The Segal-Shale-Weil representation is the following. By the Stone-von Neumann theorem there is an essentially unique irreducible unitary representation of the Heisenberg group of a given symplectic vector space. This being essentially unique implies that for each element of the symplectic group, there is a unique unitary operator such that for all
Gérard Lion, Michèle Vergne, The Weil representation, Maslov index and theta series, Progress in Math. 6, Birkhäuser 1980. vi+337 pp.
Maurice de Gosson, Maslov classes, metaplectic representation and Lagrangian quantization, Math. Research 95, Akademie-Verlag, Berlin, 1997. 186 pp.
José M. Gracia-Bondía, On the metaplectic representation in quantum field theory, Classical and quantum systems (Goslar, 1991), 611–614, World Sci. 1993
Gerald B. Folland, Harmonic analysis in phase space, Princeton Univ. Press 1989; ch. 4: Metaplectic representation