Truncation and the induction theorem

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg (ABG) gives an equivalence of the bounded derived category of finite dimensional modules for the principal block of a Lusztig quantum algebra at an $\ell^{th}$ root of unity, with am explicit full subcategory of the bounded derived category of integrable type 1 modules for a Borel part of the quantum algebra. Some restrictions on $\ell$ are required; in particular, it is assumed $\ell > h$, the Coxeter number. The same paper suggests there is an analogous result for representations of semisimple algebraic groups in characteristic $p>0$, and the authors of this paper have proved such a result (with $p>h$) in a separate manuscript, recently posted. The philosophy of the proof is a variation on that of ABG, but contains new ingredients and some missing details, even in the quantum case. The present paper continues the study of the modular case, showing the equivalence constructed (via a right derived functor of induction from a Borel part) behaves well, when $p>2h-2$, with respect to certain weight poset "truncations", making use of van der Kallen's 1989 "excellent order" highest weight categories. This implies, in particular, that the equivalence can be reformulated in terms of triangulated categories associated to derived categories of finite dimensional quasi-hereditary algebras. We expect that a similar result holds in the quantum case.