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For $(V,\omega)$ a symplectic vector space, the metaplectic group $Mp(V,\omega)$ is the Lie group which is the universal double cover of the symplectic group $Sp(V,\omega)$.
This has various more explicit presentations. One is by quadratic Hamiltonians: The metaplectic group is that subgroup of the quantomorphism group of the symplectic manifold $(V,\omega)$ whose elements are given by paths of Hamiltonians that are homogeneously quadratic Hamiltonians (due to Leray 81, section 1.1, see also Robbin-Salamon 93, sections 9-10). (The more general subgroup given by possibly inhomogeneous quadratic Hamiltonians this way is the extended affine symplectic group. The subgroup given by linear Hamiltonians is the Heisenberg group $Heis(V,\omega)$.)
There is also a nontrivial circle group-extension of the symplectic group, called $Mp^c$. This is the circle extension associated to the plain metaplectic group $Mp$ above, via the canonical action of $\mathbb{Z}_2$ on $U(1)$ (by complex conjugation): (Forger-Hess 79 (2.4))
where the last line denotes the quotient group by the diagonal action of $\mathbb{Z}_2$.
(This is in direct analogy to the group Spin^c and its relation to Spin.)
Again, this has various more explicit presentations.
The Segal-Shale-Weil representation is the following. By the Stone-von Neumann theorem there is an essentially unique irreducible unitary representation $W$ of the Heisenberg group $Heis(V,\omega)$. This being essentially unique implies that for each element $g\in Sp(V,\omega)$ of the symplectic group, there is a unique unitary operator $U_g$ such that for all $v\in V$
The group $Mp^c$ is the subgroup of the unitary group of all such $U_g$ for $g\in Sp(V,\omega)$. The map $U_g \mapsto g$ exhibits this as a group extension by the circle group
e.g. (Robinson-Rawnsley 89, p. 19, Dereziński-Gérard 13, def. 10.24)
Alternatively, there is again a characterization by quadratic Hamiltonians (Robinson-Rawnsley 89, theorem (2.4)
A symplectic vector space $(V,\omega)$ has a compatible complex structure $J$. Write
for the corresponding unitary group.
The restriction (pullback) of $Mp^c$ above to this subgroup is denoted $MU^c$ in (Robinson-Rawnsley 89, p. 22)
(beware the notational clash with the Thom spectrum MU, which is unrelated).
Inside the symplectic group $Sp(2n, \mathbb{R})$ in dimension $2n$ sits the general linear group in dimension $n$
as the subgroup that preserves the standard Lagrangian submanifold $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$. Restriction of the metaplectic group extension along this inclusion defines the metalinear group $Ml(n)$
Hence a metaplectic structure on a symplectic manifold induces a metalinear structure on its Lagrangian submanifolds.
The extension
is nontrivial (does not give a split exact sequence).
(Robinson-Rawnsley 89, theorem (2.8))
The extension
is trivial (does give a split exact sequence).
(Robinson-Rawnsley 89, theorem (2.9))
Every symplectic manifold admits a metaplectic structure.
(Robinson-Rawnsley 89, theorem (6.2))
Since the unitary group $U(V,J)$ is the maximal compact subgroup of the symplectic group (see here) every $Sp(V,\omega)$-principal bundle has a reduction to a $U(V,J)$-principal bundle. By prop. this reduction in turn lifts to a $MU^c(V,J)$-structure. By def. this induces an $Mp^c$-structure under inclusion along $MU^c \hookrightarrow Mp^c$.
Original references include
Andre Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111: 143–211. (1964).
M. Kashiwara; Michèle Vergne, On the Segal-Shale-Weil Representations and Harmonic Polynomials, Inventiones mathematicae (1978) (EuDML, pdf)
Michael Forger, Harald Hess, Universal metaplectic structures and geometric quantization, Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. (euclid:cmp/1103904723)
Jean Leray, Lagrangian analysis and quantum mechanics, MIT press 1981 pdf
Further discussion includes
P. L. Robinson, John Rawnsley, The metaplectic representation, $Mp^c$-structures and geometric quantization, 1989 (doi:10.1090/memo/0410)
Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation Math. Z. 221 (1996), no. 2, 307–335, (MR98f:58051, doi, pdf), also in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993)
John Rawnsley, On the universal covering group of the real symplectic group, Journal of Geometry and Physics 62 (2012) 2044–2058 (pdf)
Michel Cahen, Simone Gutt, $Spin^c$, $Mp^c$ and Symplectic Dirac Operators, Geometric Methods in Physics Trends in Mathematics 2013, pp 13-28 (pdf)
Jan Dereziński, Christian Gérard, Mathematics of Quantization and Quantum Fields, Cambridge University Press, 2013
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