The Steinberg representation is irreducible

Abstract

We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.

The decomposition of certain abstract-induced modules over reductive algebraic groups

“…In this section, we study certain modules over unipotent groups. The setting and arguments follow [10,Section 6]. In this section, let F be a filed of characteristic p.…”

“…Proposition 4.1 is inspired by [10,Proposition 6.7 ]. The proof is also similar and we introduce some notation and give some preliminary results before we give the proof.…”

“…Recently, A.Putman and A.Snowden showed that the Steinberg representation of a connected reductive algebraic group over an infinite field is always irreducible (see [10]). Contrast to the proofs of Xi and Yang which make essential use of the fact that Fq is a union of finite fields, the arguments of [10] can deal with more general cases, which also suggests us to consider the abstract-induced module M(θ) = kG ⊗ kB θ in a more general setting.…”

“…Recently, A.Putman and A.Snowden showed that the Steinberg representation of a connected reductive algebraic group over an infinite field is always irreducible (see [10]). Contrast to the proofs of Xi and Yang which make essential use of the fact that Fq is a union of finite fields, the arguments of [10] can deal with more general cases, which also suggests us to consider the abstract-induced module M(θ) = kG ⊗ kB θ in a more general setting. Now let G be a connected reductive algebraic group defined over an algebraically closed field F, and B be a Borel subgroup of G. We get all the composition factors of M(θ) = kG ⊗ kB θ when char F = p is poistive and char k = p (see Theorem 6.1).…”

“…In Section 3, we list some properties of the unipotent groups U and introduce and study the self-enclosed subgroup of U, which is useful in the later discussion. In Section 4, we study certain modules over unipotent groups which generalize the results of [10,Proposition 6.7]. Section 5 gives the nonvanishing property of the augmentation map.…”