Parabolic subgroup of a real reductive group - Imported Knowlpedia Content

Definition

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. These statements, including the passage between parabolic subalgebras and subgroups, are proved for the usual real reductive class in Knapp, Chapter VII, §7.

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.