Plancherel decomposition of Howe duality and Euler factorization of automorphic functionals

TL;DR

This paper develops a general Plancherel formula for the Weil representation in the context of reductive dual pairs and demonstrates how local factors of automorphic functionals are determined by the Langlands correspondence, revealing a broad Euler factorization principle.

Contribution

It establishes a comprehensive Plancherel formula for the Weil representation and connects local factors of automorphic functionals to the Langlands correspondence through this framework.

Findings

Plancherel formula for Weil representation derived

Euler factorization principle demonstrated in Howe duality context

Local factors determined by Langlands parameters via Plancherel

Abstract

There are several global functionals on irreducible automorphic representations which are Eulerian, that is: pure tensors of local functionals, when the representation is written as an Euler product $\pi = \otimes'_v \pi_v$ of local representations. The precise factorization of such functionals is of interest to number theorists and is -- naturally -- very often related to special values of $L$-functions. The purpose of this paper is to develop in full generality the Plancherel formula for the Weil or oscillator representation, considered as a unitary representation of a reductive dual pair, and to use it in order to demonstrate a very general principle of Euler factorization: local factors are determined via the Langlands correspondence by a local Plancherel formula. This pattern has already been observed and conjectured in the author's prior work with Venkatesh in the case of period integrals. Here, it is shown that the Rallis inner product formula amounts to the same principle in the setting of global Howe duality.