Reductive groups on a local field. II. Groups schemes. Existence of valuated root datum. (Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée.) (French) Zbl 0597.14041

Bruhat, François; Tits, Jacques

Publ. Math., Inst. Hautes Étud. Sci. 60, 1-194 (1984).

The work under review is the second part of a comprehensive study of reductive algebraic groups \(G\) over local fields \(K\), which had been taken up by the authors in 1965 after the appearance of a seminal paper by N. Iwahori and H. Matsumoto [Publ. Math., Inst. Haut. Étud. Sci. 25, 5-48 (1965; Zbl 0228.20015)] treating the case of Chevalley or “split” groups. The first part of this work was published in 1972 [Publ. Math., Inst. Haut. Étud. Sci. 41, 5–251 (1966; Zbl 0254.14017)]. Its main objective was the derivation, for arbitrary reductive groups \(G\), of a number of fundamental results (like the classification of maximal compact subgroups, the Bruhat-, Cartan-, and Iwasawa-decompositions, the existence of a building) from the existence of a “valuated root datum”, a structure which can be described axiomatically and which essentially consists of a valuation of the root subgroups \(U_a\) of \(G\) compatible with the valuation of the base field \(K\), the formation of natural subgroups and commutators, and the action of the normaliser \(N\) of the relevant maximal split torus \(S\subset G\). Whereas the existence of such valuated root data is easily established for split groups, this is not the case in general. The main purpose of this second part is to provide a conceptual existence proof under quite general conditions (in particular for arbitrary reductive \(G\) over henselian discretely valuated fields \(K\) with perfect residue field \(k\); but also dense valuations are dealt with). The proof proceeds by a twofold Galois descent for valuated root data, already prepared in chapter 9 of part I. A first “quasi-split descent” allows the passage from the split case to the quasi-split case. This step could be performed at a more elementary level, but the authors rely on their general principles laid out in part I (nonetheless, an appendix provides the necessary explicit commutation relations for root subgroups of quasi-split groups required in the more direct approach). The second “étale descent”, from the quasi-split to the general case relative to an étale base change, rests on a crucial result (5.1.12) about maximal tori of \(G\) which is derived as a consequence of a theory of independent interest, the development of which constitutes the main body of the article. It associates to any bounded subset \(F\) of an apartment of the building \({\mathcal J}\) of \(G\) (or more general, to any concave real-valued function on the roots of G) a group scheme \(G_ F\) over the ring \({\mathcal O}\) of integers of the field \(K\) whose generic fibre \(G_F\otimes_{\mathcal O}K\) coincides with \(G\) and whose group of \({\mathcal O}\)- rational points \(G_F({\mathcal O})\) equals the stabilizer of \(F\) with respect to the \(G(K)\)-action on \({\mathcal J}.\)
These schemes are constructed first in the quasi-split case and later, by ”étale descent” in the general case. The basic idea here is to equip a maximal torus \(T\) and the root subgroups \(U_a\) of \(G\) with \({\mathcal O}\)- scheme structures and then to “integrate” a schematic ”open Bruhat cell” to an \({\mathcal O}\)-group scheme by means of an \({\mathcal O}\)-linear representation.
In the case of a discrete valuation on \(K\) and a facet \(F\) of the building \({\mathcal J}\), the group \(G_F({\mathcal O})\) is a parabolic subgroup of a Tits system of affine type in \(G(K)\). The star of \(F\) in \({\mathcal J}\) identifies then naturally with the spherical building of the reductive part of the closed fibre \(G_F\otimes_{\mathcal O}k\), \(k\) the residue field of \(K\). There are also results on the unipotent radical of this group and generalizations to the case of dense valuations.
Since the publication of this work, another article by the authors [Bull. Soc. Math. Fr. 112, 259–301 (1984; Zbl 0565.14028)] has appeared, in which, for the special case of general linear groups and, in sequel to the work of O. Goldman and N. Iwahori [Acta Math. 109, 137–177 (1963; Zbl 0133.294)], some of the constructions and results described above are interpreted in terms of norms on the underlying vector spaces.
It is obvious that a short review can only give a very approximate picture of the contents of this work. This applies in particular to the generality and the precise formulation of the individual results. For more details, also concerning the historical development and work by other authors, we refer to the introductions of part I and II.
Editorial remark: See also the joint “Looking back” review of this article and Zbl 0254.14017.

Reviewer: P.Slodowy

MSC:

14Lxx	Algebraic groups
14L15	Group schemes
20G25	Linear algebraic groups over local fields and their integers

Keywords:

reductive algebraic groups over local fields; valuated root datum; quasi- split descent; étale descent; open Bruhat cell; spherical building

Citations:

Zbl 0228.20015; Zbl 0254.14017; Zbl 0565.14028; Zbl 0133.294

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