  Let $\pi\colon Y\to X$ denote the canonical resolution of the two dimensional
Kleinian singularity $X$ of type ADE. In the present paper, we establish
isomorphisms between the cohomological and K-theoretical Hall algebras of
$\omega$-semistable properly supported sheaves on $Y$ with fixed slope $\mu$
and $\zeta$-semistable finite-dimensional representations of the preprojective
algebra of affine type ADE of slope zero respectively, under some conditions on
$\zeta$ depending on the polarization $\omega$ and $\mu$. These isomorphisms
are induced by the derived McKay correspondence. In addition, they are
interpreted as decategorified versions of a monoidal equivalence between the
corresponding categorified Hall algebras. In the type A case, we provide finer
descriptions of the cohomological, K-theoretical and categorified Hall algebra
of $\omega$-semistable properly supported sheaves on $Y$ with fixed slope
$\mu$: for example, in the cohomological case, the algebra can be given in
terms of Yangians of finite type ADE Dynkin diagrams.
