The Schrödinger character is supported on the center:
\[
{\operatorname{Tr}}\rho(v,z)=\begin{cases}|X|\psi(z),&v=0,\\0,&v\ne0.\end{cases}
\]
A nonzero translation has no fixed coordinate. A nontrivial multiplier has vanishing character sum.
For a maximal isotropic subspace \(L\) and a subgroup lift \(\widetilde L\) with trivial central coordinate, the average
\[
P_L=\frac1{|L|}\sum_{\ell\in\widetilde L}\rho(\ell)
\]
is a self-adjoint idempotent. Its trace is \(|X|/|L|=1\), because only the identity contributes. Its range is therefore one dimensional.
For \(L=X\) this range consists of constant functions:
\[
(P_Lf)(u)=|X|^{-1}\sum_{x\in X}f(u+x).
\]
Reading the diagram
The complementary-line projector later used in arithmetic attachment is an exact operator in the Heisenberg image algebra. Its existence is finite representation theory.