Pairings of automorphic distributions

TL;DR

This paper introduces a new pairing of automorphic distributions on flag varieties, generalizing classical Rankin-Selberg integrals, enabling novel constructions and analytic continuations of automorphic L-functions.

Contribution

It presents a generalized pairing framework for automorphic distributions applicable to open orbits, extending classical integral representations of L-functions.

Findings

Provides a new integral pairing for automorphic distributions.

Enables analytic continuation of automorphic L-functions.

Generalizes classical Rankin-Selberg integrals.

Abstract

We present a pairing of automorphic distributions that applies in situations where a Lie group acts with an open orbit on a product of generalized flag varieties. The pairing gives meaning to an integral of products of automorphic distributions on these varieties. This generalizes classical integral representations or "Rankin-Selberg integrals" of L-functions, and gives new constructions and analytic continuations of automorphic L-functions. Keywords: Automorphic forms, invariant pairings, automorphic distributions, L-functions, analytic continuation, rapid decay.