Over a p-adic local field F of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group G = Gm × Sp2n. It is associated to the Langlands ?-functions attached to any irreducible admissible representations ? ? ? of G(F) and the standard representation ? 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 ?-function ??(?s) (a product of n + 1 certain abelian ?-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 Sp2n in the Grassmannian variety of Sp4n, 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 Sp2n.
The method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard L-function of GLn and is expected to work for all classical groups. We will consider the Archimedean local theory and the global theory in our future work.
Chapters
1. Introduction
2. Local Theory of Piatetski-Shapiro and Rallis
3. Functional Equation for $\beta _\psi (\chi _s)$
4. Harmonic Analysis for $\beta _\psi (\chi _s)$
5. $\eta _{\mathrm {pvs},\psi }$-Fourier Transform on $X_{P_{\Delta }}$
6. Harmonic Analysis on ${\mathbb {G}}_m\times {\mathrm {Sp}}_{2n}$
7. Multiplicity One and Gamma Functions
8. Theorems 1.2, 1.3, and 1.4
Dihua Jiang, University of Minnesota, Minneapolis, MN.
Zhilin Luo, University of Chicago, IL.
Lei Zhang, National University of Singapore, Singapore.
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