Restricted General Linear Group rGL(H)

Shale’s restricted general linear group is

Equivalently, $|T|-I$ is Hilbert–Schmidt . This group is exactly where the Gaussian measure in §3 is quasi-invariant.

Key properties:

• Stable under polar decomposition $T=u(T)|T|$.
• Carries the “change-of-variables” unitary rep 𝔘(T) on $L_2(M,n)$.

Example: In finite dimensions, $rGL(H)=GL(H)$.