Differential Operators and Highest Weight Representations

This work concerns the representation theory of semisimple Lie groups. From the algebraic perspective, the theory of unitarizable highest weight modules is highly developed. The classification was given in 1981, and, more recently, even the character and nilpotent cohomology formulas have been determined for G of classical type. However, from the analytic point of view, as originally presented by Harish-Chandra, unitarizable highest weight modules occur as subspaces of certain spaces of vector-valued polynomials, or, equivalently, as subspaces of holomorphic sections for vector bundles on G/K. The main results of this book offer characterizations of unitary highest weight representations as solutions to systems of differential operators.