# Publications of the Research Institute for Mathematical Sciences

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**Volume 44, Issue 3, 2008, pp. 749–835**

**DOI: 10.2977/prims/1216238304**

Published online: 2008-09-30

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

L. Barchini^{[1]}, Anthony C. Kable

^{[2]}and Roger Zierau

^{[3]}(1) Oklahoma State University, Stillwater, United States

(2) Oklahoma State University, Stillwater, United States

(3) Oklahoma State University, Stillwater, United States

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.

*Keywords: *Generalized Verma modules, Gyoja’s conjecture, covariant maps

Barchini L., Kable Anthony, Zierau Roger: Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type. *Publ. Res. Inst. Math. Sci.* 44 (2008), 749-835. doi: 10.2977/prims/1216238304