Contractions of the actions of reductive algebraic groups in arbitrary characteristic. (English) Zbl 0778.20018

Grosshans, Frank D.

Invent. Math. 107, No. 1, 127-133 (1992).

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

Keywords:

commutative \(k\)-algebra; reductive algebraic group; Borel subgroup; \(G\)- action; graded algebra; \(G\)-equivariant algebra homomorphism; flat deformation; good filtration

References:

[1]	Donkin, S.: Invariants of unipotent radicals. Math. Z.198, 117-125 (1988) · Zbl 0627.14013 · doi:10.1007/BF01183043
[2]	Donkin, S.: Rational representations of algebraic groups: Tensor products and filtrations. (Lect. Notes Math., vol. 1140) Berlin Heidelberg New York: Springer 1985 · Zbl 0586.20017
[3]	Grosshans, F.: Finitely generated rings of invariants having rational singularities. (Can. Math. Soc. Conf. Proc., vol. 10, pp. 53-60) Providence: Am. Math Soc. 1989 · Zbl 0716.14004
[4]	Grosshans, F.: The invariants of unipotent radicals of parabolic subgroups. Invent. Math.73, 1-9 (1983) · Zbl 0511.14006 · doi:10.1007/BF01393822
[5]	Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. J. Reine Angew. Math.317, 157-199 (1980) · Zbl 0451.20040 · doi:10.1515/crll.1980.317.157
[6]	Matsumura, H.: Commutative ring theory. (Camb. Stud. Adv. Math., vol. 8) Cambridge: Cambridge University Press 1986 · Zbl 0603.13001
[7]	Nagata, M.: Lectures on the fourteenth problem of Hilbert. Bombay: Tata Institute 1965 · Zbl 0182.54101
[8]	Newstead, P.E.: Introduction to moduli problems and orbit spaces. Bombay: Tata Institute 1978 · Zbl 0411.14003
[9]	Popov, V.L.: Contraction of the actions of reductive algebraic groups. Math. USSR Sb.58 (2), 311-335 (1987) · Zbl 0627.14033 · doi:10.1070/SM1987v058n02ABEH003106
[10]	Zariski, O. and Samuel, P.: Commutative algebra, vol. II. Princeton: Van Nostrand 1960 · Zbl 0121.27801