Über die Glattheit von Quotientenabbildungen. (On the smoothness of quotient maps) (Q1071084)

Let G be a semisimple algebraic group acting on a factorial Gorenstein algebra S defined over a field of characteristic 0. Let \(X:=Spec S^ G\) and \(\pi: X\to Y\) be the quotient map. The main results are: 1. Let x be a smooth point of X whose orbit has maximal dimension and such that \(\pi\) (x) is a smooth point of Y. Then \(\pi\) is smooth at x. 2. Let S be positively graded and let \(\chi_ S(t)\) be its generating function which is the power series expansion of a rational function. Then deg \(\chi\) \({}_ S\leq \deg \chi_ SG\). This answers a conjecture of \textit{V. L. Popov} [Math. USSSR, Izv. 20, 333-354 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 347-370 (1982; Zbl 0547.20034)].