Test vectors for finite periods and base change

Let E/F be a quadratic extension of finite fields. By a result of Gow, an irreducible representation π of G = GL_n(E) has at most one non-zero H-invariant vector, up to multiplication by scalars, when H is GL_n(F) or 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 U(n,E/F), due to Kawanaka, when H is GL_n(F) (resp. from GL_n(F), due to Shintani, when H is 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 N_G/N_H and let B_π,ψ be the (normalized) Bessel function of π on the ψ-Whittaker model. For the H-average W_π,ψ = 1/|H|∑_h∈ Hπ(h) B_π,ψ of the Bessel function, we prove that W_π,ψ(I_n) = dimρ/ dimπ·| GL_n(E)|/| GL_n(F)| | U(n,E/F)|, where ρ is the representation of U(n,E/F) (resp. GL_n(F)) that base changes to π when H is GL_n(F) (resp. U(n,E/F)). As an application we classify the members of a generic L-packet of SL_n(E) that admit invariant vectors for SL_n(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 .