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

TL;DR

This work analyzes Hecke algebra combinatorics in relation to Kostant's problem for simple highest weight modules in ${\mathcal O}$ of $\mathfrak{sl}_n$, focusing on cyclic submodules generated by (dual) KL-basis elements and the left-cell invariance of Kåhrström's conditions. It builds a bridge from KL-theoretic data to representation-theoretic categorification via category ${\mathcal O}$, Koszul grading, and Soergel tilting formulas, and proves left-cell invariance for the categorical Kåhrström condition while obtaining partial results for the combinatorial version. The paper also establishes compatibility with parabolic induction and clarifies how longest elements of parabolic subgroups behave under these constructions. It culminates in a detailed program of conjectures and open problems, outlining both proven invariances and directions for extending the left-cell invariance to combinatorial Kåhrström conditions.

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.

Theorems & Definitions (71)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 4
• proof
• Example 5
• Example 7
• Proposition 8
• Lemma 9