On the complex and convex geometry of Ol'shanskii semigroups
Let \(K\) be a connected real Lie group with compact Lie algebra and \(K_C\) its complexification which is a complex reductive Lie group. Let \(t\) be a Cartan subalgebra. Each \(K\)-biinvariant domain \(D\subset K_C\) can be written as \(D=K\exp SK\), where \(S\subset it\) is a domain which is invariant under the Weyl group of \(K\). Azad and Loeb gave a characterization of the \(K\)-biinvariant plurisubharmonic functions on \(D\) under the assumption that \(D\) is convex. The author generalizes the ``biinvariant'' results to the following setting. Let \(g\) be a finite dimensional real Lie algebra with a compactly embedded Cartan subalgebra \(t\). Under some mild additional assumptions on the corresponding root system, there exists a generating invariant closed convex cone \(W_{\max}\) in \(g\) which is maximal with respect to the property that all elements in its interior are elliptic, i.e., conjugate to elements of \(t\). In the special case of a compact Lie algebra we have \(W_{\max}=g\), but the considered setting also incorporates hermitian simple Lie algebras, a certain class of solvable Lie algebras and also a lot of mixed Lie algebras which are neither reductive nor solvable.
scientific article; zbMATH DE number 1202022
scientific article