Invariant meromorphic functions on Stein spaces

Dans ce travail nous développons des outils et des méthodes fondamentaux afin d’étudier les fonctions méromorphes invariantes sur les espaces de Stein $X$ munis d’une action holomorphe d’un groupe complexe-réductif $G$. Nous construisons des quotients à la Rosenlicht pour l’action d’un sous-groupe algébrique de $G$ sur $X$. En particulier on montre que dans cette situation les fonctions méromorphes invariantes sous ce sous-groupe algébrique séparent ses orbites en position générale. Nous donnons aussi des applications concernant les espaces presque homogènes et les types d’orbite principaux. De plus, le résultat principal est utilisé afin de clarifier la relation entre les invariants holomorphes voire méromorphes de $G$. Une étape importante de notre preuve consiste à montrer un analogue faible équivariant du théorème de Narasimhan sur les plongements propres des espaces de Stein.

In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result to investigate the relation between holomorphic and meromorphic invariants for reductive group actions. As one important step in our proof we obtain a weak equivariant analogue of Narasimhan’s embedding theorem for Stein spaces.

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