Representation theory and convexity (Q5927627): Difference between revisions

In Section I the essential results on Ol'shanskii semigroups \(S=G \operatorname {Exp} (iW)\), where \(G\) is a Lie group, and invariant convex functions are collected. These semigroups are complex domains generalizing complex reductive groups that show up for \(W=\mathfrak g\) where \(\mathfrak g\) is a compact Lie algebra. Let \(D\subset S\) be a \(G\)-biinvariant domain. Then \(G\times G\) acts on \(D\), and one has to do with harmonic analysis of Hilbert spaces of holomorphic functions \({\mathcal H}\subset Hol(D)\) on which \(G\times G\) acts unitarily. These spaces are called biinvariant Hilbert spaces. In Section II it is shown that irreducible biinvariant Hilbert spaces come from highest weight representations. For example, the unitary representation of \(G\times G\) on a biinvariant Hilbert space \({\mathcal H}\) extends to an action of the semigroup \(S_{\min}\times S_{\min}\) by contractions on \({\mathcal H}\). In Section III the author explains how to construct biinvariant Kähler structures on biinvariant Stein domains and demonstrates with the help of a certain Legendre transform that the studied symplectic manifolds are isomorphic to domains in the cotangent bundle \(T^*(G)\).