Definition
- G is a connected semisimple real Lie group.
- is the Lie algebra of G
- is the complexification of .
- θ is a Cartan involution of
- is the corresponding Cartan decomposition
- is a maximal abelian subalgebra of
- Σ is the set of restricted roots of, corresponding to eigenvalues of acting on .
- Σ+ is a choice of positive roots of Σ
- is a nilpotent Lie algebra given as the sum of the root spaces of Σ+
- K, A, N, are the Lie subgroups of G generated by and .
Then the Iwasawa decomposition of is
and the Iwasawa decomposition of G is
The dimension of A (or equivalently of ) is called the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a (disconnected) maximal compact subgroup provided the center of G is finite.
The restricted root space decomposition is
where is the centralizer of in and is the root space. The number is called the multiplicity of .
Read more about this topic: Iwasawa Decomposition
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