We prove, for primes p ≥ 5, two inequalities between the fundamental invariants of Brauer p-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero characters, and the number of conjugacy classes of a defect group and of its derived subgroup. For this, we determine these invariants explicitly, or at least give bounds for them for several classes of classical groups. 1 2 GUNTER MALLE(2) G is a covering group of an alternating group, of a sporadic group or of a simple group of Lie type in characteristic p. Then B is neither a minimal counterexample to (C1) nor to (C2).Here, we say that (G, B) is a minimal counterexample to (Ci) if (Ci) holds for all pblocks B 1 of groups G 1 with |G 1 /Z(G 1 )| strictly smaller than |G/Z(G)| and with defect groups isomorphic to those of B.While we obtain partial results for the primes p = 2, 3, the case of groups of Lie type in non-defining characteristic seems out of reach at the moment; the case p = 3 might be accessible to a similar but more tedious investigation, but the prime p = 2 will require a different approach.Let us remark that no reduction of our conjecture to the case of (quasi-)simple groups has been proposed so far.The conjectured inequalities are closely related to three long-standing conjectures in representation theory. Brauer's k(B)-conjecture claims that k(B) ≤ |D|, while the Alperin-McKay conjecture relates the number k 0 (B) to the analogous number for the Brauer correspondent block of the normaliser N G (D) of a defect group D of B. Finally, Brauer's height zero conjecture proposes that k(B) = k 0 (B) if and only if D is abelian. In fact, we show in Theorem 2.1 that the statement of the known direction of this conjecture implies (C1) in the case of abelian defect groups. Let us point out one motivation for studying these: if the celebrated Alperin-McKay conjecture holds true, then k 0 (B) ≤ |D/D ′ | by the proven kGV-conjecture. But then (C1) claims that k(B) ≤ |D|(k(D ′ )/|D ′ |), which in general is much smaller than the bound |D| stipulated by Brauer's k(B)-conjecture. Thus, if true, our conjecture would yield a better bound on k(B) than Brauer's (yet unproven) one.The paper is built up as follows: In Section 2 we present some first reductions. The covering groups of alternating groups are treated in Section 3. In Section 4 we verify the inequalities for blocks of quasi-simple groups of Lie type for the defining prime in Corollary 4.3. In Section 5 we first present some general results for groups of Lie type in cross characteristic and then show the conjecture for blocks of groups of classical type (Corollary 5.18). To this end we also derive explicit formulas for invariants of unipotent blocks which we believe to be of independent interest (see Theorems 5.11 and 5.16 and Proposition 5.12). The groups of exceptional type are then considered in Section 6, see Theorem 6.5.Acknowledgement: I thank Frank Himstedt for providing me with information on the principal 2-and 3-blocks o...
