A reducible characteristic variety in type A

TL;DR

This paper demonstrates that certain simple highest weight modules for sl_12 have reducible characteristic varieties, answering a longstanding question and linking singularities to advanced algebraic and geometric concepts.

Contribution

It provides the first example of reducible characteristic varieties in type A, connecting representation theory, singularity theory, and geometric methods.

Findings

Simple highest weight modules for sl_12 can have reducible characteristic varieties

The Kashiwara-Saito singularity is relevant to these modules

The study involves p-canonical basis, W-graphs, and decomposition numbers

Abstract

We show that simple highest weight modules for sl_12 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.