We show that a homomorphism of algebras is a categorical epimorphism if and only if all induced morphisms of the associated module varieties are immersions. This enables us to classify all minimal singularities in the subvarieties of modules from homogeneous standard tubes.
These notes present some fundamental results and examples in the theory of algebraic group actions, with special attention to the topics of geometric invariant theory and of spherical varieties. Their goal is to provide a self-contained introduction to more advanced lectures.
Let be a complex, semisimple Lie algebra, with an involutive automorphism and set , . We consider the differential operators, , on that are invariant under the action of the adjoint group of . Write for the differential of this action. Then we prove, for the class of symmetric pairs considered by Sekiguchi, that . An immediate consequence of this equality is the following result of Sekiguchi: Let be a real form of one of these symmetric pairs , and suppose that is a -invariant...
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...
Soit un groupe algébrique semi-simple complexe, un sous-groupe unipotent maximal de , un tore maximal de normalisant . Si est un -module rationnel de dimension finie, alors opère sur l’algèbre des fonctions polynomiales sur ; la structure de -module de est décrite par la -algèbre des -invariants de . Cette algèbre est de type fini et multigraduée (par le degré de et le poids par rapport à ). On donne une formule intégrale pour la série de Poincaré de cette algèbre graduée....
We consider problems in invariant theory related to the classification of four vector subspaces of an -dimensional complex vector space. We use castling techniques to quickly recover results of Howe and Huang on invariants. We further obtain information about principal isotropy groups, equidimensionality and the modules of covariants.
Let be a representation of a reductive linear algebraic group on a finite-dimensional vector space , defined over an algebraically closed field of characteristic zero. The categorical quotient carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of intrinsic? That is, does every automorphism of map each stratum to another stratum?(ii) Are the individual Luna strata in intrinsic? That is, does every automorphism...