Linear maps preserving fibers

TL;DR

This paper investigates the structure of linear maps that preserve fibers of the quotient map in the context of complex reductive groups, providing classifications and explicit descriptions of certain subgroups.

Contribution

It determines the identity component of the subgroup preserving the null cone and fibers, especially for adjoint and cofree modules, extending understanding of fiber-preserving linear maps.

Findings

H equals H_0 in many cases

Complete determination of H for adjoint representations

Description of subgroup G_F preserving fibers in cofree modules

Abstract

Let $G\subset\GL(V)$ be a complex reductive group where $\dim V<\infty$, and let $\pi\colon V\to\quot VG$ be the categorical quotient. Let $\NN:=\pi\inv\pi(0)$ be the null cone of $V$, let $H_0$ be the subgroup of $\GL(V)$ which preserves the ideal $\I$ of $\NN$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H=H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of $\GL(V)$ preserving a fiber $F$ of $\pi$ when $V$ is an irreducible cofree $G$-module.