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 C* 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-character. In this work, some results concerning this character is 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 [3, 6]

Keywords: Polarization at a representation, π-character, induced representation and acceptable Lie groups