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KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS

Published online by Cambridge University Press:  30 July 2009

JOHAN KÅHRSTRÖM
JOHAN KÅHRSTRÖM*
Affiliation:
Department of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden e-mail: johank@math.uu.se
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Abstract

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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 19 - 32
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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