AFRICA MATHEMATICS ANNALS

Reproducing Kernel Cartan Subalgebra

Anoh Yannick Kraidi ⁽¹⁾ and Kinvi Kangni ⁽²⁾
Université Félix Houphouët Boigny
⁽¹⁾ 25 BP 1245 Abidjan 25, Côte d’Ivoire
⁽²⁾ 25 BP 1214 Abidjan 25, Côte d’Ivoire

Mathematics Subject Classification: (MSC 2010) 7B20, 46E22, 17B22
Key words : Cartan subalgebra, reproducing kernel, root systems

Abstract:

Let 𝔤 be a semi-simple Lie algebra, 𝔧 a Cartan subalgebra of 𝔤, 𝔧^* the dual of 𝔧, 𝔧^∨ the bidual of 𝔧 and B(.,.) the restriction to 𝔧 of the Killing form of 𝔤. In this work, we define the inversion formula on 𝔧^* using a linear map L on 𝔧^* and the reproducing kernel defined on the bidual 𝔧^∨ of 𝔧. When we consider the transform of the reproducing kernel obtained from the image of a representation of 𝔤 on 𝔧^*, we get a relationship between these two kernels. We give a generalization of the theorem of decreasing principle for operators in the classical case. After that, we will prove some properties of the reproducing kernel Cartan subalgebra.