On an invariant bilinear form on the space of automorphic forms via asymptotics (Q1626389)

In this paper, the author defines a new bilinear invariant form $\mathcal{B}$ on the space of compacly-supported, $C^{\infty}$-, $K$-finite automorphic forms on a split reductive group $G$ over a function field. This form is defined using the so-called asymptotics maps introduced in the recent works of Bezrukavnikov, Kazhdan, Sakellaridis, Venkatesh. It turns out that this form is symmetric and it gives rise, in a natural way, to a certain invertible operator $L$ (between certain subspaces of automorphic forms). The author then proves certain functional equation that this operator satisfies (equation (6.9) in the paper; this is the ``strange'' functional equation in the Gaitsgory's work on geometric Eisenstein series from 2017.). Maybe even more interestingly, the inverse of this operator is given by the total global analog of the expression for the Aubert-Steiner-Stuhler-Zelevinsky duality for the local non-Archimedean reductive groups and representations. The role of the parabolic induction of the local case is here played by the Eisenstein series, and the role of the Jacquet modules of the local situation is played here by the constant term. Along the way, some additional results are proved -- e.g., in section 3.2, the extension of the classical Satake isomorphism to an isomorphism of certain larger algebras. Although the proofs are given mainly using explicit calculus of functions and distributions on the group $G$ (with some careful asymptotics involved, resembling some notions of Arthur in his work on the trace formula), the main motivation came from the geometric Langlands program. The relevant geometry is reviewed in Appendices A-C, which comprise the last forty pages of the paper, and are not purely a review-some results are also proved. From the Introduction: `` In Appendix A, we consider the global model for the formal arc space of a group embedding into an algebraic monoid.'' ``In Appendix B, we review the definition of the factorization algebras on the affine Grassmannian...that act on geometric Eisenstein series.'' ``In Appendix C, we study the compactification of the diagonal morphism of $\mathrm{Bun}_G$...'' (here $\mathrm{Bun}_G$ denotes the stack of $G$-bundles on $X,$ the projective curve which gives us the function field of the definition).