Parabolic subgroup

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. The phrase also has a different meaning for Coxeter groups: a subgroup generated by a subset of the simple reflections.