On an invariant bilinear form on the space of automorphic forms via asymptotics

TL;DR

This paper introduces a new invariant bilinear form on automorphic forms over function fields, connecting asymptotics, geometric compactifications, and dualities in the Langlands program, with implications for automorphic operator theory.

Contribution

It defines and analyzes a novel bilinear form using asymptotics and geometric methods, linking it to dualities and operators in automorphic forms theory.

Findings

The bilinear form $\\mathcal B$ is related to miraculous duality in the geometric Langlands program.

A new operator $L$ is constructed, connecting automorphic forms and pseudo-forms.

An explicit formula for the inverse of $L$ suggests an analogy with the Aubert-Zelevinsky involution.

Abstract

This article concerns the study of a new invariant bilinear form $\mathcal B$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathcal B$ using the asymptotics maps from Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathcal B$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We then give another definition of $\mathcal B$ using the constant term operator and the inverse of the standard intertwining operator. The form $\mathcal B$ defines an invertible operator $L$ from the space of compactly supported automorphic forms to a new space of "pseudo-compactly" supported automorphic forms. We give a formula for $L^{-1}$ in terms of pseudo-Eisenstein series and constant term operators which suggests that $L^{-1}$ is an analog of the Aubert-Zelevinsky involution.