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

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