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Journal of Lie Theory 20 (2010), No. 4, 673--687
Copyright Heldermann Verlag 2010

Principal Basis in Cartan Subalgebra

Rudolf Philippe Rohr
Dept. of Mathematics, University of Geneva, 2-4 rue de Lievre / C.P.64, 1211 Geneva 4, Switzerland


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

Keywords: Lie algebra, Cartan subalgebra, principal basis, Langlands dual.

MSC: 17B

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