Geometric invariants for representations of finite groups
Much of this lecture will focus on the special case of elementary
abelian $p$-groups, the ``source" of most work on support varieties.
This connects the work of Brian Parshall, as well as Jon Carlson,
Leonard Scott, and others to our recent work which is described below.
For a finite group $\tau$ and a field $k$ of characteristic $p$
dividing the order of $\tau$, we construct a map
$p_\tau: \bU_\tau \ \to \ \bP Y_\tau$ of varieties over $k$ with
unions of affine spaces as fibers and with $(\bP Y_\tau)/\tau$
$p$-isogenous to $\Proj H^\bu(\tau,k)$. On $\bU_\tau$, we construct
a ``universal $p$-nilpotent operator" $\bP \Theta_\tau$ which
leads to the construction of $\tau$-equivariant coherent sheaves
associated to finite dimensional $k\tau$-modules $M$. These
coherent sheaves are algebraic vector bundles on $\bU_\tau$ if
$M$ has constant Jordan type. For any finite dimensional
$k\tau$-module $M$, these coherent sheaves are vector bundles
when restricted to the complements of the generalized support
varieties of $M$.
For a linear algebraic group $\bG$, we compare these constructions
for the finite group $\bG(\bF_p)$ to previous constructions for
the infinitesimal group scheme $\bG_{(r)}$. For a rational
$\bG$-module $M$ of exponential type $< p^r$, we obtain a relationship
between classes in algebraic $K$-groups for $\bG(\bF_p)$ and
for $\bG_{(r)}$. The special case in which $\tau$ is an elementary
abelian $p$-group serves as our guide for more general groups as well as
extends the vector bundle invariants earlier obtained for such groups.