Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Abstract

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let $\mathfrak{h}$ be an algebraic subalgebra of the tangent Lie algebra $\mathfrak{g}$ of G. We find all subalgebras $\mathfrak{h}$ that have no nontrivial characters and whose centralizers $\mathfrak{U}(\mathfrak{g})^\mathfrak{h}$ and $P(\mathfrak{g})^\mathfrak{h}$ in the universal enveloping algebra $\mathfrak{U}(\mathfrak{g})$ and in the associated graded algebra $P(\mathfrak{g})$, respectively, are commutative. For all these subalgebras, we prove that $\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g}$ and $P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g}$. Furthermore, we obtain a criterion for the commutativity of $\mathfrak{U}(\mathfrak{g})^\mathfrak{h}$ in terms of representation theory.