Vol. 4, No. 3, 2010

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A new approach to Kostant's problem

Johan Kåhrström and Volodymyr Mazorchuk

Vol. 4 (2010), No. 3, 231–254
Abstract

For every involution w of the symmetric group Sn 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(sln) 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 sln, n 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/