Conformally invariant systems of differential operators associated to two-step nilpotent maximal parabolics of non-Heisenberg type

Abstract

Scope and Method of Study: The main work of this thesis concerns systems of differential operators that are equivariant under an action of a Lie algebra. We call such systems conformally invariant. The main goal of this thesis is to construct such systems of operators for a homogeneous manifold G_0/Q_0 with G_0 a Lie group and Q_0 a maximal two-step nilpotent parabolic subgroup. We use the invariant theory of a prehomogeneous vector space to build such systems.

Findings and Conclusions: We determined the complex parameters for the line bundles L_{-s} on which our systems of differential operators are conformally invariant. The systems that we construct yield explicit homomorphisms between appropriate generalized Verma modules. We also determine whether or not these homomorphisms are standard.