Basic relative invariants associated to homogeneous cones and applications (Q2718664)

Let \(\Omega\) be a homogeneous cone in a real vector space \(V\). Then a split solvable Lie group \(H\subset GL(V)\) acts simply transitively on \(\Omega\). The author determines all the polynomials on \(V\) relatively invariant under the action of \(H\) and then the basic ones, namely the polynomials by which every relatively invariant polynomial is expressed as a product of their powers. The cone \(\Omega\) is characterized as the set on which all the basic relative invariants are positive. Let \(r\) be the rank of the cone \(\Omega\). There are exactly \(r\) basic relative invariants.NEWLINENEWLINENEWLINEThe Riesz distribution on \(\Omega\) is characterized as the distribution whose Laplace transform is a relatively invariant function on the dual cone \(\Omega^*\) under the adjoint action of \(H\). It is shown that the Riesz distribution is supported by the origin if and only if its Laplace transform is a product of powers of the basic relative invariants associated to \(\Omega^*\). Next, the author defines a polynomial \(\Delta^I\) for each nonempty subset \(I\subset\{1,2,\dots,r\}\) so that \(\overline\Omega\) is characterized by the inequalities \(\Delta^I\geq 0\).NEWLINENEWLINENEWLINEHe finally examines two typical examples; the cone of real positive definite symmetric matrices and the so-called Vinberg cone. The two main references are \textit{S. G. Gindikin} [Usp. Mat. Nauk 19, 3-92 (1964; Zbl 0144.08101) and \textit{E. B. Vinberg} [Trans. Mosc. Math. Soc. 12, 340-403 (1963); translation from Tr. Mosk. Mat. O.-va 12, 303-358 (1963; Zbl 0138.43301)].