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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]] $G = KAN$ and Lie algebra $\\mathfrak{a}$ of $A$.\nLet $M = Z_K(\\mathfrak{a})$ and $M' = N_K(\\mathfrak{a})$ be the centralizer and the normalizer of $\\mathfrak{a}$ in $K$, and let $W = M'/M$ be the restricted Weyl group.\nLet $P = MAN$ be the minimal [[parabolic subgroup|Z2FEBW5MQM]].\nFor $w \\in W$, the double coset $P \\tilde w P$ does not depend on the representative $\\tilde w \\in M'$ of $w$, and it is written $PwP$.\nThen $G$ is the disjoint union\n$$G = \\bigsqcup_{w \\in W} PwP.$$\n\n## Remarks\n\nThe coset $P \\tilde w P$ does not depend on $\\tilde w$ because two representatives differ by an element of $M \\subseteq P$.\nA standard proof follows the algebraic case, with the minimal parabolic subgroup in place of a Borel subgroup.\nDisjointness uses that $W$ acts simply transitively on the positive systems of restricted roots.\n\nExample: $G = \\mathrm{SL}_2(\\R)$, where $P$ is the group of upper triangular matrices and $W = \\set{1, s}$ with $s$ represented by $\\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$.\nA matrix $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ lies in $P$ when $c = 0$ and in $PsP$ when $c \\ne 0$.\nSo $\\mathrm{SL}_2(\\R) = P \\sqcup PsP$.\n\nThe [[Bruhat decomposition for reductive algebraic groups|T91E86D0RH]] over an algebraically closed field is the analogue with a Borel subgroup in place of $P$.\nFor a real group, $W$ is the Weyl group of the restricted root system, which can be smaller than the Weyl group of the complexified group.\n\n## Explanations\n\n### L1 human\n\nThe group is cut into finitely many pieces, one for each element of a finite group $W$.\nEach piece is a set of the form $PwP$, where $P$ is the minimal parabolic subgroup.\nFor $2 \\times 2$ real matrices of determinant $1$, there are two pieces: the upper triangular matrices, and all the others.\n\n### L2 human\n\nPassing to the flag manifold $G/P$, the decomposition says that $P$ has finitely many orbits on $G/P$, one for each $w \\in W$.\nThe orbit of the longest element of $W$ is open and dense, and its complement has lower dimension.\nThis open orbit is used to realize principal series representations on functions on $N^- = \\theta(N)$.\n",
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