Geodesic
KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS
Glasgow mathematical journal, Tome 52 (2010) no. 1, pp. 19-32

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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 WI the Weyl group of I and let w○ and wI○ 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 I-module LI(x) of highest weight x ⋅ 0, x ∈ WI, as the answer for the simple highest weight -module L(xwI○w○) of highest weight xwI○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. 
KÅHRSTRÖM, JOHAN. KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS. Glasgow mathematical journal, Tome 52 (2010) no. 1, pp. 19-32. doi: 10.1017/S0017089509990127
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