Kostant’s problem and parabolic subgroups

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let gI be the corresponding semi-simple subalgebra of g. Denote by WI the Weyl group of gI and let w◦ and w I ◦ be the longest elements of W and WI , 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 gI -module LI(x) of highest weight x · 0, x ∈ WI , as the answer for the simple highest weight gmodule L(xwI ◦w◦) of highest weight xw I ◦w◦ · 0. We also give a new description of the unique quasi-simple quotient of the Verma module ∆(e) with the same annihilator as L(y), y ∈W .