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Automorphic vector bundles on connected Shimura varieties. (English) Zbl 0684.14006

To a semisimple group G over \({\mathbb{Q}}\) and a Hermitian symmetric domain X is associated a connected Shimura variety \(S^ 0(G,X)\). An automorphism \(\tau\) of \({\mathbb{C}}\) gives rise to a conjugate \(\tau S^ 0(G,X)\) which can be realized as \(S^ 0(G',X')\); \(G'\) and \(X'\) can be constructed using the Taniyama group [see M.Borovoj, Sel. Math. Sov. 3, 3-39 (1984; Zbl 0555.32020) and J. S. Milne, Arithmetic and geometry, Papers dedic. I. R.Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 239- 265 (1983; Zbl 0527.14035)]. There is also an association from automorphic functions on X to automorphic functions on \(X'.\)
This paper addresses the corresponding question for holomorphic automorphic forms. In effect this means describing what conjugating by \(\tau\) does to automorphic vector bundles, and the author describes the construction of an automorphic vector bundle which is the \(\tau\)- conjugate of a given one. The isomorphism is compatible with the association of the underlying Shimura varieties and with the Hecke operators. - Results of this sort will be needed for a theory of “algebraic automorphic forms”. They were inspired by the work of M. Harris [Invent. Math. 82, 151-189 (1985; Zbl 0598.14019)].
Reviewer: J.Repka

MSC:

14G25 Global ground fields in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
11F27 Theta series; Weil representation; theta correspondences
11R39 Langlands-Weil conjectures, nonabelian class field theory
32N10 Automorphic forms in several complex variables
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References:

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