# parabolic subgroup

Let $G$ be a complex semi-simple Lie group. Then any subgroup $P$ of $G$ containg a Borel subgroup $B$ is called parabolic. Parabolics are classified in the following manner. Let $\mathfrak{g}$ be the Lie algebra of $G$, $\mathfrak{h}$ the unique Cartan subalgebra contained in $\mathfrak{b}$, the algebra of $B$, $R$ the set of roots corresponding to this choice of Cartan, and $R^{+}$ the set of positive roots whose root spaces are contained in $\mathfrak{b}$ and let $\mathfrak{p}$ be the Lie algebra of $P$. Then there exists a unique subset $\Pi_{P}$ of $\Pi$, the base of simple roots associated to this choice of positive roots, such that $\{\mathfrak{b},\mathfrak{g}_{-\alpha}\}_{\alpha\in\Pi_{P}}$ generates $\mathfrak{p}$. In other words, parabolics containing a single Borel subgroup are classified by subsets of the Dynkin diagram, with the empty set corresponding to the Borel, and the whole graph corresponding to the group $G$.

Title parabolic subgroup ParabolicSubgroup 2013-03-22 13:28:02 2013-03-22 13:28:02 bwebste (988) bwebste (988) 4 bwebste (988) Definition msc 17B20