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  "body": "\n## Definition\n\nLet $G$ be a connected and simply connected [[nilpotent Lie group|DK87YN458D]] with Lie algebra $\\mathfrak g$ and [[exponential map|H3QX1ZFC8K]] $\\exp \\colon \\mathfrak g \\to G$.\nLet $\\mathfrak g^*$ be the [[dual space|07JBSZE0V4]] of $\\mathfrak g$, with the coadjoint action $(g \\cdot \\ell)(X) = \\ell(\\mathrm{Ad}(g^{-1}) X)$, where $\\mathrm{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].\nFor $\\ell \\in \\mathfrak g^*$, a *polarization* at $\\ell$ is a Lie subalgebra $\\mathfrak h \\subseteq \\mathfrak g$ such that $\\ell([\\mathfrak h, \\mathfrak h]) = 0$ and $\\mathfrak h$ is a maximal subspace with this property.\nFor such $\\mathfrak h$, let $H = \\exp \\mathfrak h$ and let $\\chi_\\ell$ be the unitary character $\\chi_\\ell(\\exp Y) = e^{i \\ell(Y)}$ of $H$.\nThe *Kirillov orbit method* assigns to $\\ell$ the [[induced representation|6Q7AHAZ6Q4]]\n$$\\pi_\\ell = \\mathrm{Ind}_H^G \\chi_\\ell.$$\nBy Kirillov's theorem, $\\pi_\\ell$ is irreducible, and its class depends only on the [[orbit|0YQPZB5Q1G]] $G \\cdot \\ell$, not on the choice of $\\mathfrak h$.\nThe map $G \\cdot \\ell \\mapsto \\pi_\\ell$ is a bijection from the set of coadjoint orbits onto the [[unitary dual|95Q8WZMN0W]] of $G$.\n\n## Remarks\n\nA polarization always exists for nilpotent $\\mathfrak g$, and its dimension is $\\tfrac12 (\\dim \\mathfrak g + \\dim \\mathfrak g_\\ell)$, where $\\mathfrak g_\\ell = \\set{X : \\ell([X, \\mathfrak g]) = 0}$.\n\nExample: let $\\mathfrak g$ have basis $X, Y, Z$ with $[X, Y] = Z$ and $Z$ central, the Lie algebra of the Heisenberg group.\nFor $\\ell$ with $\\ell(Z) = \\lambda \\ne 0$, the coadjoint orbit is the plane $\\set{\\ell' : \\ell'(Z) = \\lambda}$.\nThe subalgebra $\\mathfrak h = \\operatorname{span}(Y, Z)$ is abelian, so $\\ell([\\mathfrak h, \\mathfrak h]) = 0$, and it is maximal with this property since $\\ell([X, Y]) = \\lambda \\ne 0$.\nThe induced representation $\\pi_\\ell$ is the Schrödinger representation with central character $e^{i \\lambda z}$.\nFor $\\ell(Z) = 0$ the orbit is the single point $\\ell$, and $\\pi_\\ell$ is the one-dimensional character $\\exp(aX + bY + cZ) \\mapsto e^{i(a \\ell(X) + b \\ell(Y))}$.\n\nNon-example: for $\\ell(Z) = \\lambda \\ne 0$, the center $\\mathfrak h = \\operatorname{span}(Z)$ satisfies $\\ell([\\mathfrak h, \\mathfrak h]) = 0$ but is not maximal.\nThe induced representation $\\mathrm{Ind}_H^G \\chi_\\ell$ acts on $L^2(\\R^2)$ and is a direct sum of infinitely many copies of the Schrödinger representation, so it is reducible.\n",
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