SPHERICAL-FUNCTIONS ON OLSHANSKII SPACES

Ol'shanskii spaces for semisimple groups are special cases of ordered symmetric spaces. The theory of spherical functions on semisimple Ol's hanskii spaces is extended to general Ol'shanskii spaces. The detailed structure theory for Lie algebras with invariant cones is used to gen eralize geometric results for Ol'shanskii spaces. Spherical functions are then defined via a set of integral equations using a Volterra alge bra that consists of G-invariant kernels satisfying a causality condit ion. It is shown that the resulting integral spherical formulas suffic e to build a spherical Laplace transform and an inversion formula for this transform. (C) 1996 Academic Press, Inc.