Abstract
Let G be a noncompact semi-simple Lie group with Iwasawa decomposition
{G=KAN}
. For a semigroup
{S\subset G}
with nonempty interior we find a
domain of convergence of the Helgason–Laplace transform
{I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg}
,
where dg is the Haar measure of G,
{u\in K}
,
{\lambda\in\mathfrak{a}^{\ast}}
,
{\mathfrak{a}}
is the Lie algebra of A and
{gu=ke^{\mathsf{a}(g,u)}n\in KAN}
. The domain is given in terms of a flag
manifold of G written
{\mathbb{F}_{\Theta(S)}}
called the
flag type of S, where
{\Theta(S)}
is a subset of the simple
system of roots. It is proved that
{I_{S}(\lambda,u)<\infty}
if λ belongs to a convex cone defined from
{\Theta(S)}
and
{u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))}
, where
{\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}}
is a B-convex set
and
{\pi:K\rightarrow\mathbb{F}_{\Theta(S)}}
is the natural
projection. We prove differentiability of
{I_{S}(\lambda,u)}
and apply the results to construct of a Riemannian metric in
{\mathcal{D}_{\Theta(S)}(S)}
invariant by the group
{S\cap S^{-1}}
of units of S.
Gelfand pairs admit an Iwasawa decomposition
