2025/05/12 by Friedlander, Eric M.
#14L17 #20G05 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2505.08094
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 Vr(\mathbb G) of height r 1-parameter subgroups of \mathbb G or the moduli space \mathcal Cr(\mathcal Np(\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 Vr(\mathbb G) → \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(-).