×

zbMATH — the first resource for mathematics

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
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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] A. Borel, Groupes algébriques linéaires,Ann. of Math.,62 (1956), 20–80. · Zbl 0070.26104 · doi:10.2307/1969949
[2] A. Borel,Linear algebraic groups, New York, Benjamin, 1969. · Zbl 0206.49801
[3] A. Borel andT. A. Springer, Rationality properties of linear algebraic groups II,Tôhoku Math. J.,20 (1968), 443–497. · Zbl 0211.53302 · doi:10.2748/tmj/1178243073
[4] A. Borel etJ. Tits, Groupes réductifs,Publ. Math. I.H.E.S.,27 (1965), 55–150.
[5] A. Borel etJ. Tits, Compléments à l’article “ Groupes réductifs ”,Publ. Math. I.H.E.S.,41 (1972), 253–276, · Zbl 0254.14018
[6] A. Borel etJ. Tits, Homomorphismes “ abstraits ” de groupes algébriques simples,Ann. of Math.,97 (1973). 499–571. · Zbl 0272.14013 · doi:10.2307/1970833
[7] A. Borel etJ. Tits, Théorèmes de structure et de conjugaison pour les groupes algébriques linéaires,C. R. Acad. Sci.,287 (1978), 55–57.
[8] F. Bruhat, Groupes semi-simples sur un corps local,Actes du Congrès international des Mathématiciens (Nice, 1970). Paris, Gauthier-Villars, 1971. · Zbl 0238.20053
[9] F. Bruhat etJ. Tits, BN-paires de type affine et données radicielles affines,C. R. Acad. Sci.,263 (1966), 598–601. · Zbl 0214.28803
[10] F. Bruhat etJ. Tits, Groupes simples résiduellement déployés sur un corps local,C. R. Acad. Sci.,263 (1966), 766–768. · Zbl 0263.14013
[11] F. Bruhat etJ. Tits, Groupes algébriques simples sur un corps local,C. R. Acad. Sci.,263 (1966), 822–825. · Zbl 0263.14014
[12] F. Bruhat etJ. Tits, Groupes algébriques simples sur un corps local, cohomologie galoisienne, décompositions d’Iwasawa et de Cartan,C. R. Acad. Sci.,263 (1966), 867–869. · Zbl 0263.14015
[13] F. Bruhat etJ. Tits, Groupes algébriques simples sur un corps local,Proc. Conf. on local fields (Driebergen, 1966), Springer, 1967, 23–36.
[14] C. Chevalley, Sur certains groupes simples,Tohôku Math. J. (2),7 (1955), 14–66. · Zbl 0066.01503 · doi:10.2748/tmj/1178245104
[15] C. Chevalley,Séminaire sur la classification des groupes de Lie algébriques, 2 vol., Paris, Inst. H. Poincaré, 1958 (multigraphié).
[16] C. Chevalley, Certains schémas de groupes semi-simples,Séminaire N. Bourbaki,13, exposé 219 (1960–1961), New York, Benjamin, 1966.
[17] M. Demazure, Schémas en groupes réductifs,Bull. Soc. Math. Fr.,93 (1965), 369–413. · Zbl 0163.27402
[18] H. Hijikata,On the arithmetic of p-adic Steinberg groups, Yale University, 1964 (multigraphié).
[19] H. Hijikata,Maximal compact subgroups of some p-adic classical groups, Yale University, 1964 (multigraphié).
[20] H. Hijikata, On the structure of semi-simple algebraic groups over valuation fields I,Japan J. Math.,1 (1975), 225–300. · Zbl 0386.20021
[21] N. Iwahori andH. Matsumoto, On some Bruhat decomposition and the structure of the Hecke ring ofp-adic Chevalley groups,Publ. Math. I.H.E.S.,25 (1965), 5–48. · Zbl 0228.20015
[22] M. Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen überp-adischen Körpern,Math. Z.,88 (1965), 40–47, et89 (1965), 250–272. · Zbl 0143.04702 · doi:10.1007/BF01112691
[23] G. Prasad, Elementary proof of a theorem of Bruhat-Tits and Rousseau and of a theorem of Tits,Bull. Soc. Math. Fr.,110 (1982), 197–202. · Zbl 0492.20029
[24] M. Raynaud, Modèles de Néron,C. R. Acad. Sci.,262 (1966), 345–347. · Zbl 0141.18203
[25] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes,Lecture Notes in Math.,119, Springer, 1970. · Zbl 0195.22701
[26] M. Raynaud etL. Gruson, Critères de platitude et de projectivité,Inv. Math.,13 (1971), 1–89. · Zbl 0227.14010 · doi:10.1007/BF01390094
[27] G. Rousseau,Immeubles des groupes réductifs sur les corps locaux, Thèse, Université Paris XI, 1977 (multigraphié). · Zbl 0412.22006
[28] J.-P. Serre,Corps locaux, Paris, Herman, 1962. · Zbl 0137.02601
[29] J.-P. Serre, Cohomologie galoisienne,Lecture Notes in Math.,5, Springer, 1964. · Zbl 0143.05901
[30] J.-P. Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés,Publ. Math. I.H.E.S.,34 (1968), 37–52.
[31] R. Steinberg, Variations on a theme of Chevalley,Pacific J. Math.,9 (1959), 875–891. · Zbl 0092.02505
[32] R. Steinberg, Regular elements of semi-simple algebraic groups,Publ. Math. I.H.E.S.,27 (1965), 49–80.
[33] R. Steinberg,Lecture on Chevalley groups, Yale University lecture notes, 1967. · Zbl 0164.34302
[34] J. Tits, Classification of algebraic semisimple groups,Proc. Sympos. Pure Math.,9 (1966), 33–62. · Zbl 0238.20052
[35] J. Tits, Sur les constantes de structure et le théorème d’existence des algèbres de Lie semi-simples,Publ. Math. I.H.E.S.,31 (1966), 525–562. · Zbl 0145.25804
[36] J. Tits, Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque,J. für die reine und angew. Math.,247 (1971), 196–220. · Zbl 0227.20015 · doi:10.1515/crll.1971.247.196
[37] J. Tits, Buildings of spherical type and finite BN-pairs,Lecture Notes in Math.,386, Springer, 1974. · Zbl 0295.20047
[38] J. Tits, Résumé de cours,Annuaire du Collège de France, 1975–1976, 49–56.
[39] J. Tits, Résumé de cours,Annuaire du Collège de France, 1979–1980, 75–80.
[40] J. Tits, Reductive groups over local fields,Proc. Sympos. Pure Math.,33 (1979), 29–69. · Zbl 0415.20035
[41] J. Tits, Classification of buildings of spherical type and Moufang polygons: a survey,Coll. Intern. sulle Teorie Combinatoire, Rome, 1973, Accad. Naz. dei Lincei, 1976, t. 1, 229–246.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.