Corrigendum to “the Weil-steinberg Character of Finite Classical Groups”

This paper corrects the statement and the proof of Theorem 1.5 of the paper quoted in the title (Represent. Theory 13 (2009), 427–459). Theorem 1.5 of our paper [1] requires a correction. Below we provide a new statement of this theorem and correct the proof. A mistake in the original proof of Theorem 1.5 is due to missing the multiple 2 at a certain point of the proof (see [1, page 456, line 23]). Let G be a simple algebraic group of type Cn defined over a field of characteristic 2 and G = Sp(2n, q), q = 2. If μ is a dominant weight of G then φμ denotes the irreducible representation of G with highest weight μ, and Φμ is the representation of G afforded by the principal indecomposable module corresponding to (φμ)G if μ is a q-restricted weight. In addition, ω := (φ(q−1)λn)G. Let st be the 2modular Steinberg representation of G. Recall that st = (φ(q−1)(λ1+···+λn))G = Φst, where λ1, . . . , λn are the fundamental weights of G. The standard Frobenius endomorphism G → G is denoted by Fr0, and it acts on the representations and the weights of G (so Fr0(μ) = 2μ). Theorem 1.5 in [1] has to be corrected as follows: Theorem 1.5. Let λ1, . . . , λn be the fundamental weights of G, and let τ = (q − 1)(λ1 + · · ·+ λn−1). Then ω ⊗ st = st⊕ Φτ . Recall that ε1, . . . , εn denote the weights of G introduced in [2, Planchee III]. The following lemma is a refinement of [1, Lemma 7.2(1)]. Lemma. φ2λn is the only composition factor of φλn ⊗ φλn occurring with multiplicity 1. Proof. Let M be the G-module afforded by the representation φλn ⊗ φλn . Note that the weights of φλn and hence of M are known. In terms of εj the weights of φλn are ±ε1 ± · · · ± εn, so the weights of M are ∑ i∈N ±2εi, where N can be any subset of {1, . . . , n} (possibly empty; in this case the weight in question is meant to be the zero weight). It follows that 2λi = 2ε1 + · · ·+ 2εi (i = 1, . . . , n) occur as weights of M . Let H = GL(2n, F 2) and let ε ′ 1, . . . , ε ′ 2n be the weights of the natural H-module V . One can embed G into H so that a maximal torus T of G is contained in a maximal torus T′ of H, and εi = ε ′ i|T, εn+i|T = −εi for i = 1, . . . , n. Let Vi (1 ≤ i ≤ 2n) be the i-th exterior power of V , and V0 the trivial H-module. Set Received by the editors October 17, 2010. 2000 Mathematics Subject Classification. Primary 20G40, 20C33.