A new approach to Kostant’s problem

Kåhrström, Johan; Mazorchuk, Volodymyr

doi:10.2140/ant.2010.4.231

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A new approach to Kostant's problem

Johan Kåhrström and Volodymyr Mazorchuk

Vol. 4 (2010), No. 3, 231–254

DOI: 10.2140/ant.2010.4.231

Abstract

For every involution $w$ of the symmetric group $S_{n}$ we establish, in terms of a special canonical quotient of the dominant Verma module associated with $w$ , an effective criterion to verify whether the universal enveloping algebra $U ({s l}_{n})$ surjects onto the space of all ad-finite linear transformations of the simple highest weight module $L (w)$ . An easy sufficient condition derived from this criterion admits a straightforward computational check (using a computer, for example). All this is applied to get some old and many new results, which answer the classical question of Kostant in special cases; in particular we give a complete answer for simple highest weight modules in the regular block of ${s l}_{n}$ , $n \leq 5$ .

Keywords

universal enveloping algebra, Kostant's problem, Kazhdan–Lusztig combinatorics

Mathematical Subject Classification 2000

Primary: 17B10

Secondary: 17B35, 16E30

Milestones

Received: 18 June 2008

Revised: 17 October 2009

Accepted: 31 December 2009

Published: 5 February 2010

Authors

Johan Kåhrström
Department of Mathematics Uppsala University 75106 Uppsala Sweden

Volodymyr Mazorchuk
Department of Mathematics Uppsala University 75106 Uppsala Sweden
http://www.math.uu.se/~mazor/

Vol. 4, No. 3, 2010