# Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

### L. Barchini

Oklahoma State University, Stillwater, United States### Anthony C. Kable

Oklahoma State University, Stillwater, United States### Roger Zierau

Oklahoma State University, Stillwater, United States

## Abstract

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.