Bessel functions associated with representations of formally real Jordan algebras (Q1088109)

Let U be a formally real Jordan algebra, E a Euclidean space, \(\phi\) : \(U\to Sym^+(E)\) a representation of U, where \(Sym^+(E)\) denotes the real linear space of all symmetric endomorphisms of E equipped with the Jordan product \(A\circ B=(AB+BA)\). Denote by Q(\(\xi)\) the corresponding vector-valued quadratic form and by \(\Sigma\) the Stiefel manifold \(\{\xi\in E:\) \(Q(\xi)=e\}\), where e is the identity of U. The authors introduce generalized Bessel functions \[ J(r,s)=\int_{\Sigma}\exp \{-i (\phi (r^{1/2})\sigma, \phi (s^{1/2})\sigma_ 0)\quad d\beta (\sigma)\quad, \] where \(\beta\) is the Riemannian volume form on \(\Sigma\). They extend the classical results of radial Fourier analysis to this case and prove an asymptotic formula.