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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 afunction field. We define $\mathcal B$ using the asymptotics maps fromBezrukavnikov-Kazhdan and Sakellaridis-Venkatesh, which involve the geometry ofthe wonderful compactification of $G$. We show that $\mathcal B$ is naturallyrelated to miraculous duality in the geometric Langlands program through thefunctions-sheaves dictionary. In the proof, we highlight the connection betweenthe classical non-Archimedean Gindikin-Karpelevich formula and certainfactorization algebras acting on geometric Eisenstein series. We then giveanother definition of $\mathcal B$ using the constant term operator and theinverse of the standard intertwining operator. The form $\mathcal B$ defines aninvertible operator $L$ from the space of compactly supported automorphic formsto a new space of "pseudo-compactly" supported automorphic forms. We give aformula for $L^{-1}$ in terms of pseudo-Eisenstein series and constant termoperators which suggests that $L^{-1}$ is an analog of the Aubert-Zelevinskyinvolution.

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