表示空间上的伴随向量场与微分算子
Adjoint vector fields and differential operators on representation spaces
摘要：
设$G$是具有李代数$\g$的半单代数群. 1979年，J. Dixabel证明了零化$\g$上所有$G$ -不变多项式的任何向量场都位于由“伴随向量场“生成的$\bbk[\g]$ -模中，即，形式为$ς(y)(x)=[x,y]$、$x,y\in\g$的向量场$ς$。勒瓦色尔和斯塔福德发现了狄克逊定理的实质性推广。他们在$\g$上的微分算子代数中明确地描述了$\bbk[\g]^G$的中心化器。在向量场的水平上，他们的结果归结为Dixlane定理。本文的目的是在仿射代数群及其有理表示的一般背景下探讨类似的问题。
英文摘要：
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 $ς$ of the form $ς(y)(x)=[x,y]$, $x,y\in\g$. 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.
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