Brion, Michel; Dixmier, Jacques Asymptotic behaviour of the dimensions of covariants. (Comportement asymptotique des dimensions des covariants.) (French) Zbl 0769.14016 Bull. Soc. Math. Fr. 119, No. 2, 217-230 (1991). Let \(X\) be an affine algebraic variety defined over an algebraically closed field \(k\) of characteristic 0 and let \(A=\bigoplus A_ n\) be the graded algebra of regular functions on \(X\). Let \(G\) be a reductive algebraic group over \(k\) acting algebraically and effectively on \(X\). The authors study the multiplicities, \(m_{\lambda,n}\), of irreducible representations \(\lambda\) of \(G\) in \(A_ n\). They prove several results including the following: Assume the generic orbit of \(G\) in \(X\) is closed and the stabilizer of a generic point in \(X\) is trivial. Assume also that \(m_{0,n}\neq 0\) for large \(n\), where 0 denotes the trivial 1-dimensional representation of \(G\). Then, for any irreducible representation \(\lambda\) of \(G\), \(m_{\lambda,n}/m_{0,n}\to\dim\lambda\) as \(n\to\infty\). Reviewer: D.M.Snow (Notre Dame) Cited in 4 Documents MSC: 14L30 Group actions on varieties or schemes (quotients) 14L24 Geometric invariant theory 20G05 Representation theory for linear algebraic groups Keywords:representation in algebra of regular functions; algebraic group × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] BOURBAKI (N.) . - Groupes et algèbres de Lie , Chap. 9. - Masson, 1982 . MR 84i:22001 | Zbl 0505.22006 · Zbl 0505.22006 [2] BROER (B.) . - Hilbert series in invariant theory . - Thesis, Rijksuniversiteit te Utrecht, 1990 . [3] HOWE (R.) . - Asymptotics of dimensions of invariants for finite groups , J. of Algebra, t. 122, 1989 , p. 374-379. MR 90d:16001 | Zbl 0693.20005 · Zbl 0693.20005 · doi:10.1016/0021-8693(89)90223-8 [4] KRAFT (H) . - Geometrische Methoden in der Invariantentheorie . Aspects of mathematics, Vieweg & Sohn, Braunschweig, 1984 . MR 86j:14006 | Zbl 0569.14003 · Zbl 0569.14003 [5] MORALÈS (M.) . - Fonctions de Hilbert, genre géométrique d’une singularité quasi-homogène de Cohen-Macaulay , C. R. Acad. Sci. Sér. I Math., t. 301, 1985 , p. 699-702. MR 87e:14003 | Zbl 0598.14045 · Zbl 0598.14045 [6] MUMFORD (D.) and FOGARTY (J.) . - Geometric invariant theory . 2nd ed., Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, 1982 . Zbl 0504.14008 · Zbl 0504.14008 [7] POPOV V.L. - Contractions of the actions of reductive algebraic groups , Math. USSR Sbornik, t. 58, 1987 , p. 311-335. Zbl 0627.14033 · Zbl 0627.14033 · doi:10.1070/SM1987v058n02ABEH003106 [8] SPRINGER (T.A.) . - Invariant theory . - Lecture Notes in Math., 585, 1977 . MR 56 #5740 | Zbl 0346.20020 · Zbl 0346.20020 · doi:10.1007/BFb0095644 [9] SYLVESTER (J.J.) . - Tables of the generating functions and ground-forms of the binary duodecimic, with some general remarks, and tables of the irreductible syzygies of certain quantics , Amer. J. Math., t. 4, 1881 , p. 41-61. JFM 13.0100.03 · JFM 13.0100.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.