Lie Groups Beyond an Introduction pp 372-455 | Cite as

# 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*,*θ*, *B*) satisfying certain compatibility conditions. Here *G* is a Lie group, *K* is a compact subgroup, B is an involution of the Lie algebra go of *G*, and *B* is a bilinear form on go. 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 *θ* stable abelian subalgebra of the Lie algebra. The involution *θ*, which is called the “Cartan involution” of the Lie algebra, is the differential of a global Cartan involution Θ of *G.* In terms of Θ, *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 p_{0} of *θ*, 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 *K A*p *K* decomposition and the Bruhat decomposition. Here *A* _{
p} is the analytic subgroup corresponding to a maximal abelian subspace of p_{0}.

The degree of disconnectedness of the subgroup *M* _{p} = *Z* _{ K } (*A* _{p}) controls the disconnectedness of many other subgroups of *G.* The most complete description of *M* _{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* _{p} *A* _{p} *N* _{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* _{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 go of the center of the Lie algebra to of *K* is just to. In this case, *G*/*K* may be realized as a bounded domain in some 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.

## Keywords

Parabolic Subgroup Cartan Subalgebra Cartan Subgroup Iwasawa Decomposition Parabolic Subalgebra## Preview

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