Constructions for infinitesimal group schemes

TL;DR

This paper introduces a global p-nilpotent operator for infinitesimal group schemes, linking representation theory with algebraic vector bundles on projective schemes, and providing new tools for classifying modules.

Contribution

It defines a new operator that encodes local Jordan types and constructs algebraic vector bundles on the scheme of one-parameter subgroups, advancing the understanding of G-modules.

Findings

The global p-nilpotent operator encodes local Jordan types of modules.

Construction of algebraic vector bundles distinguishes representations with same local Jordan type.

Provides a method to construct vector bundles on the projectivization of the scheme of one-parameter subgroups.

Abstract

Let G be an infinitesimal group scheme over a field k of positive characteristic p. We introduce the global p-nilpotent operator $\Theta_G: k[G] \to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups of G. This operator applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain G-modules (including those of constant Jordan type), we employ the global p-nilpotent operator to associate various algebraic vector bundles on the projective scheme $\bP(G)$, the projectivization of the scheme of one-parameter subgroups of G. These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on $\bP(G)$.