| Summary: | This thesis concerns the study of a new invariant bilinear form B on the space of automorphic forms of a split reductive group G over a global field. The form B is natural from the viewpoint of the geometric Langlands program.
First, we study a certain reductive monoid M associated to a parabolic subgroup P of G. The monoid M is used implicitly in the study of the geometry of Drinfeld's compactifications of the moduli stacks BunP and BunG. We show that M is a retract of the affine closure of the quasi-affine variety G/U, and we relate M to the Vinberg semigroup of G. Second, we define B over a function field using the asymptotics maps defined in Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh using the geometry of the wonderful compactification of G. We show that B is related to the miraculous duality functor studied by Drinfeld and Gaitsgory through the functions-sheaves dictionary. In the proof, we use the work of Schieder, which concerns the singularities of Drinfeld's compactification of BunG. We then give an alternate definition of B, which extends to number fields, using the constant term operator and the inverse of the standard intertwining operator. The form 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.
Lastly, we study the Radon transform as an operator R : C+ → C - from the space of smooth K-finite functions on F n \ {0} with bounded support to the space of smooth K-finite functions on Fn \ {0} supported away from a neighborhood of 0, where F is a (possibly Archimedean) local field. When n = 2, the Radon transform coincides with the standard intertwining operator. We prove that R is an isomorphism and provide explicit formulas for R-1 . These formulas in turn give a formula for B over a number field when G = SL(2).
|
|---|
