Several systems of partial differential operators are associated to each complex simple Lie algebra of rank greater than one. Each system is conformally invariant under the given algebra. The systems so constructed yield explicit reducibility results for a family of scalar generalized Verma modules attached to the Heisenberg parabolic subalgebra of the given Lie algebra. Points of reducibility for such families lie in the union of several arithmetic progressions, possibly overlapping. For classical algebras, enough systems are constructed to account for the first point of reducibility in each progression. The relationship between these results and a conjecture of Akihiko Gyoja is explored. §1. Introduction To describe our results, it is first necessary to set the scene. In the body of the paper we shall work most of the time in an exclusively algebraic framework, but it will be useful here to take a more inclusive viewpoint, mixing the analytic and the algebraic. We shall first attempt to explain the significance of our results and place them in context. To some extent, these remarks may be taken as an introduction to a broader investigation of which this work and [2] are the first fruits. Then we shall draw a map to aid the reader in navigating on the admittedly lengthy journey through the proofs.

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Abstract. An explicit geometric description of certain components of Springer fibers for SL(n, C) s given in this article. These components are associated to closed S(GL(p) × GL(q))-orbits in the flag variety. The geometric results are used to compute the associated cycles of the discrete series representations of SU (p, q). A discussion of an alternative, and more general, computation of associated cycles is given in the appendix.