SOME PROPERTIES OF REPRODUCING KERNEL CARTAN SUBALGEBRA

Abstract

Let $\mathfrak{j}$ and $\mathfrak{j}^{'}$ be the Cartan subalgebras of the complex semi-simple Lie algebras $\mathfrak{g}$ and $\mathfrak{g}^{'}$, $\mathfrak{j}^{*}$ and $(\mathfrak{j}^{'})^{*}$ their duals, $\mathfrak{j}^{\vee}$ and $(\mathfrak{j}^{'})^{\vee}$ the biduals of $\mathfrak{j}$ and $\mathfrak{j}^{'}$ respectively. We consider $B(.,.)$, the restriction to $\mathfrak{j}$ and to $\mathfrak{j}^{'}$ of the Killing form of $\mathfrak{g}$ and $\mathfrak{g}^{'}$. In this work, using the kernel $K$ of the reproducing kernel Cartan subalgebra $\mathfrak{j}^{\vee}$ and an operator $\Phi$ from $\mathfrak{j}^{*}$ to $(\mathfrak{j}^{'})^{*}$, we construct another reproducing kernel Cartan subalgebra denoted by $\mathfrak{j}_{\Phi}^{\vee}$ obtained from the kernel $K \circ \Phi$ and study the relationships between $\mathfrak{j}^{\vee}$, $\mathfrak{j}_{\Phi}^{\vee}$ and $(\mathfrak{j}^{'})^{\vee}$.