Iwasawa decomposition

Definition

• G is a connected semisimple real Lie group.
• \mathfrak{g}_0 is the Lie algebra of G
• \mathfrak{g} is the complexification of \mathfrak{g}_0 .
• θ is a Cartan involution of \mathfrak{g}_0
• \mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0 is the corresponding Cartan decomposition
• \mathfrak{a}_0 is a maximal abelian subalgebra of \mathfrak{p}_0
• Σ is the set of restricted roots of \mathfrak{a}_0 , corresponding to eigenvalues of \mathfrak{a}_0 acting on \mathfrak{g}_0 .
• Σ+ is a choice of positive roots of Σ
• \mathfrak{n}_0 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 \mathfrak{k}_0, \mathfrak{a}_0 and \mathfrak{n}_0 .

Then the Iwasawa decomposition of \mathfrak{g}_0 is :\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{a}_0 \oplus \mathfrak{n}_0 and the Iwasawa decomposition of G is :G=KAN meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold K \times A \times N to the Lie group G , sending (k,a,n) \mapsto kan .

The dimension of A (or equivalently of \mathfrak{a}_0 ) is equal to 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 : \mathfrak{g}_0 = \mathfrak{m}0\oplus\mathfrak{a}0\oplus{\lambda\in\Sigma}\mathfrak{g}{\lambda} where \mathfrak{m}_0 is the centralizer of \mathfrak{a}_0 in \mathfrak{k}0 and \mathfrak{g}{\lambda} = {X\in\mathfrak{g}0: [H,X]=\lambda(H)X;;\forall H\in\mathfrak{a}0 } is the root space. The number m{\lambda}= \text{dim},\mathfrak{g}{\lambda} is called the multiplicity of \lambda.

Examples

If G=SLn(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices with determinant 1, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.

For the case of n = 2, the Iwasawa decomposition of G = SL(2, R) is in terms of : \mathbf{K} = \left{ \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ \theta\in\mathbf{R} \right} \cong SO(2) , : \mathbf{A} = \left{ \begin{pmatrix} r & 0 \ 0 & r^{-1} \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ r 0 \right}, : \mathbf{N} = \left{ \begin{pmatrix} 1 & x \ 0 & 1 \end{pmatrix} \in SL(2,\mathbb{R}) \ | \ x\in\mathbf{R} \right}.

For the symplectic group G = Sp(2n, R), a possible Iwasawa decomposition is in terms of

: \mathbf{K} = Sp(2n,\mathbb{R})\cap SO(2n) = \left{ \begin{pmatrix} A & B \ -B & A \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ A+iB \in U(n) \right} \cong U(n) , : \mathbf{A} = \left{ \begin{pmatrix} D & 0 \ 0 & D^{-1} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ D \text{ positive, diagonal} \right}, : \mathbf{N} = \left{ \begin{pmatrix} N & M \ 0 & N^{-T} \end{pmatrix} \in Sp(2n,\mathbb{R}) \ | \ N \text{ upper triangular with diagonal elements = 1},\ NM^T = MN^T \right}. Obtaining the matrices appearing in the decomposition above can be reduced to the calculation of matrix square roots, matrix inverses and performing a QR decomposition.

Non-Archimedean Iwasawa decomposition

There is an analog to the above Iwasawa decomposition for a non-Archimedean field F: In this case, the group GL_n(F) can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup GL_n(O_F), where O_F is the ring of integers of F.

References

References

1. Iwasawa, Kenkichi. (1949). "On Some Types of Topological Groups". [[Annals of Mathematics]].
2. (2024). "Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson". [[Canadian Journal of Physics]].
3. Bump. (1997). "Automorphic forms and representations". Cambridge University Press.