On the complex and convex geometry of Ol'shanskii semigroups
type: Item
instance of: scholarly article
title: On the complex and convex geometry of Ol'shanskii semigroups
zbMATH Open document ID: 0901.22003
publication date: 1998-09-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75274
review text: 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.
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32D05
Mathematics Subject Classification ID: 32D10
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 1202022
zbMATH Keywords: Ol'shanskii semigroup
zbMATH Keywords: classical Lie group
zbMATH Keywords: real Lie group
zbMATH Keywords: complex reductive Lie group
zbMATH Keywords: Lie algebras
author: Karl-Hermann Neeb
reviewed by: A. K. Guts
MaRDI profile type: Publication
OpenAlex ID: W2147584926
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DOI: 10.5802/AIF.1614
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1202022