On the p-defect of character degrees of finite groups of Lie type

This paper is concerned with the representation theory of finite groups. According to Robinson, the truth of certain variants of Alperin’s weight conjecture on the p-blocks of a finite group would imply some arithmetical conditions on the degrees of the irreducible (complex) characters of that group. The purpose of this note is to prove directly that one of these arithmetical conditions is true in the case where we consider a finite group of Lie type in good characteristic. According to Robinson [8, Theorem 5.1] (see also [9, §5]), the truth of certain variants of Alperin’s weight conjecture on the p-blocks of a finite group would imply some arithmetical conditions on the p-parts of the degrees of the irreducible (complex) characters of that group. The purpose of this note is to prove directly that one of these arithmetical conditions is true in the case where we consider a finite group of Lie type in good characteristic. (See Example 2 for the problems which arise in bad characteristic.) Let G be a connected reductive group defined over the finite field Fq, where q is a power of some prime p. Let F :G → G be the corresponding Frobenius map and G the finite group of fixed points. Recall that p is good for G if p is good for each simple factor involved in G, and that the conditions for the various simple types are as follows. An : no condition, Bn, Cn, Dn : p 6= 2, G2, F4, E6, E7 : p 6= 2, 3, E8 : p 6= 2, 3, 5. For the basic properties of finite groups of Lie type, see [1]. Now we can state: Theorem 1 Assume that p is a good prime for G. Let χ be an irreducible character of G . Then there exists an F -stable parabolic subgroup P ⊆ G and an irreducible character ψ of U P (where UP is the unipotent radical of P ) such that |U P |/ψ(1) equals the p-part of |G F |/χ(1). In order to prove this result, we first reduce to the case that G has a connected center. This can be done using regular embeddings (see [6]), as follows. We can embed G as a closed subgroup into some connected reductive group G with a connected center, such that G has an Fq-rational structure compatible