---
tag: M9DY1E7RKS
kind: definition
name: Parabolic subgroup
msc: ["22E46", "22E15"]
defines: Z2FEBW5MQM
uses: [XJS07AJ50Y, 4YDR9Q36DS, 5JSYA1JNCC]
sources:
  - url: https://en.wikipedia.org/wiki/Langlands_decomposition
---

## Definition

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center.
Let $G = KAN$ be an [[Iwasawa decomposition|4YDR9Q36DS]], with $\mathfrak{a}$ the Lie algebra of $A$.
Let $M = Z_K(\mathfrak{a})$ be the set of $k \in K$ with $\mathrm{Ad}(k) H = H$ for all $H \in \mathfrak{a}$.
The group $P_0 = MAN$ is the *minimal parabolic subgroup*.
A *parabolic subgroup* of $G$ is a closed subgroup $P$ of $G$ such that $g P_0 g^{-1} \subseteq P$ for some $g \in G$.

## Remarks

Example: $G = \mathrm{SL}_2(\R)$, with $K = \mathrm{SO}(2)$, $A$ the positive diagonal matrices and $N$ the upper unipotent matrices.
Then $M = \set{\pm 1}$, and $P_0 = MAN$ is the group of all upper triangular matrices in $\mathrm{SL}_2(\R)$.
So $P_0$ is parabolic, with $g = e$.

Non-example: $\mathrm{SO}(2)$ in $\mathrm{SL}_2(\R)$ is closed, but it is not parabolic.
It is compact, so each subgroup of it is relatively compact, while each conjugate of $P_0$ contains a conjugate of $A \cong \R$ and is not.

In $\mathrm{SL}_n(\R)$, the minimal parabolic subgroup is the group of upper triangular matrices, and the parabolic subgroups are the conjugates of the block upper triangular subgroups.
Equivalently, $P$ is parabolic when it is the normalizer in $G$ of a subalgebra $\mathfrak{q}$ of $\mathfrak{g}$ whose complexification contains a Borel subalgebra of $\mathfrak{g}_\C$.

For complex or algebraic groups, a parabolic subgroup is a closed subgroup containing a [[Borel subgroup|5JSYA1JNCC]].
The phrase also has a different meaning for Coxeter groups: a subgroup generated by a subset of the simple reflections.