Domains of holomorphy and representations of \(SL(n,\mathbf R)\) (Q5957334)

For \(G=SL(n,\mathbb{R})\) and \(K=SO(n)\) Akhiezer and Gindikin realized the Riemannian symmetric space \(G/K\) as a totally real \(G\)-invariant submanifold of a Stein manifold \({\mathcal A}\), where \({\mathcal A}\subset G_C/K_C\). The present article gives several other descriptions of \({\mathcal A}\) in terms of the flag variety for \(G_C\). It is also shown that \({\mathcal A}\) is a domain of holomorphy for Szegő kernels associated to interesting irreducible \(G\)-representations.