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

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.