KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS

Source abstract

Abstract Let be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S . For I ⊆ S let I be the corresponding semi-simple subalgebra of . Denote by W I the Weyl group of 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 I -module L I ( x ) of highest weight x ⋅ 0, x ∈ W I , as the answer for the simple highest weight -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 Δ( e ) with the same annihilator as L ( y ), y ∈ W .