Characteristic functions of semigroups in semi-simple Lie groups

Abstract

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G = K ⁢ A ⁢ N {G=KAN} . For a semigroup S ⊂ G {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform I S ⁢ ( λ , u ) = ∫ S e λ ⁢ ( 𝖺 ⁢ ( g , u ) ) ⁢ 𝑑 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 ∈ 𝔞 ∗ {\lambda\in\mathfrak{a}^{\ast}} , 𝔞 {\mathfrak{a}} is the Lie algebra of A and g ⁢ u = k ⁢ e 𝖺 ⁢ ( 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 𝔽 Θ ⁢ ( S ) {\mathbb{F}_{\Theta(S)}} called the flag type of S, where Θ ⁢ ( S ) {\Theta(S)} is a subset of the simple system of roots. It is proved that I S ⁢ ( λ , u ) < ∞ {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from Θ ⁢ ( S ) {\Theta(S)} and u ∈ π - 1 ⁢ ( 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ) {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ⊂ 𝔽 Θ ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and π : K → 𝔽 Θ ⁢ ( S ) {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of I S ⁢ ( λ , u ) {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in 𝒟 Θ ⁢ ( S ) ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)} invariant by the group S ∩ S - 1 {S\cap S^{-1}} of units of S.