Casselman--Shahidi conjecture on the singularity of intertwining operators: groups of exceptional type

TL;DR

This paper proves cases of the Casselman--Shahidi conjecture on the singularity of intertwining operators for exceptional groups, extending previous work on classical groups using a new algorithm for Weyl element decompositions.

Contribution

It introduces a uniform method to verify the conjecture for exceptional groups by developing an algorithm for reduced decompositions of Weyl elements.

Findings

Proved certain cases of the conjecture for exceptional groups.

Developed an algorithm for Weyl element decompositions.

Extended previous classical group results to exceptional groups.

Abstract

As a sequel to our recent work on Casselman--Shahidi's holomorphicity conjecture on half-normalized intertwining operators for quasi-split classical groups, we modify our method, based on a lemma of Heiermann--Opdam, to prove certain cases of the conjecture for groups of exceptional type uniformly. One main ingredient, established here, is to find an algorithm to produce reduced decompositions of co-rank one relative longest Weyl elements, in terms of certain "small" counterparts.