Reducible characteristic cycles of Harish-Chandra modules for $\mathrm{U}(p,q)$ and the Kashiwara-Saito singularity

TL;DR

This paper provides examples of reducible characteristic cycles in Harish-Chandra modules for U(p,q), linking singularities to Kashiwara-Saito and Williamson's work, and explaining reducibility phenomena in representation theory.

Contribution

It introduces new examples of reducible characteristic cycles for Harish-Chandra modules and connects these to known singularities and recent discoveries by Williamson.

Findings

Examples of reducible characteristic cycles for U(p,q) modules.

Relation of singularities to Kashiwara-Saito and Williamson's work.

Explanation of reducible associated varieties in certain modules.

Abstract

We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for $\mathrm{U}(p,q)$ by analyzing a four-dimensional singular subvariety of $\mathbb{C}^8$. We relate this singularity to the Kashiwara-Saito singularity arising for Schubert varieties for ${\mathrm{GL}}(8,\mathbb{C})$ with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for ${\mathfrak{g}}{\mathfrak{l}}(12,\mathbb{C})$ with reducible associated varieties that Williamson discovered.