Langlands decomposition of a parabolic subgroup

Statement

Let $G$ be a connected real reductive Lie group with a fixed Cartan involution, and let $P\subseteq G$ be a parabolic subgroup compatible with the resulting positive restricted roots. The Langlands decomposition is

P=MAN,

where $A$ is a connected real split abelian subgroup, $M$ is reductive with compact center modulo the center inherited from $G$ , and $N$ is the connected nilpotent radical. The factors $M$ and $A$ commute, $MA$ normalizes $N$ , and multiplication $M\times A\times N\to P$ is a diffeomorphism under the standard normalization of the factors.

Construction from restricted roots

Choose the subset of simple restricted roots defining $P$ . The Lie algebra $\mathfrak a_P$ is the common kernel in $\mathfrak a$ of those roots, $A=\exp(\mathfrak a_P)$ , and $\mathfrak n$ is the sum of the positive restricted-root spaces not belonging to the Levi subsystem. The remaining centralizer data determine $\mathfrak m$ . This construction and the uniqueness properties of the factors are given in Knapp, Chapter VII, §7.

Role in normalized induction

A representation $\sigma$ of $M$ and a character $e^\nu$ of $A$ define a representation of $P$ by letting $N$ act trivially. Induction from $P$ to $G$ , with the modular correction involving the half-sum $\rho_P$ of roots in $\mathfrak n$ , produces normalized parabolic induction. The three factors separate discrete representation data on $M$ , continuous spectral parameters on $A$ , and the nilpotent directions in $N$ .

Minimal and extreme cases

For a minimal parabolic, this is the familiar $P_0=MAN$ associated with an Iwasawa decomposition. At the other extreme, $P=G$ has $N=\{e\}$ ; depending on the normalization, its split central part may appear in $A$ . These cases explain why authors sometimes state the theorem only for proper parabolics or impose a specified class of real reductive groups.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progress in Mathematics 140, Birkhäuser, 2002. Author-maintained record. Relevant: Chapter VII, §7 on parabolic subgroups and their $MAN$ decomposition.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Mathematical Series 36, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on principal series and parabolic induction.