Download
Filename Size Access Description License

Abstract

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 $\overline M$ associated to a parabolic subgroup P of G. The monoid $\overline M$ is used implicitly in the study of the geometry of Drinfeld's compactifications of the moduli stacks Bun_P and Bun_G. We show that $\overline M$ is a retract of the affine closure of the quasi-affine variety G/U, and we relate $\overline M$ 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 Bun_G. 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_+ \to 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 F^n - {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).

Details

Additional Details

Actions

from
to
Download Full History