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 $G={\mathbb G}_m\times{\mathrm Sp}_{2n}$. It is associated to the Langlands $\gamma$-functions attached to any irreducible admissible representations $\chi\otimes\pi$ of $G(F)$ and the standard representation $\rho$ of the dual group $G^\vee({\mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${\rm GL}_1(F)$, which is associated to a $\gamma$-function $\beta_\psi(\chi_s)$ (a product of $n+1$ certain abelian $\gamma$-functions). Our work on ${\rm GL}_1(F)$ 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 when $n=0$. The approach is to use the compactification of ${\rm Sp}_{2n}$ in the Grassmannian variety of ${\rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${\rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${\rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.