Invariants for ${\mathbb{G}}_{(r)}$-modules

Abstract

We revisit the constructions given by J. Pevtsova and the author of refined invariants for finite dimensional representations of infinitesimal group schemes ${\mathbb{G}}_{(r)}$ over a field $k$ of characteristic $p>0$. Our focus is on the universal $p$-nilpotent operator seen as an element in the group algebra of the group scheme ${\mathbb{G}}_{(r),X}$ over $X$, where $X$ is either the moduli space $V_r(\mathbb{G})$ of height $r$ $1$-parameter subgroups of $\mathbb{G}$ or the moduli space ${\mathcal{C}}_r({\mathcal{N}}_p(\mathfrak{g}))$ of $r$-tuples of $p$-nilpotent, pair-wise commuting elements of the Lie algebra of $\mathbb{G}$. We formalize Jordan type function using several variants of the continuous function $J{T}_{\mathbb{G},r,M}(-):\mathbb{P}{V}_r(\mathbb{G})\to \mathcal{Y}$ where $\mathcal{Y}$ is the poset of Young diagrams with $p$-columns. One of these variants is designed to be more conducive to computation. The vector bundle construction given by J. Pevtsova and the author is extended to all finite dimensional ${\mathbb{G}}_{(r)}$-modules, producing coherent sheaves on $X$ which are locally free on the strata of $X$ associated to $J{T}_{\mathbb{G},r,M}(-)$.