nLab metaplectic representation

Idea

A representation of the metaplectic group.

The Segal-Shale-Weil metaplectic representation is also called the symplectic spinor representation.

Segal-Shale-Weil representations

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 $Heis(V,\omega)$ of a given symplectic vector space. This being essentially unique implies that for each element $g\in Sp(V,\omega)$ of the symplectic group, there is a unique (up to multiplication by a constant of modulus one) unitary operator $U_g$ such that for all $v\in V$

$$W(g(v)) = U_g W(v) U^{-1}_g \,.$$

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

$$U(1)\longrightarrow Mp^c(V,\omega) \longrightarrow Sp(V,\omega) \,.$$

Related concepts

• spin representation

• related Langlands program?

References

• M. Kashiwara; Michèle Vergne, On the Segal-Shale-Weil Representations and Harmonic Polynomials, Inventiones mathematicae (1978) (EuDML, pdf)

• Gérard Lion, Michèle Vergne, The Weil representation, Maslov index and theta series, Progress in Math. 6, Birkhäuser 1980. vi+337 pp.

• Joel Robbin, Dietmar Salamon, Feynman path integrals and the metaplectic representation, Math. Z. 221 (1996), no. 2, 307–-335, (MR98f:58051, doi, pdf)

• 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

• R. Ranga Rao, The Maslov index on the simply connected covering group and the metaplectic representation, J. Funct. Anal. 107 (1992), no. 1, 211–233, MR93g:22013, doi

• G. Burdet, M. Perrin, Weyl quantization and metaplectic representation, Lett. Math. Phys. 2 (1977/78), no. 2, 93–99, MR473105, doi

• Y. Flicker, D. Kazhdan, G. Savin, Explicit realization of a metaplectic representation, J. Analyse Math. 55 (1990), 17–39, MR92c:22036, doi

• Gerald B. Folland, Chapter 4 in: A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]