geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
A representation of the metaplectic group.
The Segal-Shale-Weil metaplectic representation is also called the symplectic spinor representation.
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 $v\in V$
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
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, Harmonic analysis in phase space, Princeton Univ. Press 1989; ch. 4: Metaplectic representation
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