---
tag: Z2FEBW5MQM
kind: concept
name: Parabolic subgroup
msc: ["22E46", "22E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Langlands_decomposition
---

## Summary

A parabolic subgroup of a real [[semisimple Lie group|XJS07AJ50Y]] is a closed subgroup that contains a conjugate of the minimal parabolic subgroup $MAN$.
In $\mathrm{SL}_n(\R)$ the parabolic subgroups are the conjugates of the groups of block upper triangular matrices.
They are the stabilizers of partial flags and the building blocks of the Langlands decomposition.
Inducing representations from parabolic subgroups is the main way to build representations of $G$.

## Explanations

### L1 human

In the group of invertible matrices, look at the matrices that are upper triangular in blocks.
Such a matrix maps the span of the first few basis vectors to itself.
So it keeps a chain of subspaces in place, a partial flag.
The groups of this shape, and their conjugates, are the parabolic subgroups.
For $2 \times 2$ matrices of determinant $1$, the only proper ones are the conjugates of the upper triangular matrices.