metaplectic representation


Symplectic geometry

Representation theory



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 WW of the Heisenberg group Heis(V,ω)Heis(V,\omega) of a given symplectic vector space. This being essentially unique implies that for each element gSp(V,ω)g\in Sp(V,\omega) of the symplectic group, there is a unique unitary operator U gU_g such that for all vVv\in V

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

The group Mp cMp^c is the subgroup of the unitary group of all such U gU_g for gSp(V,ω)g\in Sp(V,\omega). The map U ggU_g \mapsto g exhibits this as a group extension by the circle group

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


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Revised on January 5, 2015 14:33:35 by Urs Schreiber (