The Penrose Transform in the Split Signature

A version of the Penrose transform is introduced in the split signature. It relates the cohomological data with supports on the open subsets of the complex 3-projective space and kernel of differential operators on the (real) Grassmannian of 2-planes in the Euclidean 4-space. As an example we derive a cohomological interpretation of the so-called X-ray transform. Furthermore, a cohomological realization of the so-called "minimal" representation of SL(4,R) is given. We also present the split Penrose transform in split instanton backgrounds.