Arithmetic vector bundles and automorphic forms on Shimura varieties. I
type: Item
instance of: scholarly article
title: Arithmetic vector bundles and automorphic forms on Shimura varieties. I
zbMATH Open document ID: 0598.14019
DOI: 10.1007/BF01394784
author: Michael Harris
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143282
review text: This is the first installment of a two-part paper in which the author studies various approaches to defining automorphic forms on Shimura varieties with the aim of proving that the different notions actually give rise to the same objects. The author associates, to a representation \((\rho,V_{\rho})\) of the reductive algebraic group \(G\) attached to a Shimura variety \(M\), a vector bundle \([\tilde V_{\rho}]\) on \(M\). He shows that \([\tilde V_{\rho}]\) contains a canonical local system with coefficients in the reflex field. He then applies this fact to prove a conjecture of Deligne that there is a functor \({\mathcal V}\to [{\mathcal V}]\), defined over the reflex field, from the category of homogeneous vector bundles on \(\check M\) to the category of vector bundles on M with continuous \(G({\mathbb{A}}^ f)\)-action such that \([{\mathcal V}]_{{\mathbb{C}}}\) is complex-analytically isomorphic to the analytic vector bundle \[ {\mathcal V}^{\#}=\lim_{\overset \leftarrow K}G({\mathbb{Q}})\setminus (\beta^*_ X{\mathcal V})\times G({\mathbb{A}}^ f)/K, \] where \(\beta_ X\) is the Borel imbedding, and the limit is taken over compact open subgroups of \(G({\mathbb{A}}^ f)\). Moreover, as Deligne conjectured, if \(G'\subset G\) is attached to the Shimura variety \(M'\) then \([{\mathcal V}|_{\check M'}]=[{\mathcal V}]|_{M'}\), and any homogeneous differential operator \(\delta\) from \({\mathcal V} \to {\mathcal V}'\) gives rise, in a functorial way, to a \(G'({\mathbb{A}}^ f)\)-invariant differential operator \([\delta]\) from \([{\mathcal V}]\) to \([{\mathcal V}']\).
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 32N15
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 32N10
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 3964018
zbMATH Keywords: representation of reductive algebraic group attached to a Shimura variety
zbMATH Keywords: automorphic forms on Shimura varieties
zbMATH Keywords: homogeneous vector bundles
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3964018