SOME PROPERTIES OF REPRODUCING KERNEL CARTAN SUBALGEBRA

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