Parabolic subgroup

Summary

A parabolic subgroup of a real semisimple Lie group 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 $G$.