Adjoint vector fields and differential operators on representation spaces

TL;DR

This paper extends classical results about vector fields annihilating invariant polynomials from semisimple groups to more general affine algebraic groups, exploring the structure of differential operators and vector fields in this broader context.

Contribution

It generalizes Dixmier's theorem and Levasseur-Stafford's results to affine algebraic groups and their rational representations, broadening the understanding of invariant differential operators.

Findings

Generalization of Dixmier's theorem to affine algebraic groups

Explicit description of centralizers of invariant polynomial rings

Insights into vector fields on representation spaces

Abstract

Let $G$ be a semisimple algebraic group with Lie algebra $\g$. In 1979, J. Dixmier proved that any vector field annihilating all $G$-invariant polynomials on $\g$ lies in the $\bbk[\g]$-module generated by the "adjoint vector fields", i.e., vector fields $\varsigma$ of the form $\varsigma(y)(x)=[x,y]$, $x,y\in\g$. A substantial generalisation of Dixmier's theorem was found by Levasseur and Stafford. They explicitly described the centraliser of $\bbk[\g]^G$ in the algebra of differential operators on $\g$. On the level of vector fields, their result reduces to Dixmier's theorem. The purpose of this paper is to explore similar problems in the general context of affine algebraic groups and their rational representations.