Tube domain and an orbit of a complex triangular group
type: Item
instance of: scholarly article
title: Tube domain and an orbit of a complex triangular group
zbMATH Open document ID: 1143.22005
author: Hideyuki Ishi
author: Takaaki Nomura
published in: Mathematische Zeitschrift
publication date: 2008-05-27T00:00:00Z
full work available at URL: http://hdl.handle.net/2324/3334
review text: Let \(V\) be the real vector space \(\text{Sym}(r,\mathbb R)\) of \(r\times r\) real symmetric matrices. Denote by \(\Omega\) the cone of positive definite matrices in \(V\). We have the corresponding tube domain \(\Omega + i V\) in the complexification \(W := V_{\mathbb C}\), the space of \(r\times r\) complex symmetric matrices. Let \(A_\mathbb C\) denote the subgroup of \(GL(r,\mathbb C)\) consisting of diagonal matrices, and \(N_\mathbb C\) the subgroup of strictly lower triangular matrices. The complex triangular group mentioned in the title is \(T_\mathbb C := N_\mathbb CA_\mathbb C\). Let \(\Delta_1(w),\dots, \Delta_r(w)\) be the principal minors of \(w\in W\). The following proposition is known. Proposition 1.1. Let \(w\in W\) and \(w\in \Omega+iV\). Then \[ \text{Re}\,\frac{\Delta_k(w)}{\Delta_{k-1}(w)}> 0,\quad \text{for}\quad k = 1,\dots, r, \] where we understand \(\Delta_0(w)\equiv1\). Since Proposition 1.1 can be generalized to any symmetric cone \(\Omega\), the question arises if the validity of Proposition 1.1 is characteristic of symmetric cones (or of symmetric tube domains). The following lemmas are proved. Lemma 1.2. Let \(w\in W\) and suppose \(w\in T_\mathbb C\cdot I_r\). Then, writing \(w\) as \(w = na^{\text{t}}n\) with \(n\in N_\mathbb C\) and \(a := \text{diag}[a_1,\dots, a_r]\in A_\mathbb C\), one has \[ a_k =\frac{\Delta_k(w)}{\Delta_{k-1}(w)}, \quad k = 1,\dots, r. \] The authors present a proof of this Lemma in the general case where \(W\) is the complexification of a Euclidean Jordan algebra. Lemma 1.3. Suppose \(na^{\text t}n\in \Omega + iV\) for \(n\in N_\mathbb C\) and \(a =\text{diag}[a_1,\dots, a_r]\in A_\mathbb C\). Then \[ \text{Re}\,a_1> 0,\dots,\text{Re}\,a_r> 0. \] This Lemma is generalized to the case of homogeneous convex cones. It turns out that the validity of Proposition 1.1 depends solely on the validity of Lemma 1.2 for general homogeneous convex cones. Experiments say that the failure of Lemma 1.2 is a frequent occurr…
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32A07
zbMATH DE Number: 5279029
zbMATH Keywords: subgroup of strictly lower triangular matrices
zbMATH Keywords: space of complex symmetric matrix
zbMATH Keywords: tube
zbMATH Keywords: (non-)symmetric cones
zbMATH Keywords: complexification of Euclidean Jordan algebra
zbMATH Keywords: homogeneous convex cones
zbMATH Keywords: complexification of clan
MaRDI profile type: Publication
OpenAlex ID: W2057928936
cites work: Q4321748
cites work: Basic relative invariants associated to homogeneous cones and applications
cites work: On symplectic representations of normal \(j\)-algebras and their application to Xu's realizations of Siegel domains
cites work: Classification of homogeneous bounded domains of lower dimension
cites work: Q4791726
cites work: On Penney's Cayley transform of a homogeneous Siegel domain
cites work: Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group
cites work: The construction of homogeneous convex cones
cites work: Q5513583
cites work: Q4115417
DOI: 10.1007/S00209-008-0337-2
Recommended article: TUBE DOMAINS OVER Cn
Recommended article: A remark on tube domains
Recommended article: TUBE DOMAINS AND RESTRICTIONS OF MINIMAL REPRESENTATIONS
Recommended article: Q4549288
Recommended article: Characterizations of \(G\)-tube domains
Recommended article: Envelope of holomorphy of a tube domain over a complex Lie group
Recommended article: Q4865845
Recommended article: Structure and equivalence of a class of tube domains with solvable groups of automorphisms
Recommended article: Q3987878
Recommended article: Q3701768
description: scientific article; zbMATH DE number 5279029