Invariants of finite linear groups on relatively free algebras

It is shown that if G is a finite group of degree preserving automorphisms of R, the ring of n×n generic matrices over a field of characteristic zero generated by d > 1 elements, then the fixed ring R^G can never be generated by d elements unless n = 1 and G is a quasireflection group. As a consequence, for n > 1, R^G is never a generic matrix ring.