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

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.

63 pages

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