Basic relative invariants of homogeneous cones and their Laplace transforms
type: Item
instance of: scholarly article
title: Basic relative invariants of homogeneous cones and their Laplace transforms
zbMATH Open document ID: 1451.22002
author: Hideto Nakashima
published in: Journal of the Mathematical Society of Japan
publication date: 2018-04-13T00:00:00Z
review text: In this paper a characterization of symmetric cones \(\Omega\) among irreducible homogeneous cones is given by using the Laplace transforms of relatively \(H\)-invariant functions on \(\Omega\). Actually, \(\Omega\) is symmetric if and only if there exists a non-constant \(H\)-invariant polynomial \(\Delta^{\underline{\nu}}(x)\) such that its Laplace transform is the reciprocal of a certain polynomial. Let \(\Omega\) be a homogeneous cone of rank \(r\) in a finite dimensional real vecor space \(V\) and \(\bigtriangleup\) the Vinberg algebraic structure arisen from the action of a split solvable Lie group on \(\Omega\). Let \(\Delta_1, \dots,\Delta_r\) be the basic relative invariants of \(\Omega\), that is, irreducible relatively \(H\)-invariant polynomial functions on \(\Omega\). For \(\underline{\nu}=(\nu_1,\dots, \nu_r)\in{\mathbb Z}^r\), let \(\Delta^{\underline{\nu}}(x)\) be a rational function on \(\Omega\) defined by \(\Delta_1^{\nu_1}(x)\dots\Delta_r^{\nu_r}(x)\). For \(y\in \Omega^*\), the dual of \(\Omega\), we define a rational function \(\Delta_*^{\underline{\mu}}(y)\) similarly. Then Gindikin shows that for a suitable \(\underline{\nu}\), the Laplace transform of \(\Delta^{\underline{\nu}}\) is given by the inverse of \(\Delta_*^{\underline{\nu'}}(y)\). To prove the property that the existence of \(\Delta^{\underline{\nu}}\) and \(\Delta_*^{\underline{\nu'}}(y)\) such that both are non-constant polynomials characterizes symmetric cones, the author uses the inductive structure of \(\Delta_i\). We note that \((V, \Delta)\) has the normal decomposition \(V=\bigoplus_{1\leq j\leq k\leq r}V_{kj}\) and for \(p+q=r\). We put \[ V_-=\bigoplus_{1\leq j\leq k\leq p}V_{kj},\ \ E=\bigoplus_{1\leq j\leq p<k}V_{kj},\ \ V_+=\bigoplus_{p<j\leq k \leq r}V_{kj}. \] We denote by \(\Delta_1^-, \dots,\Delta_p^-\) and \(\Delta_+, \dots,\Delta_q^+\) the basic invariants of \(\Omega_-\) and \(\Omega_+\) respectively. Moreover we denote by \(R\) and \(R_\pm\) the right multipli…
reviewed by: Takeshi Kawazoe
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 11S90
zbMATH DE Number: 6859854
zbMATH Keywords: Laplace transforms
zbMATH Keywords: symmetric cones
zbMATH Keywords: homogeneous cones
zbMATH Keywords: basic relative invariants
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/07017447
OpenAlex ID: W2784661908
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DOI: 10.2969/JMSJ/07017447
description: scientific article; zbMATH DE number 6859854