Duke Mathematical Journal

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

Jonathan Wang

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This article concerns the study of a new invariant bilinear form B on the space of automorphic forms of a split reductive group G over a function field. We define B using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of G. We show that 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 B 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 pseudocompactly supported automorphic forms. We give a formula for L1 in terms of pseudo-Eisenstein series and constant term operators which suggests that L1 is an analogue of the Aubert–Zelevinsky involution.

Article information

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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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

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