Iwasawa decomposition

The unique representation of an arbitrary element $g$ of a non-compact connected semi-simple real Lie group $G$ as a product $g = k an$ of elements $k,\ a,\ n$ of analytic subgroups $K,\ A,\ N$ , respectively, where $K$ , $A$ and $N$ are defined as follows. Let $\mathfrak g = \mathfrak k + \mathfrak P$ be a Cartan decomposition of the Lie algebra $\mathfrak g$ of $G$ ; let $\mathfrak a$ be the maximal commutative subspace of the space $\mathfrak P$ , and let $\mathfrak N$ be a nilpotent Lie subalgebra of $\mathfrak g$ such that $\mathfrak N$ is the linear hull of the root vectors in some system of positive roots with respect to $\mathfrak a$ . The decomposition of the Lie algebra as the direct sum of the subalgebras $\mathfrak k$ , $\mathfrak a$ and $\mathfrak N$ is called the Iwasawa decomposition [1] of the semi-simple real Lie algebra $\mathfrak g$ . The groups $K$ , $A$ and $N$ are defined to be the analytic subgroups of $G$ corresponding to the subalgebras $\mathfrak k$ , $\mathfrak a$ and $\mathfrak N$ , respectively. The groups $K$ , $A$ and $N$ are closed; $A$ and $N$ are simply-connected; $K$ contains the centre of $G$ , and the image of $K$ under the adjoint representation of $G$ is a maximal compact subgroup of the adjoint group of $G$ . The mapping $(k,\ a,\ n) \rightarrow kan$ is an analytic diffeomorphism of the manifold $K \times A \times N$ onto the Lie group $G$ . The Iwasawa decomposition plays a fundamental part in the representation theory of semi-simple Lie groups. The Iwasawa decomposition can be defined also for connected semi-simple algebraic groups over a $p$ - adic field (or, more generally, for groups of $p$ - adic type) (see [4], [5]).

Comments

An example of an Iwasawa decomposition is $\mathop{\rm SL}\nolimits _{n} ( \mathbf R ) = K A N$ with $K = \mathop{\rm SO}\nolimits _{n} ( \mathbf R )$ , $A$ the subgroup of diagonal matrices of $\mathop{\rm SL}\nolimits _{n} ( \mathbf R )$ and $N$ a lower triangular matrix with $1$ ' s on the diagonal. So, in particular, every element of $\mathop{\rm SL}\nolimits _{n} ( \mathbf R )$ gets written as a product of a special orthogonal matrix and a lower triangular matrix.

References

[1]	K. Iwasawa, "On some types of topological groups" Ann. of Math. , 50 (1949) pp. 507–558 MR0029911 Zbl 0034.01803
[2]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[3]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[4]	F. Bruhat, "Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley sur un corps $\mathfrak{p}$-adique" Publ. Math. IHES , 23 (1964) pp. 45–74 MR179298
[5]	N. Iwahori, H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of $\mathfrak{p}$-adic Chevalley groups" Publ. Math. IHES , 25 (1965) pp. 5–48 MR185016
[a1]	S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 MR0754767 Zbl 0543.58001