Principal basis in Cartan subalgebra

Abstract

Let $\Lie{g}$ be a simple complex Lie algebra and $\Lie{h}$ a Cartan subalgebra. In this article we explain how to obtain the principal basis of $\Lie{h}$ starting form a set of generators $\{p_1,...,p_r\}$,$r=\rank(\Lie{g})$, of the invariants polynomials $\Sgg$. For each invariant polynomial $p$, we define a $G$-equivariant map $Dp$ form $\Lie{g}$ to $\Lie{g}$. We show that the Gram-Schmidt orthogonalization of the elements $\{Dp_1(ρ^\vee), ... Dp_r(ρ^\vee) \}$ gives the principal basis of $\Lie{h}$. Similarly the orthogonalization of the elements $\{Dp_1(ρ), >... Dp_r(ρ) \}$ produces the principal basis of the Cartan subalgebra of $\lLie{g}$, the Langlands dual of $\Lie{g}$.