Kahler differentials, the T-functor, and a theorem of Steinberg

William G. Dwyer and Clarence W. Wilkerson
(Notre Dame University and Purdue University)

It is shown that any component of the Lannes T-functor
applied to a finitely generated connected graded polynomial
algebra with an unstable action of the mod p Steenrod algebra
yields a finitely generated polynomial algebra.

This is a purely algebraic analogue of the observation that
for a connected Lie group with no homology p-torsion, each
centralizer of a sub-elementary abelian p-group also has no
homology p-torsion.

It has the algebraic consequence, previously proved by Nakajima,
that if W is a subgroup of GL(V) such that Symm[V#]^W is
polynomial algebra, then for any subset U of V, the
stabilizer group W_U has the same property.

Examples are given of reflection groups in positive characteristic
for which stabilizer subgroups are not reflection groups.
These include  $W(F_4)$ at $p=3$, $W(E_8)$ for $p=5$,
and $W( SU(pN)/Z/pZ)$ at $p=p$, for most N.