Let $M$ be a ALE $n$-manifold. Then it is known a folklore result that the $L^2$ Hodge cohomology is given by:

$L^2\mathcal H^k=H^k(M,\partial M)$ if $k < n$,

$L^2\mathcal H^{n/2}=Im(H^{n/2}(M,\partial M)\to H^{n/2}(M))$, and

$L^2\mathcal H^k=H^k(M)$ if $k>n$.

A proof can be found in "HODGE COHOMOLOGY OF GRAVITATIONAL INSTANTONS" by Hunsicker, Hausel & Mazzeo. Their proof is very involved and it seems to me that a simpler proof should exist. Is a there a good simple proof? Maybe even published in the literature somewhere.