Join operation for the Bruhat order and Verma modules

TL;DR

This paper explores the relationship between the join operation in the Bruhat order and intersections of Verma modules, establishing exact results in type A and proposing conjectures for other types, with implications for Ext spaces.

Contribution

It introduces a conjectural framework linking the join operation in Weyl groups with Verma module intersections beyond type A, and describes the poset structure and Kazhdan-Lusztig polynomials.

Findings

Join operation matches Verma module intersections in type A.

Conjectural subsets of Weyl groups where the correspondence holds.

Complete description of socles of cokernels and Ext spaces.

Abstract

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.