Penrose Transform for Flag Domains

Abstract It is a great pleasure for me to talk at this conference in Roger’s honor on the Penrose transform with which I have been friends for the last 18 years of my mathematical life. I chose to talk about a generalized Penrose transform. There are two sides of the Penrose transform which are at the focus of possible generalizations: cohomological representations of solutions of a class of differential equations and representations of higher 8-cohomology in a holomorphic language. We need to understand on which equations and complex manifolds it is possible to generalize Penrose’s constructions. It is not necessary to connect them with group actions, but it is reasonable to start experimental work with homogeneous manifolds, and flag domains (homogeneous domains on flag manifolds) are an appropriate class for such a consideration. Some results of W. Schmid on realizations of the discrete series of representations are already fragments of the theory. We will not review here known results on the Penrose transform on some flag domains. I prefer to describe what the final theory is supposed to look like and to formulate a chain of conjectures without technical details. In this theory the central role must be played by explicit formulas for some intertwining operators and that is why I will start off the introduction with a short review of formulas for the usual Penrose transform.