Hypergeometric functions associated to cones of convex type in a Euclidean Jordan algebra
The paper investigates properties of a family of functions introduced by \textit{S. Sahi} [Invent. Math. 110, No.2, 409-418 (1992; Zbl 0779.22006)]. Let \(V\) be a simple Euclidean Jordan algebra of rank \(r\) and identity element \(e\). For a complete system of idempotents \(\left( X_{1},X_{2},\ldots ,X_{r}\right) \) of \(V\) and \(0\leq p+q\leq r\), let \[ \mathcal{E}_{p,q}=X_{1}+\cdots+X_{p}-X_{p+1}-\cdots-X_{p+q}. \] Let \(L\) be the structure group of \(V\), \(\mathcal{O}_{p,q}=L\mathcal{E}_{p,q}\) (\(0\leq p+q\leq r\)) the orbits of \(L\) in \(V\), \(S_{p,q}\;\)the stabilizer of \(\mathcal{E}_{p,q}\) in \(L\) and \(\Omega_{p,q}=S_{p,q}.\mathcal{E}_{p+q,0}\). Then \(\Omega_{p,q}\) is a Riemannian symmetric space admitting a \(S_{p,q} \)-invariant measure \(\text{d}\lambda_{p,q}\). Sahi's \(\mathbf{e}_{p,q}\) Gaussian functions are defined on \(\mathcal{O}_{p,q}\) by \[ \mathbf{e}_{p,q}\left( \ell.\mathcal{E}_{p,q}\right) =\int_{\Omega_{p,q} }\text{e}^{-\text{tr}\left( \ell.y\right) }\text{d}\lambda_{p,q}(y)\qquad\left( \ell\in L\right) , \] where \(\text{tr}\) is the trace of the Jordan algebra \(V\). When \(p+q<r\), the non-open orbit \(\mathcal{O}_{p,q}\) admits a unique (up to a scalar) \(L\)-quasi-invariant measure \(\text{d}\mu_{p,q}\); the authors give explicit estimations for the functions \(\mathbf{e}_{p,q}\) and prove that \[ \mathbf{e}_{p,q}\in L^{2}\left( \mathcal{O}_{p,q},\text{d}\mu _{p,q}\right) \qquad\left( p+q<r\right) , \] correcting a proof of S.Sahi (loc.cit.). When \(p+q=r\) and \(\text{d}X_{p,q}\) denotes the Euclidean measure of the Jordan algebra \(V\) restricted to the open orbit \(\mathcal{O}_{p,q}\), they prove \[ \mathbf{e}_{p,q}\in L^{2}\left( \mathcal{O}_{p,q},\left| \Delta\left( \ell.\mathcal{E}_{p,q}\right) \right| ^{d-2}\text{d}X_{p,q}\right) , \] where \(\Delta\) is the determinant of the Jordan algebra \(V\), and \(d\) the numerical invariant related to the dimension \(n\) of \(V\) and to the rank by \(n=r+r(r-1)d/2\). Despite the title of the paper, the proofs use mainly root systems and homogeneous spaces, rather than the Jordan theory. Estimates and square integrability are proved in a case-by-case investigation, using the classification of simple Euclidean Jordan algebras. For example, the study of \(\mathbf{e}_{1,1}\) in the exceptional Jordan algebra is done by reduction to the classical Jordan subalgebra of rank \(2\) and dimension \(10\). One should ask for an extension of such a reduction to the general case, as well as for a unified treatment of all cases in the frame of Jordan theory.
scientific article; zbMATH DE number 1836690