Arithmetic vector bundles and automorphic forms on Shimura varieties. I
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 \([{\mathcal V}']\).
representation of reductive algebraic group attached to a Shimura variety
scientific article; zbMATH DE number 3964018
scientific article