Real Forms of the Radon–Penrose Transform

Abstract

The complex Radon correspondence relates an n -dimensional projective space with the Grassmarm manifold of its p -dimensional planes. This is the geometric background of the Radon–Penrose transform, which intertwines cohomology classes of homogeneous line bundles with holomorphic solutions to the generalized massless field equations. A good framework to deal with such problems is provided by the recently developed theory of integral transforms for sheaves and D -modules. In particular, an adjunction formula describes the range of transforms acting on general function spaces, associated with constructible sheaves. The linear group SL ( n + 1,C) naturally acts on the Radon correspondence. A distinguished family of function spaces is then the one associated with locally constant sheaves along the closed orbits of the real forms of SL ( n + 1,C). In this paper, we systematically apply the above-mentioned adjunction formula to such function spaces. We thus obtain in a unified manner several results concerning the complex, conformal, or real Radon transforms.