Holomorphic discrete models of semisimple Lie groups and their symplectic constructions (Q1580798)

Let \(G\) be a connected real semisimple Lie group. An irreducible unitary representation of \(G\) is in the discrete series if some non-zero matrix coefficient function is in \(L^2(G)\). The author defines the holomorphic discrete model as a unitary \(G\)-representation consisting of all holomorphic discrete series with multiplicity one. He constructs such an object when \(G\) has a compact Cartan subgroup (e.g. \(G=SL(2,\mathbb{R})\), \(SU(p,q)\) etc.) by using symplectic techniques. In this case, if \(B\) is a Borel subgroup of the complexification \(G^\mathbb{C}\) of \(G\) which contains a compact Cartan subgroup of \(G\), the set \(G\cdot B\) is open in \(G^\mathbb{C}\) and the \(G\)-orbit of \(eP\) in \(G^C/P\), where \(P\supset B\) is a parabolic subgroup of \(G^\mathbb{C}\), is an open \(G\)-orbit. The construction of the holomorphic discrete model is achieved by studying line bundles over the inverse images of these distinguished \(G\)-orbits in flag manifolds \(G^\mathbb{C}/P\) via the fibration \(G^\mathbb{C}/(P,P) \to G^\mathbb{C}/P\), and by using the machinery of geometric quantization. The paper has many nice ideas and the author has taken pains in explaining these ideas and the methodology of the proofs.