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Let W be a finite group acting on a finite dimensional vector space V over the field k, U a subset of V , and WU the subgroup of W which fixes U pointwise. Suppose that the algebra Sym(V #)W is a polynomial algebra. Steinberg has shown that if k has characteristic zero the algebra Sym(V #)WU is also a polynomial algebra. We extend this result to the case k = Fp (see also [14]) by proving that Lannes’s functor “T” preserves polynomial algebras. We give examples to show that if k = Fp and W is generated by reflections, it is not necessarily the case that WU is generated by reflections.

Kähler Differentials, the $T$-functor, and a Theorem of Steinberg