Iwasawa Decomposition - Definition

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 .