Characteristic cycles of Harish-Chandra modules are a subtle geometric invariant of considerable interest. For example, they are at the heart of the Adams-Barbasch-Vogan approach to the Arthur conjectures at the archimedian places. In this talk, we give some new examples of reducible characteristic cycles for representations of SU(p,q). These examples are closely related to (though in many ways simpler than) the examples that Kashiwara-Saito found for Schubert varieties for SL(n). This is joint work with L. Barchini and P. Somberg.