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 $\mathfrak{B}$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathfrak{B}$ using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathfrak{B}$ 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 $\mathfrak{B}$ using the constant term operator and the inverse of the standard intertwining operator. The form $\mathfrak{B}$ defines an invertible operator $L$ from the space of compactly supported automorphic forms to a new space of pseudocompactly 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 analogue of the Aubert–Zelevinsky involution.