Homomorphisms of semi-holonomic verma modules : an exceptional case.

Abstract

Verma modules play an important part in the theory of invariant operators on homogeneous spaces. If G is a semisimple Lie group and P a parabolic subgroup of G, then there is often a differential geometry for which the homogeneous space G/P represents the flat model. An example is conformal geometry, where G is the special orthogonal group SO(n, C). A Verma module homomorphism will corresponds to an invariant operator on the flat space. The obvious question is: how can we generalize these operators to cases where there is curvature? In this thesis we will look at a variation of Verma modules called semiholonomic Verma modules, introduced by Eastwood and Slovák. They have studied the conformal case in detail, but here we will investigate instead the exceptional case of G = E₆. We will investigate when a Verma module homomorphism lifts to a semi-holonomic Verma module homomorphism. When this happens, we can deduce that there is a curved analogue of the corresponding invariant operator.