Harmonic Analysis and Gamma Functions on Symplectic Groups

Over a p p -adic local field F F of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group G = G m × S p 2 n G={G}_m× {Sp}₂ₙ . It is associated to the Langlands γ γ -functions attached to any irreducible admissible representations χ ⊗ π χ ⊗ π of G ( F ) G(F) and the standard representation ρ ρ of the dual group G ∨ ( C ) 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 G L 1 ( F ) {GL}_1(F) , which is associated to a γ γ -function β ψ ( χ s ) β _ψ (χ _s) (a product of n + 1 n+1 certain abelian γ γ -functions). Our work on G L 1 ( F ) {GL}_1(F) plays an indispensable role in the development of our work on G ( F ) 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 n=0 . The approach is to use the compactification of S p 2 n {Sp}₂ₙ in the Grassmannian variety of S p 4 n {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 L -functions of S p 2 n {Sp}₂ₙ . The method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard L L -function of G L n {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.