Gaussian projection in a Weyl representation ($P={\pi }^{-1}\int e^{-|a{|}^2/2}W(a)da$)

The integrated Gaussian in any nonzero regular representation of the Weyl relations is an orthogonal projection and obeys $PW(a)P=e^{-|a{|}^2/2}P$. Gaussian integration with the Weyl phase proves both identities. It cannot be zero: conjugating it by all Weyl operators would otherwise make the Fourier transform of every Gaussian-weighted matrix coefficient vanish, including its nonzero value at the identity. In an irreducible representation its range has dimension one, and the orbit of a unit vector there gives a dense cyclic space with universal Gaussian inner products.