Unequal Characteristic: Generalities

Abstract

The purpose of this chapter is to assemble results valid in all unequal characteristics. We determine, for example, the decomposition into $\mathcal{O}$-blocks as well as the Brauer correspondents. We also introduce modular (that is over $\mathcal{O}$, or even over ℤ _ℓ ) and structural versions of Harish-Chandra and Deligne-Lusztig induction. These are functors between categories, rather than being simply maps between Grothendieck groups. The preliminary work on these two functors will be useful in the next chapter, where we study the equivalences of categories which they induce (which turn out to be either Morita or derived equivalences).