Invariants for $\mathbb G_{(r)}$-modules
Eric M. Friedlander
TL;DR
The paper develops refined invariants for finite dimensional modules of infinitesimal group schemes $\mathbb{G}_{(r)}$ by studying the universal $p$-nilpotent operator $\Theta_{\mathbb{G},r}$ and the Jordan type function $JT_{\mathbb{G},r,M}(-)$, mapping height-$r$ 1-parameter subgroups to Young diagrams with $p$ columns. It introduces exponential-type variants $JT^{\mathrm{exp}}_{\mathfrak{g},r,M}(-)$ that simplify computation while preserving refined support data, and extends these invariants to vector bundle and sheaf constructions on moduli spaces such as $\mathbb{P}V_r(\mathbb{G})$ and $\mathbb{P}\mathcal{C}_r(\mathcal{N}_p(\mathfrak{g}))$, yielding stratifications by Jordan type. The work also relates the invariants to classical support varieties via the $\psi$-map of Suslin–Bendel–Kämpf, and establishes exponential-type simplifications, including results for $\mathbb{G}_{a(r)}$, $\mathfrak{sl}_2$-modules, and general linear groups, with a coherent treatment of stabilization to infinite height. These invariants provide finer, computable diagnostics of module structure beyond traditional cohomological supports, with implications for classification and geometry of representations of infinitesimal and linear algebraic groups.
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 $JT_{\mathbb G,r,M}(-): \mathbb P V_r(\mathbb G) \to \mathcal Y$ 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 $JT_{\mathbb G,r,M}(-)$.