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

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.