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

**Gaël NGAMBALI NDZAKIMA **

Institute of Mathematics and Physics Sciences

Porto-Novo, Republic of Bénin

**Kinvi KANGNI **

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

Abstract:

Let G be a connected semi-simple Lie group, 𝔤 its Lie algebra, j a Cartan
subalgebra of 𝔤, 𝔧_{c} be a complexification of 𝔧 and J_{c} the analytic Cartan subgroup
associated with 𝔧_{c}. Let Φ denote the set of roots of the pair . If α is an element
of Φ, then there exists a holomorphic homomorphism ξ_{α} of J_{c} into ℂ^{*} such
that :

Let π be a representation of 𝔧_{c} in a finite-dimensional vector space V. The
homomorphism ξ_{π} 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 𝔧_{c} has been studied in [6].