User:Maximilian Janisch/latexlist/Algebraic Groups/Iwasawa decomposition

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The unique representation of an arbitrary element of a non-compact connected semi-simple real Lie group $k$ as a product $g = k a n$ of elements $k , \alpha , n$ of analytic subgroups $K , A , N$, respectively, where $K$, $4$ and $M$ are defined as follows. Let $\mathfrak { g } = \mathfrak { k } + \mathfrak { P }$ be a Cartan decomposition of the Lie algebra $8$ of $k$; let $1$ be the maximal commutative subspace of the space $7$, and let $57$ be a nilpotent Lie subalgebra of $8$ such that $57$ is the linear hull of the root vectors in some system of positive roots with respect to $1$. The decomposition of the Lie algebra as the direct sum of the subalgebras $3$, $1$ and $57$ is called the Iwasawa decomposition [1] of the semi-simple real Lie algebra $8$. The groups $K$, $4$ and $M$ are defined to be the analytic subgroups of $k$ corresponding to the subalgebras $3$, $1$ and $57$, respectively. The groups $K$, $4$ and $M$ are closed; $4$ and $M$ are simply-connected; $K$ contains the centre of $k$, and the image of $K$ under the adjoint representation of $k$ is a maximal compact subgroup of the adjoint group of $k$. The mapping $( k , a , n ) \rightarrow k a n$ is an analytic diffeomorphism of the manifold $K \times A \times N$ onto the Lie group $k$. 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 $D$-adic field (or, more generally, for groups of $D$-adic type) (see [4], [5]).

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 $t$-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 $t$-adic Chevalley groups" Publ. Math. IHES , 25 (1965) pp. 5–48 MR185016

Comments

An example of an Iwasawa decomposition is $SL _ { n } ( R ) = K A N$ with $K = SO _ { n } ( R )$, $4$ the subgroup of diagonal matrices of $SL _ { \gamma } ( R )$ and $M$ a lower triangular matrix with $1$'s on the diagonal. So, in particular, every element of $SL _ { \gamma } ( R )$ gets written as a product of a special orthogonal matrix and a lower triangular matrix.

References

[a1]	S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 MR0754767 Zbl 0543.58001