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Les formes automorphes feuilletées

  • Jean-Pierre OtalEmail author
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Abstract

We introduce a family \(\{\mathcal {B}_s(\Gamma )\}_{s\in \mathbb C}\) of line bundles over the unit tangent bundle \(\text {T}^1X_\Gamma \) of a hyperbolic surface \(X_\Gamma \). These bundles vary in an holomorphic way when restricted to the leaves of the stable foliation and to the leaves of the unstable foliation. A stable (resp. unstable) foliated automorphic form of weight s on \(\text {T}^1X_\Gamma \) is defined as a continuous section of \(\mathcal {B}_s(\Gamma )\rightarrow \text {T}^1X_\Gamma \) which is holomorphic along the leaves of the stable (resp. unstable) foliation. We study mainly the case when \(X_\Gamma \) is compact. In that situation we construct, from any \(s(1-s)\)-eigenfunction of the Laplace operator on \(X_\Gamma \), two foliated automorphic forms, one of weight s and one of weight \(1-s\). We give some general properties of the foliated automorphic forms ; in particular, we construct when \(0<\mathfrak {R}s<1\), an isomorphism between the spaces \(\mathcal {A}_s (\Gamma )\) et \(\mathcal {A}_{1-s}(\Gamma )\) of the stable automorphic forms of respective weights s and \(1-s\).

Notes

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRS, UMR 5219, Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance

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