This article concerns the study of a new invariant bilinear form on the space of automorphic forms of a split reductive group over a function field. We define using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of . We show that 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 using the constant term operator and the inverse of the standard intertwining operator. The form defines an invertible operator from the space of compactly supported automorphic forms to a new space of pseudocompactly supported automorphic forms. We give a formula for in terms of pseudo-Eisenstein series and constant term operators which suggests that is an analogue of the Aubert–Zelevinsky involution.
"On an invariant bilinear form on the space of automorphic forms via asymptotics." Duke Math. J. 167 (16) 2965 - 3057, 1 November 2018. https://doi.org/10.1215/00127094-2018-0025