#### 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 ${S}_{n}$ 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 $U\left({\mathfrak{s}\mathfrak{l}}_{n}\right)$ surjects onto the space of all ad-finite linear transformations of the simple highest weight module $L\left(w\right)$. 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 ${\mathfrak{s}\mathfrak{l}}_{n}$, $n\le 5$.

##### Keywords
universal enveloping algebra, Kostant's problem, Kazhdan–Lusztig combinatorics
##### Mathematical Subject Classification 2000
Primary: 17B10
Secondary: 17B35, 16E30