  We study the interaction between the block decompositions of reduced
universal enveloping algebras in positive characteristic, the PBW filtration,
and the nilpotent cone. We provide two natural versions of the PBW filtration
on the block subalgebra $A_\lambda$ of the restricted universal enveloping
algebra $\mathcal{U}_\chi(\mathfrak{g})$ and show these are dual to each other.
We also consider a shifted PBW filtration for which we relate the associated
graded algebra to the algebra of functions on the Frobenius neighbourhood of
$0$ in the nilpotent cone and the coinvariants algebra corresponding to
$\lambda$. In the case of $\mathfrak{g}=\mathfrak{sl}_2(k)$ in characteristic
$p>2$ we determine the associated graded algebras of these filtrations on block
subalgebras of $\mathcal{U}_0(\mathfrak{sl}_2)$. We also apply this to
determine the structure of the adjoint representation of
$\mathcal{U}_0(\mathfrak{sl}_2)$.
