A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

For odd-dimensional hyperbolic space H, we construct transforms between the cohomology of certain line bundles on T (a twister space for H) and eigenspaces of the Laplacian Delta and of the Dirac operator D on H. The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of Delta or D on H extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of H. We also obtain vanishing theorems for the cohomology of a class of line bundles on T.