Proposition: Finite-character covariance

If $\alpha =1/N$ and $\beta =p/N$, then

where

${\mathcal{R}}_k$ rotates the covering transverse plane by $2\pi k/N$, and

Consequently,

Proof. Apply the character-weighted image sum first to the flat heat kernel. Its angular Fourier sum projects covering modes $m\in \mathbb{Z}$ to

because

The projected angular factor is $e^{\mathrm{i}(n+p/N)(\phi -{\phi }^{\prime })}$, and the Bessel order is $|Nn+p|=|n+\beta |/\alpha$. The factor $N$ supplies precisely the quotient normalization $1/\alpha$ relative to the conical volume. Thus the result is the Friedrichs heat kernel (eq:heat). Euclidean integration and spectral continuation give (eq:finiteimages).

Every image has the ordinary flat null wavefront relation. If its straight segment avoids the origin, it gives a regular geodesic on the quotient. If it crosses the origin, its endpoint covector is a limit of such geodesics. Thus the finite sum has wavefront contained in ${\mathcal{G}}^{+}$. Proposition prop:geolower gives the reverse inclusion. This argument uses the full finite image sum, including boundary representatives when $N$ is even.