1 November 2018 On an invariant bilinear form on the space of automorphic forms via asymptotics
Jonathan Wang
Duke Math. J. 167(16): 2965-3057 (1 November 2018). DOI: 10.1215/00127094-2018-0025


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.


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Jonathan Wang. "On an invariant bilinear form on the space of automorphic forms via asymptotics." Duke Math. J. 167 (16) 2965 - 3057, 1 November 2018. https://doi.org/10.1215/00127094-2018-0025


Received: 15 April 2017; Revised: 28 May 2018; Published: 1 November 2018
First available in Project Euclid: 5 October 2018

zbMATH: 06985300
MathSciNet: MR3870080
Digital Object Identifier: 10.1215/00127094-2018-0025

Primary: 11F70
Secondary: 22E50

Keywords: asymptotic , automorphic form , constant term , Drinfeld’s compactification , Eisenstein series , functions-sheaves dictionary , geometric Langlands program , intertwining operator , miraculous duality , trace of Frobenius , Vinberg semigroup , wonderful compactification

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 16 • 1 November 2018
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