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Sur la théorie des invariants. (French) Zbl 0544.14003
Publ. Math. Univ. Pierre Marie Curie 45, 92 p. (1981).
Let X be an affine variety defined over \({\mathbb{C}}\) on which there acts a semi-simple group G. Then the maximal unipotent subgroup U of the group G acts on X and on \({\mathbb{C}}[X]\), the ring of regular functions. In the paper the properties of the algebra \({\mathbb{C}}[X]^ U\), the regular U invariant functions, and that of \(X_ U\), the affine variety with ring of regular functions \({\mathbb{C}}[X]^ U\) are studied. The main result is proved by H. Kraft and by the author and is formulated in the following theorem: X has regular singularities if and only if \(X_ U\) has regular singularities. - Whence it follows the corollary: If V is a G module, then \({\mathbb{C}}[V]^ U\) is a Cohen-Macaulay algebra. The properties of the Poincaré series f(z) of the algebra \({\mathbb{C}}[V]^ U\), where V is a module over G, are studied also. In particular, it is proved that for almost every G-module V \(f(z^{-1})=(-1)^{\dim V-\dim U}\cdot z^{\dim V}f(z).\) With the help of this result the classification of all simple G-modules of simple groups G, for which \({\mathbb{C}}[V]^ U\) is the algebra of polynomials, is given.
Reviewer: A.G.Elashvili

14L24 Geometric invariant theory
14B05 Singularities in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)