Title: On Modular Analogs of Theorems of Chevalley, Springer, and Steinberg
          concerning Algebras of Coinvariants.

Larry Smith (Goettingen)

Abstract: Let \rho : G \longrightarrow GL(n, F) be a (faithful)
representation of a finite group G over the field F.  Denote by V = F^n
the representation space for \rho and F[V] the algebra of polynomial
functions on V.  The action of G on V extends to F[V] be defining
(g\cdot f)(v) = f(\rho(g)^{-1} (v)).  The subalgebra of polynomials
invariant under the action this denoted by F[V]^G and called the
algebra of invariants.  The algebra of coinvariants is less familiar
and is defined by F[V]_G = F \tensor_{F[V]^G} F[V].  Steinberg's
Theorem says (amongst other things) that in the case F= C the complex
numbers, that C[V]_G is a Poincar'e duality algebra (for the
algebraists : a zero dimensional Gorenstein algebra) iff G is a
reflection group.  Chevalley's Theorem tells us that for a complex
reflection group C[V]_G is the regular representation of G.  If Z/m is
cyclic group acting on V and commuting with the action of G then
Springer's Theorem relates the values of certain Molien/Poincar'e
series at roots of unity in C to the dimensions of the fixed point sets
of subgroups of Z/m acting on C(G) the complex group ring of G.  In
this talk I will describe some joint work with Abraham Broer, Vic
Reiner, and Peter Webb that extends these results to arbitrary fields
F.