Hecke combinatorics, Kåhrström’s conditions and Kostant’s problem

This paper discusses various aspects of the Hecke algebra combinatorics that are related to conditions appearing in Kåhrström’s conjecture, which addresses Kostant’s problem for simple highest weight modules in the Bernstein-Gelfand-Gelfand category 𝒪 for the complex Lie algebra 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 Kåhrström’s conditions.