Kostant ’ s problem and parabolic subgroups Johan

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semi-simple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w 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 g I-module L I (x) of highest weight x · 0, x ∈ W I , as the answer for the simple highest weight g-module L(xw I • 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 .