Segal/Shale Representation 𝔘(T) on L₂(M,n)

For $T\in rGL(M)$ , the unitary operator $\mathfrak U(T)$ on $L_2(M,n)$ is

(\mathfrak U(T)f)(x)=X(T)^{1/2}\,f(T^*x),

where $X(T)$ is the Jacobian.

Remarks

Key properties (paper use):

$M_{\phi(T)g}=\mathfrak U(T)M_g\mathfrak U(T)^{-1}$ .

Examples

For 1D scaling $x\mapsto \lambda x$ , $\mathfrak U$ is a Gaussian-weighted dilation.