Semisimple Graded Lie Algebras

Abstract

Let $\mathfrak{g}$ be a real semisimple Lie algebra and τ be a Cartan involution of $\mathfrak{g}$ and let $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ be the Cartan decomposition by τ, where $\tau {{|}_{\mathfrak{k}}} = 1$ and $\tau {{|}_{\mathfrak{p}}} = - 1$. Let a be $\mathfrak{a}$ maximal abelian subspace of $\mathfrak{p}$ and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ containing $\mathfrak{a}$. Then we have $\mathfrak{h} = {{\mathfrak{h}}^{ + }} + \mathfrak{a}$, where ${{\mathfrak{h}}^{ + }} = \mathfrak{h} \cap \mathfrak{k}$ and $\mathfrak{a} = \mathfrak{h} \cap \mathfrak{p}$. Let ${{\mathfrak{g}}^{\mathbb{C}}}$ and ${{\mathfrak{h}}^{\mathbb{C}}}$ be the complexifications of $\mathfrak{g}$ and $\mathfrak{h}$. Then ${{\mathfrak{h}}^{\mathbb{C}}}$ is a Cartan subalgebra of ${{\mathfrak{g}}^{\mathbb{C}}}$. Let $\tilde{\Delta } = \Delta ({{\mathfrak{g}}^{\mathbb{C}}},{{\mathfrak{h}}^{\mathbb{C}}})$ be the root system for the pair left $({{\mathfrak{g}}^{\mathbb{C}}},{{\mathfrak{h}}^{\mathbb{C}}})$. If we put ${{\mathfrak{h}}_{\mathbb{R}}} = i{{\mathfrak{h}}^{ + }} + \mathfrak{a}$ then any root is real-valued on the real subspace ${{\mathfrak{h}}_{\mathbb{R}}}$ of ${{\mathfrak{h}}^{\mathbb{C}}}$. Since the Killing form B of $\mathfrak{g}$ is positive-definite on ${{\mathfrak{h}}_{\mathbb{R}}}$, a root $\alpha \in \tilde{\Delta }$ can be viewed as an element of ${{\mathfrak{h}}_{\mathbb{R}}}$. We have thus $\tilde{\Delta } \subset {{\mathfrak{h}}_{\mathbb{R}}}$. Let σ be the conjugation of ${{\mathfrak{g}}^{\mathbb{C}}}$ with respect to $\mathfrak{g}$ . Then $\sigma {{|}_{\mathfrak{a}}} = 1$ and $\sigma {{|}_{{i{{\mathfrak{h}}^{ + }}}}} = - 1$, and hence σ leaves ${{\mathfrak{h}}_{\mathbb{R}}}$ stable. Therefore σ permutes roots in $\tilde{\Delta }$. Let us put ${{\tilde{\Delta }}_{ \bullet }} = \tilde{\Delta } \cap i{{\mathfrak{h}}^{ + }}$, the set of imaginary roots with respect to $\mathfrak{h}$. We then have ${{\tilde{\Delta }}_{ \bullet }} = \{ \alpha \in \tilde{\Delta }:\sigma (\alpha ) = - \alpha \}$. A lexicographic order > on $\tilde{\Delta }$ is called a σ-order, if σis order-preserving on $\tilde{\Delta } - {{\tilde{\Delta }}_{ \bullet }}$, or σ (α)> 0, as long as α > 0, $\alpha \in \tilde{\Delta } - {{\tilde{\Delta }}_{ \bullet }}$. Such an order is given by choosing a basis $\{ {{H}_{1}}, \ldots {{H}_{r}},{{H}_{{r + 1}}}, \ldots ,{{H}_{l}}\}$ of ${{\mathfrak{h}}_{\mathbb{R}}}$ such that $\{ {{H}_{1}}, \ldots ,{{H}_{r}}\}$ is a basis of $\mathfrak{a}$. Now let us fix a σ-order in $\tilde{\Delta }$. The simple root system $\tilde{\prod } = \{ {{\alpha }_{1}}, \ldots ,{{\alpha }_{l}}\}$ of $\tilde{\Delta }$ with respect to this σ-order is called aσ -fundamental system of $\tilde{\Delta }$. The subset ${{\tilde{\prod }}_{ \bullet }} = \tilde{\prod } \cap {{\tilde{\Delta }}_{ \bullet }}$ of $\tilde{\prod }$ is a basis for ${{\tilde{\Delta }}_{ \bullet }}$.