On the Character of Certain Irreducible Modular Representations

Let G be an almost simple, simply connected algebraic group over an algebraically closed field of characteristic p > 0. In this paper we restate our conjecture from 1979 on the characters of irreducible modular representations of G so that it is now directly applicable to any dominant highest weight. 1. Let G be an almost simple, simply connected algebraic group over k, an algebraically closed field of characteristic p > 1. Let RepG be the category of finite dimensional k-vector spaces with a given rational linear action of G and let IrrG be a set of representatives for the simple objects of RepG. We fix a Borel subgroup B of G and a maximal torus T of B. Let Y = Hom(k∗, T ), X = Hom(T,k∗) (with group operation written as +) and let 〈, 〉 : Y ×X → Z be the obvious pairing. If V ∈ IrrG, then there is a well-defined λV ∈ X with the following property: the T -action on the unique B-stable line in V is through λV ; according to Chevalley, E → λE is a bijection from IrrG to a subset of X of the form X = {λ ∈ X; 〈α̌i, λ〉 ∈ N ∀i ∈ I} for a well-defined basis {α̌i; i ∈ I} of Y . For λ ∈ X we shall denote by Vλ the object of IrrG corresponding to λ. If V ∈ RepG, then for any μ ∈ X we denote by nμ(V ) the multiplicity of μ in V |T ; we set [V ] = ∑ μ∈X nμ(V )e μ ∈ Z[X] where Z[X] is the group ring of X (the basis element of Z[X] corresponding to μ ∈ X is denoted by e so that eμeμ = eμ+μ for μ, μ′ ∈ X). It is of considerable interest to compute explicitly the element [Vλ] ∈ Z[X] for any λ ∈ X. Let h be the Coxeter number of G. A conjectural formula for [Vλ] (assuming that p ≥ cG where cG is a constant depending only on the root datum of G) was stated in [L1, p. 316]. In the early 1990’s it was proved (see [AJS] and the references there) that there exists a (necessarily unique) prime number cG ≥ cG depending only on the root datum of G such that the conjectural formula in [L1, p. 316] is true if p ≥ cG and cG is minimum possible (but cG was not explicitly determined). In [F], Fiebig showed that cG ≤ cG where cG is an explicitly known but very large constant. In [Wi], Williamson, partly in collaboration with Xuhua He, showed that for infinitely many G, cG is much larger than c 0 G. Now the conjecture in [L1, p. 316] had an unsatisfactory aspect: it applied only to a finite set of λ ∈ X which, after application of Jantzen’s results [J] on translation functors, becomes a larger but still finite set (including all λ in X red = {λ ∈ X; 〈α̌i, λ〉 ≤ p − 1 ∀i ∈ I}); then Received by the editors Received by the editors October 7, 2014 and, in revised form, February 2, 2015. 2010 Mathematics Subject Classification. Primary 20G99. This work was supported in part by National Science Foundation grant DMS-1303060 and by a Simons Fellowship. c ©2015 American Mathematical Society 3