On the Centralizer of $K$ in $U(\mathfrak{g})$

Abstract

Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a complexified Cartan decomposition of a complex semisimple Lie algebra $\mathfrak{g}$ and let $K$ be the subgroup of the adjoint group of $\mathfrak{g}$ corresponding to $\mathfrak{k}$. If $H$ is an irreducible Harish-Chandra module of $U(\mathfrak{g})$, then $H$ is completely determined by the finite-dimensional action of the centralizer $U(\mathfrak{g}{)}^K$ on any one fixed primary $\k$ component in $H$. This original approach of Harish-Chandra to a determination of all $H$ has largely been abandoned because one knows very little about generators of $U(\mathfrak{g}{)}^K$. Generators of $U(\mathfrak{g}{)}^K$ are given by generators of the symmetric algebra analogue $S(\mathfrak{g}{)}^K$. Let $S_m(\mathfrak{g}{)}^K,m\in {\mathbb{Z}}_{+}$, be the subalgebra of $S(\mathfrak{g}{)}^K$ defined by $K$-invariant polynomials of degree at most $m$. Let $Q$ and $Q_m$ be the respective quotient fields of $S(\mathfrak{g}{)}^K$ and $S_m(\mathfrak{g}{)}^K$. We prove that if $n=dim\mathfrak{g}$ one has $Q=Q_{2n}$. We also determine the variety, $Ni{l}_K$, of unstable points with respect to the action $K$ on $\mathfrak{g}$ and show that $Ni{l}_K$ is already defined by $A_{2n}$. As pointed out to us by Hanspeter Kraft, this fact together with a result of Harm Derksen (See [D]) implies, indeed, that $A=A_r$ where $r=\left(\genfrac{}{}{0ex}{}{2n}{2}\right)dim\mathfrak{p}$.

19 pages, plain tex