Harmonic Analysis and Gamma Functions on Symplectic Groups

Over a p-adic local field F of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group G = G_m × Sp_2n. It is associated to the Langlands γ-functions attached to any irreducible admissible representations χ ⊗ π of G(F) and the standard representation ρ of the dual group G^∨(C), and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal (Braverman and Kazhdan, 2000) for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on GL₁(F), which is associated to a γ-function β_ψ(χ_s) (a product of n + 1 certain abelian γ-functions). Our work on GL₁(F) plays an indispensable role in the development of our work on G(F). These two types of harmonic analyses both specialize to the well-known local theory developed in Tate’s thesis (Tate, 1950) when n = 0. The approach is to use the compactification of Sp_2n in the Grassmannian variety of Sp_4n, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis (1986) and many other works) on the doubling local zeta integrals for the standard L-functions of Sp_2n. The method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard L-function of GL_n and is expected to work for all classical groups. We will consider the Archimedean local theory and the global theory in our future work.