Holomorphic discrete models of semisimple Lie groups and their symplectic constructions
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.
scientific article; zbMATH DE number 1507744
scientific article