Tube domain and an orbit of a complex triangular group
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 occurrence for non-symmetric cases, and the examples given in the paper are rather exceptional. In fact, the authors have verified that there is only one cone (i.e., an example in dimension 8) that has the property of Lemma 1.2 among the irreducible non-symmetric homogeneous convex cones for dimensions up to 10. As a byproduct of this study, the authors show that the basic relative invariants associated to a homogeneous regular open convex cone \(\Omega\) studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to \(\Omega\).
subgroup of strictly lower triangular matrices
complexification of Euclidean Jordan algebra
scientific article; zbMATH DE number 5279029