A new approach to Kostant's problem

TL;DR

The paper introduces a new criterion based on canonical quotients of Verma modules to determine when the universal enveloping algebra surjects onto the ad-finite transformations of certain simple modules, providing computational tools and new results.

Contribution

It develops an effective criterion for Kostant's problem using canonical quotients, enabling verification and computation for specific modules in $rak{sl}_n$.

Findings

Complete solution for simple modules in the regular block of $rak{sl}_n$, $n extless=5$

New criteria and computational methods for Kostant's problem

Extension of classical results with new cases and insights

Abstract

For every involution $\mathbf{w}$ of the symmetric group $S_n$ we establish, in terms ofa special canonical quotient of the dominant Verma module associated with $\mathbf{w}$, an effective criterion, which allows us to verify whether the universal enveloping algebra $U(\mathfrak{sl}_n)$ surjects onto the space of all ad-finite linear transformations of the simple highest weight module $L(\mathbf{w})$. An easy sufficient condition derived from this criterion admits a straightforward computational check for example using a computer. 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 $\mathfrak{sl}_n$, $n\leq 5$.