Filtrations on block subalgebras of reduced enveloping algebras

We study the interaction between the block decompositions of reduced enveloping algebras in positive characteristic, the Poincare-Birkhoff-Witt (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 u(x) (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 neighborhood of 0 in the nilpotent cone and the coinvariants algebra corresponding to lambda. In the case of g = sl(2)(k) in characteristic p > 2 we determine the associated graded algebras of these filtrations on block subalgebras of U-0(sl(2)). We also apply this to determine the structure of the adjoint representation of U-0(sl(2)).