Principal Blocks and the Steinberg Character

We determine the finite simple groups of Lie type of characteristic p, for which the Steinberg character lies in the principal `-block for every prime ` 6= p dividing the order of the group. In [1, Corollary 4.4], Bessenrodt, Malle, and Olsson determine the finite simple groups of Lie type having a non-trivial irreducible character which is in the principal `-block for all primes ` dividing the group order. Motivated by this and their subsequent investigations on block separations and inclusions [2], Bessenrodt and Zhang asked the following question. Which are the finite simple groups of Lie type of characteristic p, such that the Steinberg character is in the principal `-block for all primes ` 6= p dividing the group order? In the notation of [1], this asks for those groups G of this class for which the trivial character and the Steinberg character of G are not separated by π(G) \ {p}. Here we answer the question of Bessenrodt and Zhang. It turns out that the answer is generic in the sense that it only depends on the Lie type of the group and not on the underlying characteristic. This genericity suggests that there could be a more uniform proof than the one presented here. Theorem. Let G be a finite simple group of Lie type of characteristic p. Then the Steinberg character of G lies in the principal `-block of G for all primes ` 6= p dividing the order of G, if and only if G is one of the groups in the following list. (1) PSLn(q) with 2 ≤ n ≤ 4, (n, q) 6= (2, 2), (2, 3). (2) PSUn(q) with 3 ≤ n ≤ 4, (n, q) 6= (3, 2). (3) PSp4(q), q 6= 2. (4) PΩ8 (q). (5) G2(q), q 6= 2. (6) F4(q). (7) D4(q). (8) B2(q), q = 22m+1 > 2. (9) G2(q), q = 32m+1 > 3. (10) 2F 4(q), q = 22m+1 > 2. Date: April 29, 2008. 2000 Mathematics Subject Classification. 20C15, 20C33.