A reducible characteristic variety in type a

Abstract

We show that simple highest weight modules for sl12(ℂ) may have reducible characteristic variety. This answers a question of Borho-Brylinski and Joseph from 1984. The relevant singularity under Beilinson-Bernstein localization is the (in)famous Kashiwara-Saito singularity. We sketch the rather indirect route via the p-canonical basis, W -graphs and decomposition numbers for perverse sheaves that led us to examine this singularity.