On the congruence kernel of isotropic groups over rings

Abstract

Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ≥ 2. The elementary subgroup E(R) of G(R) is the subgroup generated by U P + (R) and U P − (R), where U P ± are the unipotent radicals of two opposite parabolic subgroups PWe prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism E(R) → E(R) between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central in E(R). In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E(R) if R is a local ring. , and the research program 6.38.74.2011 "Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic K-theory" at St. Petersburg State University. Proof. See [St14, Theorem 2.1]. Definition 2.5. Under the assumptions of Theorem 2.4, we call the subgroup E(A) = E P (A), where P is a strictly proper parabolic subgroup of G, the elementary subgroup of G(A).2.2. Abstract relative roots. In order to better understand systems of relative roots Φ(S, G) of Definition 2.2, we consider the abstract systems of relative roots derived from abstract root systems in the sense of [Bou]. The corresponding notion was first introduced in [PeSt09].Let Φ be a reduced root system in the sense of [Bou] with an inner product (−, −). Let Π = {a 1 , . . . , a l } be a fixed system of simple roots of Φ; if Φ is irreducible, we assume that the numbering follows Bourbaki [Bou]. Let D be the Dynkin diagram of Φ. We identify nodes of D with the corresponding simple roots in Π.Definition 2.6. Let Γ ≤ Aut (D) be a subgroup, and let J ⊆ Π be a Γ-invariant subset. Consider the projectionProof. The proof of (i) and (ii) for the case of maximal roots is literally the same as in [KaSt, Lemma 1], which deals with the case where α is a simple relative root. The case of minimal roots is symmetric. To establish (iii), we observe that Ψ = Φ Π,Γ is naturally a root system, and Ψ π Π,Γ (J),{id} = Φ J,Γ . Applying (i) to Ψ π Π,Γ (J),{id} , we conclude that the images under π Π,Γ of all Π-maximal (respectively, Π-minimal) roots in π −1 J,Γ (α) coincide. By [PeSt09, Lemma 3] this implies that Γ acts transitively on such roots. The claim (iv) follows similarly from (ii).Lemma 2.8. Assume that rank(Φ J,Γ ) ≥ 2 and it is irreducible. Let α, β ∈ Φ J,Γ be two relative roots. Let a max ∈ π −1 J,Γ (α) be a Π-maximal root, and let b min ∈ π −1 J,Γ (β) be a Π-minimal root. Let a 1 , . . . , a n ∈ Φ + be a sequence of roots such that b min = a max − a 1 − . . . − a n ,

On abstract homomorphisms of some special unitary groups

“…As shown in [15], the algebraic ring A plays a central role in proving that ρ| G 0 (k) has a standard description; it turns out that A also suffices for the analysis of ρ. More precisely, following the general strategy of [15] and [16], we first show that ρ lifts to a representation σ : G(A) → GL m (K), where G(A) is the generalized Steinberg group introduced by Stavrova [18] (which builds on an earlier construction due to Deodhar [7]). Then, using the fact that the kernel of the canonical map G(A) → G(A) is central (which extends a result of Stavrova to the present situation), together with our assumption that the unipotent radical R u (H) is commutative, we establish the existence of the required algebraic representation σ :…”

“…We begin by recalling in §2 several key definitions and statements pertaining to elementary subgroups of isotropic reductive group schemes (defined by Petrov and Stavrova [12]) that are needed for our purposes. We also discuss some relevant aspects of Stavrova's generalization in [18] of the classical theory of Steinberg groups and central extensions. Next, in §3, we introduce the algebraic ring A and, after establishing some preliminary statements, show that ρ lifts to a representation σ of G(A).…”

“…In this section, we recall several aspects of the theory of elementary subgroups of isotropic reductive group schemes, developed by Petrov and Stavrova [12], as well as relevant points of Stavrova's [18] generalization of the classical Steinberg group. In view of our applications of this material later in subsequent sections, we will focus mainly on the case where G = SU 2n (L, h) (n ≥ 2) as in §1.…”

“…In this subsection, we recall Stavrova's [18] generalization of the usual Steinberg group (which, in turn, is inspired by a construction of Deodhar [7]) and establish several properties that will be needed in subsequent sections. Although Stavrova works in the general context of reductive group schemes, for the sake of concreteness, we will restrict ourselves to the case G = SU 2n (L, h).…”

“…Applying [18,Theorem 1.3] to each local factor R i , we see that ker π R i is central in G(R i ). Consequently, ker π R is central in G(R), as claimed.…”