Linear maps preserving invariants

TL;DR

This paper investigates the structure of linear maps that preserve invariants under group actions, showing that such maps are generally the group itself or a small extension, with explicit calculations for specific cases.

Contribution

It proves that the group of linear maps preserving invariants is typically equal to the original group, or a small extension in special cases, and provides explicit calculations for \\SL_2 representations.

Findings

For a complex reductive group G, the invariant-preserving linear maps form G itself.

When G is the adjoint group of a simple Lie algebra, the invariant-preserving maps form an order 2 extension of G.

Explicit calculations of G' for all representations of \SL_2.

Abstract

Let $G\subset\GL(V)$ be a complex reductive group. Let $G'$ denote $\{\phi\in\GL(V)\mid p\circ\phi=p\text{for all} p\in\C[V]^G\}$. We show that, in general, $G'=G$. In case $G$ is the adjoint group of a simple Lie algebra $\lieg$, we show that $G'$ is an order 2 extension of $G$. We also calculate $G'$ for all representations of $\SL_2$.