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

2016/09/30 by Jonathan Wang · 14 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Automorphic L-function #Automorphic form #Bilinear form #Eisenstein series #Geometry and complex manifolds #Invariant (physics) #Invertible matrix #Langlands program #Langlands–Shahidi method #Mathematical physics #Mathematics #Modular form #Pure mathematics #Symmetric bilinear form #math.AG #math.NT #math.RT #msc:11F70 #msc:14D24 #msc:22E50

paper · pdf · doi:10.1215/00127094-2018-0025

published in Duke Mathematical Journal 167(16) (Duke University Press) · 63 pages

arxiv created 2017/04/16 · openalex publication_date 2018/10/05 · arxiv updated 2018/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 L−1 in terms of pseudo-Eisenstein series and constant term operators which suggests that L−1 is an analogue of the Aubert–Zelevinsky involution.

Citations

• On the Langlands retraction
• On the cohomology groups of moduli spaces of vector bundles on curves
• Geometric constant term functor(s)
• On the reductive monoid associated to a parabolic subgroup
• An introduction to affine Grassmannians and the geometric Satake equivalence
• Geometric Bernstein Asymptotics and the Drinfeld-Lafforgue-Vinberg degeneration for arbitrary reductive groups
• Picard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for SL2
• The Harder-Narasimhan stratification of the moduli stack of G-bundles via Drinfeld's compactifications
• Outline of the proof of the geometric Langlands conjecture for GL(2)
• Periods and harmonic analysis on spherical varieties
• A "strange" functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles

Cited by

• Cuspidal cohomology of stacks of shtukas