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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, with an [[Iwasawa decomposition|4YDR9Q36DS]] $G = KAN$.\nLet $\\mathfrak{a}$ be the Lie algebra of $A$, and let $\\Sigma^+$ be the positive restricted roots used to define $N$.\nLet $\\Pi \\subseteq \\Sigma^+$ be the simple ones.\nFix a subset $F \\subseteq \\Pi$ and let $\\Sigma_F$ be the set of restricted roots in the linear span of $F$.\nPut $\\mathfrak{a}_F = \\set{H \\in \\mathfrak{a} : \\alpha(H) = 0 \\text{ for all } \\alpha \\in F}$ and $\\mathfrak{n}_F = \\bigoplus_{\\lambda \\in \\Sigma^+ \\setminus \\Sigma_F} \\mathfrak{g}_\\lambda$.\nLet $A_F = \\exp \\mathfrak{a}_F$, let $N_F = \\exp \\mathfrak{n}_F$, and let $L_F = Z_G(\\mathfrak{a}_F)$ be the centralizer of $\\mathfrak{a}_F$ under the adjoint action.\nLet $M_F$ be the set of $x \\in L_F$ with $\\abs{\\chi(x)} = 1$ for every continuous homomorphism $\\chi \\colon L_F \\to \\R^\\times$.\nThen $P_F = L_F N_F$ is a [[parabolic subgroup|Z2FEBW5MQM]] of $G$, and:\n\n1. $M_F$ and $A_F$ are closed subgroups that commute with each other, $M_F \\cap A_F = \\set{e}$, and $L_F = M_F A_F$;\n2. $N_F$ is a closed, simply connected, [[nilpotent|DK87YN458D]] normal subgroup of $P_F$;\n3. the map $M_F \\times A_F \\times N_F \\to P_F$, $(m, a, n) \\mapsto man$, is a diffeomorphism.\n\nThe factorization $P_F = M_F A_F N_F$ is the *Langlands decomposition* of $P_F$.\n\n## Remarks\n\nEvery parabolic subgroup that contains $MAN$ equals $P_F$ for exactly one $F$, and every parabolic subgroup is conjugate to such a $P_F$.\nThe 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 $k = \\abs{\\Pi \\setminus F}$.\n\nFor $F = \\emptyset$, the group $M_\\emptyset$ is $M = Z_K(\\mathfrak{a})$ and $P_\\emptyset = MAN$ is the minimal parabolic subgroup.\nFor $F = \\Pi$, 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 $\\R_{\u003e0}$.\n\nExample: in $\\mathrm{SL}_n(\\R)$, a subset $F$ corresponds to block sizes $n_1 + \\cdots + n_r = n$.\nThen $P_F$ is the group of block upper triangular matrices, and $N_F$ is the group of block unipotent ones.\nThe group $A_F$ consists of the matrices $\\operatorname{diag}(t_1 I_{n_1}, \\dots, t_r I_{n_r})$ with $t_i \u003e 0$ and determinant $1$.\nThe group $M_F$ consists of the block diagonal matrices of determinant $1$ whose blocks have determinant $\\pm 1$.\n\n## Explanations\n\n### L1 human\n\nA parabolic subgroup splits into three pieces in a unique way.\nOne piece $M$ is a smaller reductive group, similar in kind to $G$.\nOne piece $A$ is a group of scalings, a copy of $\\R^k$.\nOne piece $N$ is a group of shears, and it is normal.\nFor block upper triangular matrices, $M$ and $A$ together give the diagonal blocks, and $N$ gives the part above the blocks.\n\n### L2 human\n\nThe decomposition is the basis of parabolic induction.\nA 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 $G$.\nThe principal series come from $F = \\emptyset$, and Langlands used general $F$ to classify the irreducible admissible representations of $G$.\nThe groups $M_F$ are again reductive groups, which makes induction on the size of the group possible.\n",
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