Spherical Functions on Ol'shanskii Spaces

Abstract

Ol'shanski@$ spaces for semisimple groups are special cases of ordered symmetric spaces. The theory of spherical functions on semisimple Ol'shanski@ $ spaces is extended to general Ol'shanski@$ spaces. The detailed structure theory for Lie algebras with invariant cones is used to generalize geometric results for Ol'shanski@$ spaces. Spherical functions are then defined via a set of integral equations using a Volterra algebra that consists of G-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.1996 Academic Press, Inc.

Spherical functions on mixed symmetric spaces

“…We note that the Factorization Theorem gives a concrete formula for ϕ χ λ on solvable symmetric spaces. Our product formula generalizes the one in [HiNe96] for spherical functions on Ol'shanskiȋ spaces G\G C . Spherical functions on reductive symmetric spaces and their asymptotic expansions have been studied extensively by the third author in [Ól97].…”

“…For a fixed χ ∈ X(H/H 0 ) there exists an interesting family of spherical functions (ϕ χ λ ) λ , where λ runs over a certain subset of a * C , whose construction we describe below. The construction is motivated by the special cases considered in [FHÓ94], [HiNe96] and [Ól97].…”

Positive definite spherical functions on Olshanskii domains

“…Generalizing results from [FHÓ94], we constructed in [HiNe96] a family of spherical functions ϕ µ on M + via an integral formula similar to Harish-Chandra's integral representation of spherical functions on non-compact Riemannian symmetric spaces. These functions are parametrized by a subset t * + E L ⊆ t * C , where t is a suitable Cartan subalgebra of h. In this paper we introduce a concept of positive definite spherical functions on positive domains, and in the case described above we determine all of these.…”

“…These functions are parametrized by a subset t * + E L ⊆ t * C , where t is a suitable Cartan subalgebra of h. In this paper we introduce a concept of positive definite spherical functions on positive domains, and in the case described above we determine all of these. In particular we determine which of the integral spherical functions constructed in [HiNe96] are positive definite spherical functions (Theorem 5.2).…”

“…In [HiNe96] we studied a family of functions ϕ µ : Γ(C 0 ) → C with µ ∈ t * h + E L ⊆ a * C , where −iE L turned out to be the interior of the dual cone C min,h of the minimal cone in t h defined by (3.3). These functions were defined by the integrals…”