Let \(G\) be a connected reductive algebraic group with Lie algebra \(\mathfrak g\), considered over an algebraically closed field of characteristic \(p > 0\). Given a \(p\)-character \(\chi \in \mathfrak{g}^*\), the corresponding reduced universal enveloping algebra is the finite dimensional associative algebra \[ \mathfrak{U}_\chi(\mathfrak{g}) := \mathfrak{U}(\mathfrak{g}) \big/ \langle X^p - X^{[p]} - \chi(X)^p : X \in \mathfrak{g} \rangle. \] It has a block decomposition of the form \[ \mathfrak{U}_\chi(\mathfrak{g}) = \bigoplus_\lambda A_\lambda, \] where the index \(\lambda\) runs over a dot-orbit of the Weyl group acting on a certain \(\chi\)-compatible weight lattice in the dual of a Cartan subalgebra.
The authors investigate three different PBW filtrations on the \(A_\lambda\). The first two are obtained directly from the PBW filtration on \(\mathfrak{U}_\chi(\mathfrak{g})\) and turn out to be mutually dual. The third is obtained by a shifting process involving the generators of the Harish-Chandra center of \(\mathfrak{U}(\mathfrak{g})\) itself.
These constructions are exhibited explicitly for \(\mathrm{SL}(2)\): for each \(A_\lambda\), the central idempotents are given and the simple modules are described. In the case \(\chi = 0\), the associated graded algebras are given and the adjoint representation is analyzed.