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Invariant differential operators on the tangent space of some symmetric spaces

Thierry Levasseur, J. Toby Stafford (1999)

Annales de l'institut Fourier

Let 𝔤 be a complex, semisimple Lie algebra, with an involutive automorphism ϑ and set 𝔨 = Ker ( ϑ - I ) , 𝔭 = Ker ( ϑ + I ) . We consider the differential operators, 𝒟 ( 𝔭 ) K , on 𝔭 that are invariant under the action of the adjoint group K of 𝔨 . Write τ : 𝔨 Der 𝒪 ( 𝔭 ) for the differential of this action. Then we prove, for the class of symmetric pairs ( 𝔤 , 𝔨 ) considered by Sekiguchi, that d 𝒟 ( 𝔭 ) : d 𝒪 ( 𝔭 ) K = 0 = 𝒟 ( 𝔭 ) τ ( 𝔨 ) . An immediate consequence of this equality is the following result of Sekiguchi: Let ( 𝔤 0 , 𝔨 0 ) be a real form of one of these symmetric pairs ( 𝔤 , 𝔨 ) , and suppose that T is a K 0 -invariant...

Invariant meromorphic functions on Stein spaces

Daniel Greb, Christian Miebach (2012)

Annales de l’institut Fourier

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...

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