A new approach to Kostant's problem

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 ${\mathfrak{s}\mathfrak{l}}_n$, $n\le 5$.