Chapter VI. Structure Theory of Semisimple Groups

Abstract

Every complex semisimple Lie algebra has a compact real form, as a consequence of a particular normalization of root vectors whose construction uses the Isomorphism Theorem of Chapter II. If $\mathfrak g_0$ is a real semisimple Lie algebra, then the use of a compact real form of $(\mathfrak g_0)^{\Bbb C}$ leads to the construction of a “Cartan involution” $\theta$ of $\mathfrak g_0$. This involution has the property that if $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ is the corresponding eigenspace decomposition or “Cartan decomposition,” then $\mathfrak k_0\oplus i\mathfrak p_0$ is a compact real form of $(\mathfrak g_0)^{\Bbb C}$. Any two Cartan involutions of $\mathfrak g_0$ are conjugate by an inner automorphism. The Cartan decomposition generalizes the decomposition of a classical matrix Lie algebra into its skew-Hermitian and Hermitian parts.

If $G$ is a semisimple Lie group, then a Cartan decomposition $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ of its Lie algebra leads to a global decomposition $G=K\exp\mathfrak p_0$, where $K$ is the analytic subgroup of $G$ with Lie algebra $\mathfrak k_0$. This global decomposition generalizes the polar decomposition of matrices. The group $K$ contains the center of $G$ and, if the center of $G$ is finite, is a maximal compact subgroup of $G$.

The Iwasawa decomposition $G=KAN$ exhibits closed subgroups $A$ and $N$ of $G$ such that $A$ is simply connected abelian, $N$ is simply connected nilpotent, $A$ normalizes $N$, and multiplication from $K\times A\times N$ to $G$ is a diffeomorphism onto. This decomposition generalizes the Gram-Schmidt orthogonalization process. Any two Iwasawa decompositions of $G$ are conjugate. The Lie algebra $\mathfrak a_0$ of $A$ may be taken to be any maximal abelian subspace of $\mathfrak p_0$, and the Lie algebra of $N$ is defined from a kind of root-space decomposition of $\mathfrak g_0$ with respect to $\mathfrak a_0$. The simultaneous eigenspaces are called “restricted roots,” and the restricted roots form an abstract root system. The Weyl group of this system coincides with the quotient of normalizer by centralizer of $\mathfrak a_0$ in $K$.

A Cartan subalgebra of $\mathfrak g_0$ is a subalgebra whose complexification is a Cartan subalgebra of $(\mathfrak g_0)^{\Bbb C}$. One Cartan subalgebra of $\mathfrak g_0$ is obtained by adjoining to the above $\mathfrak a_0$ a maximal abelian subspace of the centralizer of $\mathfrak a_0$ in $\mathfrak k_0$. This Cartan subalgebra is $\theta$ stable. Any Cartan subalgebra of $\mathfrak g_0$ is conjugate by an inner automorphism to a $\theta$ stable one, and the subalgebra built from $\mathfrak a_0$ as above is maximally noncompact among all $\theta$ stable Cartan subalgebras. Any two maximally noncompact Cartan subalgebras are conjugate, and so are any two maximally compact ones. Cayley transforms allow one to pass between any two $\theta$ stable Cartan subalgebras, up to conjugacy.

A Vogan diagram of $\mathfrak g_0$ superimposes certain information about the real form $\mathfrak g_0$ on the Dynkin diagram of $(\mathfrak g_0)^{\Bbb C}$. The extra information involves a maximally compact $\theta$ stable Cartan subalgebra and an allowable choice of a positive system of roots. The effect of $\theta$ on simple roots is labeled, and imaginary simple roots are painted if they are “noncompact,” left unpainted if they are “compact.” Such a diagram is not unique for $\mathfrak g_0$, but it determines $\mathfrak g_0$ up to isomorphism. Every diagram that looks formally like a Vogan diagram arises from some $\mathfrak g_0$.

Vogan diagrams lead quickly to a classification of all simple real Lie algebras, the only difficulty being eliminating the redundancy in the choice of positive system of roots. This difficulty is resolved by the Borel and de Siebenthal Theorem. Using a succession of Cayley transforms to pass from a maximally compact Cartan subalgebra to a maximally noncompact Cartan subalgebra, one readily identifies the restricted roots for each simple real Lie algebra.