Langlands decomposition

Remarks

Every parabolic subgroup that contains $MAN$ equals $P_F$ for exactly one $F$, and every parabolic subgroup is conjugate to such a $P_F$. The group $L_F = M_F A_F$ is a Levi factor of $P_F$; it is reductive, and $A_F$ is a vector group, isomorphic to $\R^k$ with $k = \abs{\Pi \setminus F}$.

For $F = \emptyset$, the group $M_\emptyset$ is $M = Z_K(\mathfrak{a})$ and $P_\emptyset = MAN$ is the minimal parabolic subgroup. For $F = \Pi$, one gets $\mathfrak{a}_F = 0$, $N_F = \set{e}$ and $P_F = M_F = G$, since a semisimple group has no nontrivial continuous homomorphism to $\R_{>0}$.

Example: in $\mathrm{SL}_n(\R)$, a subset $F$ corresponds to block sizes $n_1 + \cdots + n_r = n$. Then $P_F$ is the group of block upper triangular matrices, and $N_F$ is the group of block unipotent ones. The group $A_F$ consists of the matrices $\operatorname{diag}(t_1 I_{n_1}, \dots, t_r I_{n_r})$ with $t_i > 0$ and determinant $1$. The group $M_F$ consists of the block diagonal matrices of determinant $1$ whose blocks have determinant $\pm 1$.

Going deeper

For a student

The decomposition is the basis of parabolic induction. A representation $\sigma$ of $M_F$ and a character $\nu$ of $A_F$ give a representation $\sigma \otimes \nu \otimes 1$ of $P_F$, which one induces to $G$. The principal series come from $F = \emptyset$, and Langlands used general $F$ to classify the irreducible admissible representations of $G$. The groups $M_F$ are again reductive groups, which makes induction on the size of the group possible.