Duke Mathematical Journal
- Duke Math. J.
- Volume 167, Number 16 (2018), 2965-3057.
On an invariant bilinear form on the space of automorphic forms via asymptotics
This article concerns the study of a new invariant bilinear form on the space of automorphic forms of a split reductive group over a function field. We define using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of . We show that 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 using the constant term operator and the inverse of the standard intertwining operator. The form defines an invertible operator from the space of compactly supported automorphic forms to a new space of pseudocompactly supported automorphic forms. We give a formula for in terms of pseudo-Eisenstein series and constant term operators which suggests that is an analogue of the Aubert–Zelevinsky involution.
Duke Math. J., Volume 167, Number 16 (2018), 2965-3057.
Received: 15 April 2017
Revised: 28 May 2018
First available in Project Euclid: 5 October 2018
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
automorphic form Eisenstein series constant term intertwining operator asymptotic wonderful compactification Vinberg semigroup trace of Frobenius functions-sheaves dictionary geometric Langlands program miraculous duality Drinfeld’s compactification
Wang, Jonathan. On an invariant bilinear form on the space of automorphic forms via asymptotics. Duke Math. J. 167 (2018), no. 16, 2965--3057. doi:10.1215/00127094-2018-0025. https://projecteuclid.org/euclid.dmj/1538726432