Truncation and the Induction Theorem

Abstract

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg compares the bounded derived category Db(block(U)) of finite dimensional modules for the principal block of a Lusztig quantum enveloping algebra U at an ℓth root of unity with a special full subcategory Dtriv(B) of the bounded derived category of integrable type 1 modules for a Borel part B⊂U. Specifically, according to this “Induction Theorem” [1, Theorem 3.5.5] the right derived functor of induction IndBU yields an equivalence of categories RIndBU:Dtriv(B)→∼Db(block(U)). Some restrictions on ℓ are required–e.g., ℓ>h, the Coxeter number. It is suggested briefly [1, Remark 3.5.6] that an analog of this equivalence carries over to characteristic p>0 representations of algebraic groups. Indeed, the authors of the present paper have verified, in a separate preprint [6], that there is such an equivalence RIndBG:Dtriv(B)→∼Db(block(G)) relating an analog of Dtriv(B), defined using a Borel subgroup B of a simply connected semisimple algebraic group G, to the bounded derived category of the principal block of finite dimensional rational G-modules. The proof is not without difficulty and supplies new, previously missing details even in the quantum case. The present paper continues the study of the modular case, taking the derived category equivalence as a starting point. The main result here is that, assuming p>2h−2, the equivalence behaves well with respect to certain weight poset “truncations,” making use of a variation by Woodcock [12] on van der Kallen's “excellent order” [10]. This means, in particular, that the equivalence can be reformulated in terms of derived categories of finite dimensional quasi-hereditary algebras. We expect that a similar result holds in the quantum case.In [9, Thm. 2.1] J. Rickard proves a theorem stating existence, in an algebraic groups context, of some derived equivalences with specified corresponding character isometries. In an appendix to this paper, we discuss a similar more recent result [6, Lem. 3.2]. It gives additional information on the behavior of the derived equivalences, giving their action on some natural right-derived induced objects, lifting the given character isometries.