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  "body": "\n## Definition\n\nLet $W$ be a finite-dimensional real [[symplectic vector space|PQ7JJY0VPH]] with [[symplectic group|X12FS2JAP0]] $\\mathrm{Sp}(W)$ and [[metaplectic group|8B0JM2YDE2]] $p \\colon \\mathrm{Mp}(W) \\to \\mathrm{Sp}(W)$.\nA *reductive dual pair* in $\\mathrm{Sp}(W)$ is a pair $(G, G')$ of [[reductive|42ZCE14AD2]] subgroups of $\\mathrm{Sp}(W)$ with two properties.\nThe centralizer of $G'$ in $\\mathrm{Sp}(W)$ is $G$, and the centralizer of $G$ is $G'$.\nLet $\\tilde G = p^{-1}(G)$ and $\\tilde G' = p^{-1}(G')$, and let $\\omega^\\infty$ be the space of smooth vectors of the [[oscillator representation|CCR7BAQ5N5]] of $\\mathrm{Mp}(W)$, as a representation of $\\tilde G \\times \\tilde G'$.\nLet $\\mathcal R(\\tilde G)$ be the set of classes of [[irreducible|FX3NEFYFFN]] [[admissible representations|SZV8Y806Y7]] $\\pi$ of $\\tilde G$ that occur as quotients of $\\omega^\\infty$.\nDefine $\\mathcal R(\\tilde G')$ in the same way.\nBy Howe duality, for each $\\pi \\in \\mathcal R(\\tilde G)$ there is exactly one $\\pi' \\in \\mathcal R(\\tilde G')$ such that $\\pi \\otimes \\pi'$ is a quotient of $\\omega^\\infty$, and $\\pi \\mapsto \\pi'$ is a bijection $\\mathcal R(\\tilde G) \\to \\mathcal R(\\tilde G')$.\nThis bijection is the *theta correspondence* for the pair $(G, G')$, written $\\pi' = \\theta(\\pi)$.\n\n## Remarks\n\nThe theorem used in the definition is Howe's duality theorem for real groups; the same definition over $p$-adic fields rests on the work of several later authors.\n\nExample: let $W = \\R^2$, so $\\mathrm{Sp}(W) = \\mathrm{SL}_2(\\R)$, and let $G = \\set{\\pm 1}$, which is the orthogonal group $\\mathrm{O}(1)$.\nThe centralizer of $\\set{\\pm 1}$ is all of $\\mathrm{SL}_2(\\R)$, and the centralizer of $\\mathrm{SL}_2(\\R)$ is its center $\\set{\\pm 1}$, so $(\\mathrm{O}(1), \\mathrm{SL}_2(\\R))$ is a dual pair.\nIn the Schrödinger model on $L^2(\\R)$, the element $-1$ acts by $\\psi(x) \\mapsto \\psi(-x)$, up to a constant depending on the lift.\nSo the two characters of $\\mathrm{O}(1)$ correspond to the two irreducible pieces of the oscillator representation of $\\mathrm{Mp}_2(\\R)$, on even and on odd functions.\n\nNon-example: in $\\mathrm{SL}_2(\\R)$, let $A = \\set{\\operatorname{diag}(a, a^{-1}) : a \u003e 0}$.\nThe centralizer of $A$ is $\\set{\\operatorname{diag}(a, a^{-1}) : a \\ne 0}$, which is larger than $A$, so $(A, A)$ is not a dual pair.\n\nThe standard dual pairs include $(\\mathrm{O}(p, q), \\mathrm{Sp}_{2m}(\\R))$ in $\\mathrm{Sp}(\\R^{p+q} \\otimes \\R^{2m})$ and $(\\mathrm{U}(p, q), \\mathrm{U}(r, s))$.\n",
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