Spherical functions on mixed symmetric spaces (Q2761175)

In this article the authors investigate spherical functions on simply connected symmetric spaces \({\mathcal M}:=H\setminus G\) that are hyperbolically ordered. Recall that this means that with respect to the corresponding symmetric decomposition \({\mathfrak g}={\mathfrak h}+ {\mathfrak q}\) of the Lie algebra of \(G\), there exists an open convex cone \(C\) in \({\mathfrak q}\) that is Ad\((H)\)-invariant and hyperbolic, that is, all operators ad\(X\), \(X\in C\), are diagonalizable over the real numbers. Such spaces are not generally semisimple, a prominent example being the Jacobi group \(G=\text{HSp}(n,{\mathbb R}):=H_n\rtimes\text{Sp}(n,{\mathbb R})\) with \(H_n\) the \((2n+1)\)-dimensional Heisenberg group and subgroup \(H={\mathbb R}^n\rtimes\text{Gl}(n,{\mathbb R})\). Using the rather detailed structure theory that one has available in this setting, the authors first develop a suitable notion of a spherical function in their context, a necessity since the group \(H\) no longer need be unimodular. The main focus and result of the paper is a fundamental factorization formula for these spherical functions.