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

Unitary action of rGL(M) on Gaussian L₂ by change of variables and a Jacobian

For T∈rGL(M)T\in rGL(M), the unitary operator U(T)\mathfrak U(T) on L2(M,n)L_2(M,n) is

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

where X(T)X(T) is the .

Remarks

Key properties (paper use):

  • Implements pullback automorphisms on multiplication operators:

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

  • Theorem 3.1: T↦U(T)T\mapsto \mathfrak U(T) is weakly continuous.
Examples
  • For 1D scaling x↦λxx\mapsto \lambda x, U\mathfrak U is a Gaussian-weighted dilation.