This project provides support for continuing work on boundary problems associated with families of first-order systems of overdetermined, elliptic differential equations. The corresponding differential operators originate in the generalized potential theory in harmonic analysis on both Euclidean and Riemannian symmetric spaces. Applications to Euclidean Hardy-space theory and to explicit realization of unitarizable exceptional representations of semi-simple Lie groups will be made. Because of emphasis on group invariances, polynomial invariant theory will play an increasingly important role in these investigations. Further work on the group-theoretic connections will also be carried out. The ellipticity of the systems ensures the completeness of the normed spaces in which the solutions can be found while the over-determinedness allows one to study just the harmonic real parts of the (complex) functions under consideration.
