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Contractions of the actions of reductive algebraic groups in arbitrary characteristic. (English) Zbl 0778.20018
Let $$k$$ be an algebraically closed field, and $$A$$ a commutative $$k$$- algebra. Let $$G$$ be a reductive algebraic group which acts rationally on $$A$$, and $$B=TU$$ a Borel subgroup of $$G$$, and $$B^ -=TU^ -$$ the Borel subgroup of $$G$$ opposite to $$B$$. Denote by $$A^ U$$ the set of fixed points of $$A$$ under the action of $$U$$. The author of this paper relates properties of $$A$$ to those of $$A^ U$$. The author also constructs a $$G$$- action on $$R=(A^ U \otimes_ k k[G/U^ -])^ T$$, and shows that there is a graded algebra $$\text{gr }A$$ and an injective $$G$$-equivariant algebra homomorphism $$\Phi:\text{gr }A\to R$$. Moreover, the action of $$G$$ on $$\text{Spec}(A)$$ is a flat deformation of the action of $$G$$ on $$\text{Spec(gr }A)$$, and $$\Phi$$ is an isomorphism if and only if $$A$$ has a good filtration.
Reviewer: Li Fuan (Beijing)

##### MSC:
 20G15 Linear algebraic groups over arbitrary fields 14L30 Group actions on varieties or schemes (quotients) 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 20G05 Representation theory for linear algebraic groups
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