Basic relative invariants of homogeneous cones and their Laplace transforms
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 multiplication operators on \(V\) and \(V_\pm\) respectively. Then by calculating the irreducible factors in the determinants of \(R\) and \(R_\pm\) we see that \(\Delta_i\) can be written as a product of \(\Delta_s^-\) and \(\Delta_t^+\). Then, applying this relation for \((p,q)=(r-1,1)\) and using the induction on \(r\), we can deduce the desired result.
scientific article; zbMATH DE number 6859854