Explicit realizations of certain representations of \(\text{Sp}(n,\mathbf R)\) via the double fibration transform (Q1587579)

Fix a symplectic form on \({\mathbb{C}}^{2n}\) and let \(X\) denote the Grassmannian of isotropic \(r\)-planes. It is a generalised flag variety under the action of \(\text{Sp}(n,{\mathbb{C}})\) and therefore contains finitely many open orbits for the action of the real form \(\text{Sp}(n,{\mathbb{R}})\). In general, such orbits are known as `flag domains' and their analytic cohomology with coefficients in a homogeneous vector bundle provides a geometric construction of representations. For compact Lie groups the Bott-Borel-Weil theorem identifies these representations. For non-compact semi-simple groups, all discrete series, for example, may be realised in this way. More generally, for a flag manifold~\(D\), the representations constructed in this way can often be realised as solutions of invariant differential equations on an associated space~\(M_D\), parameterising the maximal compact cycles in~\(D\) (orbits of the maximal compact subgroup). The `double fibration transform' linking these two realisations is sometimes known as the Penrose transform. Though the general machinery behind the Penrose transform is well-known, its application in particular examples can be most interesting. In this article, the author analyses the Penrose transform for \(\text{Sp}(n,{\mathbb{R}})\) acting on any flag domain as above and with coefficients in any sufficiently negative line bundle. There are new aspects to this analysis because these cases are Hermitian symmetric but `nonholomorphic' (so \(M_D\) is the Cartesian product of a Hermitian symmetric space with its complex conjugate). The treatment is thorough and instructive. (The injectivity of the transform, shown in this article, has just been extended to all flag domains by A.T.~Huckleberry and J.A.~Wolf.) The author also briefly discusses the case of a homogeneous vector bundle.