Kostant's problem for parabolic Verma modules

TL;DR

This paper provides a complete combinatorial classification of parabolic Verma modules in the principal block of category O for special linear Lie algebras, identifying when Kostant's problem has a positive answer.

Contribution

It offers the first comprehensive combinatorial criteria for Kostant's problem in the context of parabolic Verma modules in type A Lie algebras.

Findings

Classification of modules with positive Kostant's problem answer

Complete combinatorial criteria established

Applicable to minimal and maximal parabolic subalgebras

Abstract

We give a complete combinatorial classification of those parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated to a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant's problem is positive.