Unequal Characteristic: Equivalences of Categories

Abstract

The purpose of this chapter is to verify Broué’s abelian defect conjecture (see Subsection B.2.2). In the case of non-principal blocks (which all have abelian defect group), the equivalences of categories predicted by Broué’s conjecture are always Morita equivalences (see Sections 8.1 and 8.2). While it is possible to obtain this result using Brauer trees and Brauer’s theorem B.4.2, we give instead a concrete construction of these equivalences using Harish-Chandra and Deligne-Lusztig induction. In the case of principal blocks, treated in Section 8.3, the situation is more interesting. If ℓ is odd and divides q−1, then the principal block is Morita equivalent to its Brauer correspondent and Harish-Chandra induction induces an equivalence. If ℓ is odd and divides q+1, then the principal block is Rickard equivalent to its Brauer correspondent and Deligne-Lusztig induction induces the required equivalence. If ℓ=2, the situation is more complicated: when $q\equiv \pm 3\mod 8$, then the principal block of G is Rickard equivalent to its Brauer correspondent; when $q\equiv \pm 1\mod 8$, the derived category of the principal block is equivalent to the derived category of an A _∞-algebra. These two final results are due to Gonard 2002.

The last section is dedicated to Alvis-Curtis duality, viewed as an endofunctor of the homotopy category K^b(ℤ _ℓ G) or of the derived category D^b(ℤ _ℓ G). For an arbitrary finite reductive group it was shown by Cabanes and Rickard 2001 that this duality is an equivalence of the derived category. More recently, Okuyama 2006 improved this result by showing that it was in fact an equivalence of the homotopy category (see also the work of Cabanes 2008 for a simplified treatment of Okuyama’s theorem). We give a very concrete proof of Okuyama’s result in the case of our little group $G=\mathrm{SL}_{2}(\mathbb{F}_{\!q})$.