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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $\\mathrm{Sp}(2n, \\R)$ be the real [[symplectic group|X12FS2JAP0]].\nIt is a connected Lie group, and $U(n)$ is a maximal compact subgroup of it, so its [[fundamental group|S1XK3HXHFX]] is isomorphic to $\\pi_1(U(n)) \\cong \\Z$.\nThe group $\\Z$ has exactly one subgroup of index $2$, so $\\mathrm{Sp}(2n, \\R)$ has exactly one connected two-sheeted [[covering|3XZRRE165N]], up to isomorphism.\nThat covering space carries a unique Lie group structure for which the covering map is a homomorphism.\n\nThe *metaplectic group* $\\mathrm{Mp}(2n, \\R)$ is this connected double cover of $\\mathrm{Sp}(2n, \\R)$.\nSo there is an exact sequence\n$$1 \\to \\set{\\pm 1} \\to \\mathrm{Mp}(2n, \\R) \\to \\mathrm{Sp}(2n, \\R) \\to 1.$$\nFor $n = 1$, $\\mathrm{Sp}(2, \\R) =$ [[SL(2,R)|Q0GBCB9RA4]], and $\\mathrm{Mp}(2, \\R)$ is the connected double cover of $SL(2, \\R)$.\n\n## Remarks\n\n$\\mathrm{Mp}(2n, \\R)$ is not a matrix group: it has no faithful finite-dimensional representation.\nIt does have a faithful unitary representation on the infinite-dimensional Hilbert space $L^2(\\R^n)$, the oscillator representation, also called the Segal–Shale–Weil representation.\nThis representation does not come from a representation of $\\mathrm{Sp}(2n, \\R)$; it is defined only up to sign on $\\mathrm{Sp}(2n, \\R)$.\nThe group $\\mathrm{Mp}(2, \\R)$ is the setting for modular forms of half-integral weight, such as the Jacobi theta function.\n\n## Explanations\n\n### L1 human\n\nThe metaplectic group is a twofold cover of the real symplectic group, just as the spin group is a twofold cover of the rotation group.\nEach symplectic matrix corresponds to two elements of the metaplectic group, which differ by a sign.\nIt appears because some natural constructions, such as the quantization of linear symplectic maps in quantum mechanics, are only defined up to a sign on the symplectic group.\nPassing to the metaplectic group removes this sign ambiguity.\n",
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