Restricted Symplectic Group rSp(K)

The restricted symplectic group is

where $\mathrm{rGL}(K)$ is the restricted general linear group .

In Shale’s Theorem 4.1 (Fock–Cook quantization), $T$ is unitarily implementable iff $T\in rSp(K)$, equivalently $(T^*T)^{1/2}-I$ is Hilbert–Schmidt .

Key properties:

• Closed under polar decomposition .
• Carries a continuous projective unitary representation $\overline{Y}$.

Example: Finite-dimensional case: $rSp(K)=Sp(K)$.