Equivariant embeddings of Stein domains sitting inside of complex semigroups (Q1305104)

If \(G\) is a connected Lie group sitting in its complexification \(G_{\mathbb{C}}\) , if \(D_{h}\) is a Ad(\(G\))-invariant convex domain consisting of elliptic elements in the Lie algebra of \(G\) , the author shows that, under some quite natural assumptions, there exists a \(G \times G\) equivariant embedding into some complex Hilbert space (endowed with a unitary \(G \times G\)-action) of the biinvariant Stein domain \(D=G\exp_{G_{\mathbb{C}}}(iD_{h})\). The construction of such a closed embedding is done by means of some holomorphic positive definite kernels tending to infinity at the boundary. Some interesting examples and open problems are also presented.