Hecke combinatorics, Kåhrstr{ö}m's conditions and Kostant's problem

Abstract

This paper discusses various aspects of the Hecke algebra combinatorics that are related to conditions appearing in Kåhrstr{ö}m's conjecture that addresses Kostant's problem for simple highest weight modules in the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ for the complex Lie algebra ${\mathfrak{sl}}_n$. In particular, we study cyclic submodules of the regular Hecke module that are generated by the elements of the (dual) Kazhdan-Lusztig basis as well as the problem of left cell invariance for both categorical and combinatorial Kåhrstr{ö}m's conditions.

Revised version to appear in Alg. Comb