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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


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.

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