# Geometric Invariants of Representations of Finite Groups

## TL;DR

This paper develops geometric invariants for finite group representations, extending previous invariants for infinitesimal group schemes, and constructs vector bundles to better understand the relationship between different types of group representations.

## Contribution

It introduces new invariants for finite groups using refined $ au$-point classes, generalizes vector bundle constructions for semi-direct products, and enhances understanding of support varieties in representation theory.

## Key findings

- Constructed vector bundles for semi-direct products of group schemes and finite groups.
- Extended invariants to recover earlier results for elementary abelian p-groups.
- Sharpened the comparison of support varieties for different group representations.

## Abstract

J. Pevtsova and the author constructed a ``universal $p$-nilpotent operator" for an infinitesimal group scheme $G$ over a field $k$ of characteristic $p > 0$ which led to coherent sheaves on the scheme of 1-parameter subgroups of $G$ associated to a $G$-module $M$. Of special interest is the fact that these coherent sheaves are vector bundles if $M$ is of constant Jordan type. In this paper, we provide similar invariants for a finite group $\tau$ which recover the invariants earlier obtained for elementary abelian $p$-groups. To do this, we replace the analogue of 1-parameter subgroups by a refined version of equivalence classes of $\pi$-points for $k\tau$. More generally, we provide a construction of vector bundles for the semi-direct product $G\rtimes \tau$ of an infinitesimal group scheme $G$ and a finite group $\tau$. A major motivation for this study is to further our understanding of the relationship between representations of $\mathbb G(\mathbb F_p)$ and $\mathbb G_{(r)}$ associated to a finite dimensional rational $\mathbb G$-module $M$, where $\mathbb G$ is a reductive group with $r$-th Fobenius kernel $\mathbb G_{(r)}$. Using vector bundles, we extend and sharpen earlier results comparing support varieties.