Kostant's problem and parabolic subgroups

Abstract

Let $\mathfrak{g}$ be a finite dimensional complex semi-simple Lie algebra with Weyl group $W$ and simple reflections $S$. For $I\subseteq S$ let ${\mathfrak{g}}_I$ be the corresponding semi-simple subalgebra of $\mathfrak{g}$. Denote by $W_I$ the Weyl group of ${\mathfrak{g}}_I$ and let $w_o$ and $w_o^I$ be the longest elements of $W$ and $W_I$, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight ${\mathfrak{g}}_I$-module $L_I(x)$ of highest weight $x\cdot 0$, $x\in W_I$, as the answer for the simple highest weight $\mathfrak{g}$-module $L(x{w}_o^I{w}_o)$ of highest weight $(x{w}_o^I{w}_o)\cdot 0$. We also give a new description of the unique quasi-simple quotient of the Verma module $\Delta (e)$ with the same annihilator as $L(y)$, $y\in W$.