Bessel Distributions and Automorphic Spectral Reciprocity for General Linear Groups

Abstract

In this thesis, we focus on the local and global theory of automorphic forms and relative trace formulae. In the local aspect, I work on the GL(n) Bessel distributions (functions) over non-archimedean local fields and the local Kloosterman-type orbital integrals. We prove the local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [Ba03] [Ba04] [Ba05], the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [JY96] [JY99] and G. Stevens’ approach [Ste87] on estimating certain GL(n) generalized Kloosterman sums. For the global theory, I study the automorphic spectral reciprocity formulae for certain L-functions. We prove a spectral reciprocity formula for the product of GL(n + 1) × GL(n) and GL(n) × GL(n − 1) Rankin-Selberg L-functions (n ≥ 3). This generalizes the work of V. Blomer and R. Khan in [BK17] where the case of n = 2 was established. We extend the method develooped by R. Nunes in [Nun20] from n = 2 to general n ≥ 3, by using the global zeta integrals for the GL(n + 1) × GL(n) Rankin-Selberg L-functions, the spectral theory of square-integrable automorphic forms, the language of automorphic representations and representation theoretic view.