Positive definite spherical functions on Olshanskii domains

Abstract

Let G G be a simply connected complex Lie group with Lie algebra g \mathfrak {g} , h \mathfrak {h} a real form of g \mathfrak {g} , and H H the analytic subgroup of G G corresponding to h \mathfrak {h} . The symmetric space M = H ∖ G {\mathcal {M}}=H\backslash G together with a G G -invariant partial order ≤ \le is referred to as an Ol ′ ’ shanskiĭ space. In a previous paper we constructed a family of integral spherical functions ϕ μ \phi _{\mu } on the positive domain M + := { H x : H x ≥ H } {\mathcal {M}}^{+} := \{Hx\colon Hx\ge H\} of M {\mathcal {M}} . In this paper we determine all of those spherical functions on M + {\mathcal {M}}^{+} which are positive definite in a certain sense.