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

Abstract

Let $\frak{g} = \frak{k} +\frak{p}$ be a complexified Cartan decomposition of a complex semisimple Lie algebra $\frak{g}$ and let $K$ be the subgroup of the adjoint group of $\frak{g}$ corresponding to $\frak{k} $. If $H$ is an irreducible Harish-Chandra module of $U(\frak{g})$, then $H$ is completely determined by the finite-dimensional action of the centralizer $U(\frak{g})^K$ 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 $U(\frak{g})^K$. Generators of $U(\frak{g})^K$ are given by generators of the symmetric algebra analogue $S(\frak{g})^K$. Let $S_m(\frak{g})^K, m\in {\Bbb Z}_+$, be the subalgebra of $S(\frak{g})^K$ defined by $K$-invariant polynomials of degree at most $m$. Let $Q$ and $Q_m$ be the respective quotient fields of $S(\frak{g})^K$ and $S_m(\frak{g})^K$. We prove that if $n= dim \frak{g}$ one has $Q= Q_{2n}$. We also determine the variety, $Nil_K$, of unstable points with respect to the action $K$ on $\frak{g}$ and show that $Nil_K$ 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 $r = {2n\choose 2} dim {\frak p}$.