Hanspeter Kraft 1 and Claudio Procesi 2 Sonderforschungsbereich Theoretische Mathematik, Universit~it Bonn D-5300 Bonn, Federal Republic of Germany 2 Istituto di Matematica, Universitfi di Roma, I-Rome, Italy O. Introduction 0.1. The purpose of this paper is to prove the following theorem: Let A be an n “ n matrix over an algebraically closed field K of characteristic zero, C A the conjugacy class of A and C A its (Zariski-) closure. Theorem. C a is normal, Cohen-Macaulay with rational singularities. tf a variety X with a G-action (G reductive) is the closure of an orbit (9 and dim(X--. (9)< dim X-2, it is a crucial question for the geometry of X to decide whether the singularity (in X \ (9) is normal. In fact the normality of X allows to identify the ring K[X] of regular functions on X with the functions on the orbit (9 and so, by Frobenius reciprocity, to analyse K [X] as a representation of G (cf. [11], [1]). In our case this is closely related to the "multiplicity conjecture" of Dixmier; we refer the reader to the paper [1] for a detailed description of this connection and some applications. A different proof of this theorem will appear in [23]. 0.2. The theorem has also another interesting application, shown to us by Th. Vust, in the spirit of the classical theory of Schur. If U is a finite dimensional vector space one has the classical relation between the action of GL(U) and of the symmetric group ~,, on the tensor space U | If we restrict to the subgroup