Geometric invariants of representations of finite groups

Abstract

For a finite group τ and a field k of characteristic p dividing the order of τ, we construct a map pτ:Uτ→PYτ of varieties over k with unions of affine spaces as fibers and with (PYτ)/τp-isogenous to ProjH•(τ,k). On Uτ, we construct a “universal p-nilpotent operator” PΘτ which leads to the construction of τ-equivariant coherent sheaves associated to finite dimensional kτ-modules M. These coherent sheaves are algebraic vector bundles on Uτ if M has constant Jordan type. For any finite dimensional kτ-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 G, we compare these constructions for the finite group G(Fp) to previous constructions for the infinitesimal group scheme G(r). Our comparison of bundle invariants for a rational G-module M of exponential type <pr upon restriction to G(Fp) and to G(r) is a strengthened form of earlier comparisons of support varieties for M. Our constructions extend to semi-direct products G⋊τ of an infinitesimal group scheme G and a finite group τ, thus to all finite group schemes if k is algebraically closed.