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On deformation method in invariant theory. (English) Zbl 0878.14008
Summary: We relate the deformation method in invariant theory to spherical subgroups. Let \(G\) be a reductive group, \(Z\) an affine \(G\)-variety and \(H\subset G\) a spherical subgroup. We show that whenever \(G/ H\) is affine and its semigroup of weights is saturated, the algebra of \(H\)-invariant regular functions on \(Z\) has a \(G\)-invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of \(G\). The deformation method in its usual form, as developed by Luna and others, is a particular case of this construction. Our result also applies to the description of invariants of some reducible representations of reductive groups. New applications of the deformation method are given which concern the property of being complete intersection for algebras of invariants. We also give some applications of the deformation method to doubled actions.

MSC:
14L24 Geometric invariant theory
13A50 Actions of groups on commutative rings; invariant theory
14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
14M10 Complete intersections
13D10 Deformations and infinitesimal methods in commutative ring theory
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