# 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  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 , ).