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

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