English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Pro

Presents the zaia usage as premium feature

ZAIA

Presents AI Discovery search usage as a premium feature

AI Discovery

Presents AI Discovery explain and summary usage as a premium feature

Content Analysis

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Presents the pdf analysis as premium feature

PDF Analysis

1500 Top-up

3000 Top-up

Zendy Basic

2500 Top-up

Zendy Unlimited

Presents the access of premium content as premium feature

Premium Content

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