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  "body": "\n## Definition\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center.\nLet $G = KAN$ be an [[Iwasawa decomposition|4YDR9Q36DS]], with $\\mathfrak{a}$ the Lie algebra of $A$.\nLet $M = Z_K(\\mathfrak{a})$ be the set of $k \\in K$ with $\\mathrm{Ad}(k) H = H$ for all $H \\in \\mathfrak{a}$.\nThe group $P_0 = MAN$ is the *minimal parabolic subgroup*.\nA *parabolic subgroup* of $G$ is a closed subgroup $P$ of $G$ such that $g P_0 g^{-1} \\subseteq P$ for some $g \\in G$.\n\n## Remarks\n\nExample: $G = \\mathrm{SL}_2(\\R)$, with $K = \\mathrm{SO}(2)$, $A$ the positive diagonal matrices and $N$ the upper unipotent matrices.\nThen $M = \\set{\\pm 1}$, and $P_0 = MAN$ is the group of all upper triangular matrices in $\\mathrm{SL}_2(\\R)$.\nSo $P_0$ is parabolic, with $g = e$.\n\nNon-example: $\\mathrm{SO}(2)$ in $\\mathrm{SL}_2(\\R)$ is closed, but it is not parabolic.\nIt 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.\n\nIn $\\mathrm{SL}_n(\\R)$, the minimal parabolic subgroup is the group of upper triangular matrices, and the parabolic subgroups are the conjugates of the block upper triangular subgroups.\nEquivalently, $P$ is parabolic when it is the normalizer in $G$ of a subalgebra $\\mathfrak{q}$ of $\\mathfrak{g}$ whose complexification contains a Borel subalgebra of $\\mathfrak{g}_\\C$.\n\nFor complex or algebraic groups, a parabolic subgroup is a closed subgroup containing a [[Borel subgroup|5JSYA1JNCC]].\nThe phrase also has a different meaning for Coxeter groups: a subgroup generated by a subset of the simple reflections.\n",
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