  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.
