Spherical Functions on Olʹshanskii Spaces

Olʹshanskii spaces for semisimple groups are special cases of ordered symmetric spaces. The theory of spherical functions on semisimple Olʹshanskii spaces is extended to general Olʹshanskii spaces. The detailed structure theory for Lie algebras with invariant cones is used to generalize geometric results for Olʹshanskii spaces. Spherical functions are then defined via a set of integral equations using a Volterra algebra that consists ofG-invariant kernels satisfying a causality condition. It is shown that the resulting integral spherical formulas suffice to build a spherical Laplace transform and an inversion formula for this transform.