A new approach to Kostant's problem

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 ${\mathfrak{sl}}_n$, $n\le 5$.

References in corpus (3)

• Projective-injective modules, Serre functors and symmetric algebras
• Categorification of (induced) cell modules and the rough structure of generalized Verma modules
• Categorification of Wedderburn's basis for \mathbb{C}[S_n]

Cited by in corpus (1)

• Some homological properties of the category $\mathcal{O}$, II