at most one non-zero H-invariant vector, up to multiplication by scalars, when H is \rm GLn(F) or \rm U(n,E/F). If π does have an H-invariant vector it is said to be H-distinguished. It is known, from the work of Gow, that H-distinction is characterized by base change from \rm U(n,E/F), due to Kawanaka, when H is \rm GLn(F) (resp. from \rm GLn(F), due to Shintani, when H is \rm U(n,E/F)). Assuming π is generic and H-distinguished, we give an explicit description of the H-invariant vector in terms of the Bessel function of π. Let ψ be a non-degenerate character of NG/NH and let Bπ,ψ be the (normalized) Bessel function of π on the ψ-Whittaker model. For the H-average Wπ,ψ = (1)/(|H|) ∑h∈ H π(h) Bπ,ψ of the Bessel function, we prove that Wπ,ψ(In) = \frac\rm dimρ\rm dim π ⋅ \frac|\rm GLn(E)||\rm GLn(F)| |\rm U(n,E/F)|, where ρ is the representation of \rm U(n,E/F) (resp. \rm GLn(F)) that base changes to π when H is \rm GLn(F) (resp. \rm U(n,E/F)). As an application we classify the members of a generic L-packet of \rm SLn(E) that admit invariant vectors for \rm SLn(F). Finally we prove a p-adic analogue of our result for square-integrable representations in terms of formal degrees by employing the formal degree conjecture of Hiraga-Ichino-Ikeda \citehii08.