Clans defined by representations of Euclidean Jordan algebras and associated basic relative invariants (Q2846882)

Given a finite dimensional simple Euclidean Jordan algebra \((V, \langle \cdot| \cdot\rangle)\) along with a representation \(\varphi\neq 0\) with self-adjoint operators in a Euclidean space \((E,\langle\cdot| \cdot\rangle_E)\), the authors introduce the operation \((\xi+x)\Delta(\eta+y):=\underline{\varphi}(\eta)+(Q(\xi,\eta)+x\Delta_0 y)\) on the space \(V_E:= E\oplus V\) where \(\underline{\varphi}(x)\) is the lower triangular part \(\varphi(x)\) with respect to a fixed Jordan frame in \(V\) whose sum is the unit element of \(V\), \(\Delta_0\) is the the clan product associated by Vinberg's canonical construction to the self-dual cone \(\Omega:=\text{Int}\{ x^2:\;x\in V\}\) described in terms of an Iwasava solvable subgroup of the linear automorphism group of \(\Omega\), and \(Q: E\times E\to V\) is the \(\Omega\)-positive symmetric bilinear map determined by the requirement \(\langle \varphi(x)\xi | \eta\rangle_E =\langle Q(\xi,\eta)| x \rangle\).NEWLINENEWLINEThe first goal of the paper is Theorem 3.2 establishing that the algebra \((V_E,\Delta)\) is indeed a clan structure, necessarily without unit element. The main efforts the are paid to the explicit description of the clan \(V_E^0:={\mathbb R}\oplus V_E\) obtained from \(V_E\) by adjoining a unit. In this case, the Siegel domain \(D(\Omega,Q)\) corresponding to \(V_E\) by Vinberg's construction is the cross section of the hyperplane \(e-e_0+V_E\) with the cone \(\Omega^0\) associated with \(V_E^0\) where \(e\) and \(e_0\) are the respective unit elements in \(V\) and \(V_E^0\). According to previous results by H. Ishi and the second author (cited in Theorems 1.2--3), the cone \(\Omega^0\) associated with \((V_E^0,\Delta)\) is the intersection of the positivity domains of the irreducible factors of the determinant \(\det R^0(v)\) of the right multiplication operators \(R^0(v)\) on \(V_E^0\), which are the basic relative invariants of \(\Omega^0\). Hence the technical second half of main interest is devoted to a complete explicit parametric description of these relative invariants along the lines of the classification of all simple Euclidean Jordan algebras treating the Hermitian and Lorentzian cases, summarized in the tables of Theorem 5.4 with Corollaries 5.5-6 and those of Proposition 5.7, with Theorem 5.8, respectively. The work is concluded by a similar description of the dual clan of \(V_E^0\).