Linearizing certain reductive group actions

Quotients by non-reductive algebraic group actions

Geometric invariant theory (GIT) was developed in the 1960s by Mumford in order to construct quotients of reductive group actions on algebraic varieties and hence to construct and study a number of moduli spaces, including, for example, moduli spaces of bundles over a nonsingular projective curve [26, 28]. Moduli spaces often arise naturally as quotients of varieties by algebraic group actions,...

Stratifications Associated to Reductive Group Actions on Affine Spaces

For a complex reductive group G acting linearly on a complex affine space V with respect to a characterρ, we show two stratifications ofV associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal compact subgroup ofG) coincide. The first is Hesselink’s stratification by adapted 1-parameter subgroups and the second is the Morse theoretic stratification ...

A Harish-Chandra Homomorphism for Reductive Group Actions

Consider a semisimple complex Lie algebra g and its universal enveloping algebra U(g). In order to study unitary representations of semisimple Lie groups, Harish-Chandra ([HC1] Part III) established an isomorphism between the center Z(g) of U(g) and the algebra of invariant polynomials C[t] . Here, t ⊆ g is a Cartan subspace and W is the Weyl group of g. This is one of the most basic results in...

Linear Models for Reductive Group Actions on Affine Quadrics

RÉSUMÉ. — Nous étudions les actions des groupes réductifs sur les quadriques affines complexes dont le quotient est de dimension 1. Une telle action est dite linéarisable si elle est équivalente à la restriction d’une action linéaire orthogonale dans l’espace affine ambiant de la quadrique. Une action linéaire satisfait à certaines conditions topologiques. Nous recherchons si ces conditions son...

Topological Centers of Certain Banach Module Actions