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

TL;DR

This paper refines invariants for finite-dimensional modules over infinitesimal group schemes, formalizing Jordan type functions and extending vector bundle constructions to produce coherent sheaves with stratified local freeness.

Contribution

It formalizes the Jordan type function for $G_{(r)}$-modules and extends vector bundle constructions to all such modules, enabling stratified coherent sheaves.

Findings

Defined variants of the Jordan type function for $G_{(r)}$-modules.

Extended vector bundle constructions to all finite-dimensional $G_{(r)}$-modules.

Produced coherent sheaves that are locally free on specific strata.

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 $JT_{\mathbb G,r,M}(-)$.