Metaplectic

Metaplectic refers to structures connected with the metaplectic group Mp(2n, F), a nontrivial twofold cover of the symplectic group Sp(2n, F) over a field F of characteristic not 2. It fits into a short exact sequence 1 → μ2 → Mp(2n, F) → Sp(2n, F) → 1, where μ2 ≅ {±1} in the real case. This central extension encodes the twofold nature of certain projective representations of Sp(2n, F).

The metaplectic group arises naturally in harmonic analysis via the Weil representation (also called the oscillator representation), a linear representation of Mp(2n, F) on spaces of square-integrable functions. This representation lifts the projective representation of Sp(2n, F) that comes from quadratic forms and the Fourier transform, yielding genuine linear operators.

Concretely, Mp(2n, F) can be realized by pairs (g, ξ) with g ∈ Sp(2n, F) and ξ a choice of square root of a certain 2-cocycle, or by generators corresponding to Fourier transforms, phase modulations, and symplectic shear transformations. In the Schrödinger model, these operators act on L^2(F^n) in a way that intertwines linear symplectic geometry with analysis.

Applications of the metaplectic framework include theta functions and the theta correspondence in number theory, automorphic forms, and aspects of geometric quantization, where the metaplectic correction lifts symplectic transformations to unitary operators on Hilbert spaces. The concept also has local and global manifestations in representation theory and the theory of automorphic forms.

Historically, the metaplectic group was introduced by André Weil in the 1960s to explain the double-valued nature of certain projective representations of Sp(2n). The term reflects its role as a lift of the symplectic group to a covering group used in analysis and representation theory.