A finiteness theorem for representations with a free algebra of invariants

It is proved that for any connected semisimple algebraic group $G$ defined over an algebraically closed field of characteristic zero there exist (up to isomorphism) only a finite number of finite-dimensional rational $G$-modules containing no nonzero fixed vectors and having a free algebra of invariants. The proof is constructive and makes it possible in principle to indicate these $G$-modules explicitly. It is also proved that for all irreducible $G$-modules $V$, except for a finite number, the degree of the Poincaré series of the algebra of invariants (regarded as a rational function) equals $-\dim V$.
Bibliography: 21 titles.