Chapter VII. Advanced Structure Theory

Abstract

The first main results are that simply connected compact semisimple Lie groups are in one-one correspondence with abstract Cartan matrices and their associated Dynkin diagrams and that the outer automorphisms of such a group correspond exactly to automorphisms of the Dynkin diagram. The remainder of the first section prepares for the definition of a reductive Lie group: A compact connected Lie group has a complexification that is unique up to holomorphic isomorphism. A semisimple Lie group of matrices is topologically closed and has finite center.

Reductive Lie groups $G$ are defined as 4-tuples $(G,K,\theta,B)$ satisfying certain compatibility conditions. Here $G$ is a Lie group, $K$ is a compact subgroup, $\theta$ is an involution of the Lie algebra $\mathfrak g_0$ of $G$, and $B$ is a bilinear form on $\mathfrak g_0$. Examples include semisimple Lie groups with finite center, any connected closed linear group closed under conjugate transpose, and the centralizer in a reductive group of a $\theta$ stable abelian subalgebra of the Lie algebra. The involution $\theta$, which is called the “Cartan involution” of the Lie algebra, is the differential of a global Cartan involution $\Theta$ of $G$. In terms of $\Theta$, $G$ has a global Cartan decomposition that generalizes the polar decomposition of matrices.

A number of properties of semisimple Lie groups with finite center generalize to reductive Lie groups. Among these are the conjugacy of the maximal abelian subspaces of the $-1$ eigenspace $\mathfrak p_0$ of $\theta$, the theory of restricted roots, the Iwasawa decomposition, and properties of Cartan subalgebras. The chapter addresses also some properties not discussed in Chapter VI, such as the $KA_{\frak p}K$ decomposition and the Bruhat decomposition. Here $A_{\frak p}$ is the analytic subgroup corresponding to a maximal abelian subspace of $\mathfrak p_0$.

The degree of disconnectedness of the subgroup $M_{\frak p}=Z_K(A_{\frak p})$ controls the disconnectedness of many other subgroups of $G$. The most complete description of $M_{\frak p}$ is in the case that $G$ has a complexification, and then serious results from Chapter V about representation theory play a decisive role.

Parabolic subgroups are closed subgroups containing a conjugate of $M_{\frak p}A_{\frak p}N_{\frak p}$. They are parametrized up to conjugacy by subsets of simple restricted roots. A Cartan subgroup is defined to be the centralizer of a Cartan subalgebra. It has only finitely many components, and each regular element of $G$ lies in one and only one Cartan subgroup of $G$. When $G$ has a complexification, the component structure of Cartan subgroups can be identified in terms of the elements that generate $M_{\frak p}$.

A reductive Lie group $G$ that is semisimple has the property that $G/K$ admits a complex structure with $G$ acting holomorphically if and only if the centralizer in $\mathfrak g_0$ of the center of the Lie algebra $\mathfrak k_0$ of $K$ is just $\mathfrak k_0$. In this case, $G/K$ may be realized as a bounded domain in some $\mathbb C^n$ by means of the Harish-Chandra decomposition. The proof of the Harish-Chandra decomposition uses facts about parabolic subgroups. The spaces $G/K$ of this kind may be classified easily by inspection of the classification of simple real Lie algebras in Chapter VI.