A New Proof of Steinberg’s Fixed-Point Theorem

This theorem has numerous applications to representation theory and other areas of mathematics (cf. [2, 3, 4, 5, 13]). The proof originally given by Steinberg in [15] involved the algebra of holomorphic functions on V and a subtle characterisation of reflection groups. Another proof has been given by Serre [1, Chapter V, Example 8, page 139], which is no more elementary, and which is algebraic geometric in nature. It is our purpose here to give a short elementary proof of the theorem. We start with three elementary results upon which our proof is based. Let V∗ be the dual space of V and S∗ its coordinate ring identified with the symmetric algebra on V∗. Then G acts on S∗ via gF(v) := F(gv) for g ∈ G, F ∈ S∗, and v ∈ V, and it is well known that the ring S∗G of polynomial invariants of G is free. If F1, F2, . . . , F is a set of homogeneous free generators of S∗G, then the degrees di = deg Fi (i = 1, . . . , ) are determined by G, and are called the invariant degrees of G. The symmetric algebra