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On a character associated to a representation of Cartan
subgroup of an acceptable Lie group

Institute of Mathematics and Physics Sciences
Porto-Novo, Republic of Bénin

Department of Mathematics and computer
Université Félix Houphouët Boigny
Abidjan Côte d'Ivoire

Mathematics Subject Classification: (MSC 2010) 22D30, 22E45, 43A40
Key words : Polarization at a representation, π-character, induced representation and acceptable Lie groups


Let G be a connected semi-simple Lie group, 𝔤 its Lie algebra, j a Cartan subalgebra of 𝔤, 𝔧c be a complexification of 𝔧 and Jc the analytic Cartan subgroup associated with 𝔧c. Let Φ denote  the set of roots of the pair (𝔤c,𝔧c). If α is an element of Φ, then there exists a holomorphic homomorphism ξα of Jc into * such that :

ξα(expH ) = eα(H) ∀ H ∈ 𝔧c

Let π be a representation of 𝔧c in a finite-dimensional vector space V. The homomorphism ξπ associated to the representation π will be called a π-characte.

In this work, some results concernig this character are obtained and proved and after defining a polarization at π, the irreducibility of an induced representation is computed when G is simply connected nilpotent Lie group. The particular case where π is a linear form of 𝔧c has been studied in [6].

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