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Let  G  be a noncompact semi-simple Lie group with Iwasawa decompositionG  =   K  ⁢  A  ⁢  N{G=KAN}   . For a semigroupS  ⊂  G{S\subset G}    with nonempty interior we find a domain of convergence of the Helgason–Laplace transformI  S   ⁢   (  λ  ,  u  )    =    ∫  Se   λ  ⁢   (     ⁢   (  g  ,  u  )    )⁢     g{I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg}   , where  dg  is the Haar measure of  G ,u  ∈  K{u\in K}   ,λ  ∈     ∗{\lambda\in\mathfrak{a}^{\ast}}   ,{\mathfrak{a}}    is the Lie algebra of  A  andg  ⁢  u   =   k  ⁢   e     ⁢   (  g  ,  u  )⁢  n   ∈   K  ⁢  A  ⁢  N{gu=ke^{\mathsf{a}(g,u)}n\in KAN}   . The domain is given in terms of a flag manifold of  G  written   Θ  ⁢   (  S  ){\mathbb{F}_{\Theta(S)}}    called the flag type of  S , whereΘ  ⁢   (  S  ){\Theta(S)}    is a subset of the simple system of roots. It is proved thatI  S   ⁢   (  λ  ,  u  )    &lt;  ∞{I_{S}(\lambda,u)&lt;\infty}    if λ belongs to a convex cone defined fromΘ  ⁢   (  S  ){\Theta(S)}    andu  ∈    π   -  1    ⁢   (       Θ  ⁢   (  S  )⁢   (  S  )    ){u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))}   , where   Θ  ⁢   (  S  )⁢   (  S  )    ⊂      Θ  ⁢   (  S  ){\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}}    is a  B -convex set andπ  :   K  →      Θ  ⁢   (  S  ){\pi:K\rightarrow\mathbb{F}_{\Theta(S)}}    is the natural projection. We prove differentiability ofI  S   ⁢   (  λ  ,  u  ){I_{S}(\lambda,u)}    and apply the results to construct of a Riemannian metric in   Θ  ⁢   (  S  )⁢   (  S  ){\mathcal{D}_{\Theta(S)}(S)}    invariant by the groupS  ∩   S   -  1{S\cap S^{-1}}    of units of  S .

Characteristic functions of semigroups in semi-simple Lie groups