Parabolic subgroup of a real reductive group

Let $G$ be a connected real reductive Lie group with Lie algebra $\mathfrak g$ . A parabolic subgroup of $G$ is a closed subgroup $P$ whose Lie algebra $\mathfrak p$ is a real parabolic subalgebra. Concretely, after choosing a Cartan decomposition, a maximal abelian subspace $\mathfrak a$ of its noncompact part, and positive restricted roots, $\mathfrak p$ must contain $\mathfrak p_0=\mathfrak m\oplus\mathfrak a\oplus\mathfrak n$ , where $\mathfrak m=Z_{\mathfrak k}(\mathfrak a)$ and $\mathfrak n$ is the sum of the positive restricted-root spaces. In the standard linear real-reductive setting, $P=N_G(\mathfrak p)$ . Those parabolics containing the chosen $\mathfrak p_0$ are the standard parabolic subgroups.

Classification by simple restricted roots

Fix a positive restricted-root system with simple roots $\Delta$ . Subsets $F\subseteq\Delta$ parametrize standard parabolic subgroups $P_F$ ; inclusion of subsets gives inclusion of the corresponding parabolics. Every parabolic subgroup is conjugate to a standard one. For the stated real reductive class, this classification is compatible with the passage between parabolic Lie subalgebras and parabolic subgroups.

Internal structure and representation theory

Every parabolic has a Langlands decomposition $P=MAN$ . Its nilpotent factor $N$ is assembled from positive restricted root spaces not belonging to the Levi part, while $MA$ is reductive. This decomposition is the structural input for parabolic induction: a representation of $MA$ , extended trivially across $N$ , can be induced to $G$ .

Examples and conventions

For $G=\operatorname{SL}(n,\mathbb R)$ , the subgroups of block upper triangular matrices are parabolic; the upper triangular subgroup is minimal, and $G$ itself is the parabolic corresponding to all simple roots.

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.
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 parabolic subgroups and induced representations.