Schémas en groupes et immeubles des groupes classiques sur un corps local. (French) Zbl 0565.14028
This is the first of two papers in which the authors intend to give concrete interpretations of the building of a reductive group G over a local field K and of the group scheme \(G_ x\) attached to a point x of that building [cf. Publ. Math., Inst. Hautes Étud. Sci. 41, 5-251 (1972; Zbl 0254.14017) and 60, 5-184 (1984)] in the special case where G is a classical group. Here, ”concrete” means expressed in terms of the natural representation. The present paper deals with the (general and special) linear groups over a division ring D endowed with a discrete valuation and for finite rank over its center K. Unitary groups will be considered in a forthcoming part 2.
Section 1 presents more or less classical results on ”splittable” norms (”normes scindables”, i.e., in the classical terminology, norms admitting an orthogonal basis) on \(D^ n\), and on the order \(M_{\alpha}\) in \(M_ n(D)\) associated to such a norm \(\alpha\) ; here, D is not assumed to be complete. - Section 2 shows that the ”enlarged building” (”immeuble élargi”) \({\mathcal I}\) of \(GL_ n(D)\) can be canonically identified with the set \({\mathcal N}\) of splittable norms and that this provides an isomorphism of the abstract simplicial complex of all facets of \({\mathcal I}\) with the set of all hereditary orders in \(M_ n(D)\), made into a simplicial complex by the inclusion relation. - To every \(x\in {\mathcal I}={\mathcal N}\), section 3 associates a smooth and connected group scheme \({\mathfrak G}_ x\) with generic fiber G, namely the multiplicative group scheme of \(M_ x\); it has a ”big cell”, and its group of integral points is the stabilizer of x in \(GL_ n(D)\). By section 4 all that behaves well under étale Galois extension. It follows that if K is Henselian and D is split by an étale extension of K, then the schemes \({\mathfrak G}_ x\) are those of the general theory of the cited papers. This provides an easy proof of a theorem of V. P. Platonov and V. I. Jančevskij [Soviet Math., Doklady 16, 424-427 (1975); translation from Dokl. Akad. Nak SSSR 221, 784-787 (1975; Zbl 0333.20035)]. - Section 5 deals with \(SL_ n(D)\); the results are similar except that the schemes are no longer always smooth if D does not split over an etable extension of K. Most results of sections 1 and 2 are extended to non- discrete valuations in an appendix.
Section 1 presents more or less classical results on ”splittable” norms (”normes scindables”, i.e., in the classical terminology, norms admitting an orthogonal basis) on \(D^ n\), and on the order \(M_{\alpha}\) in \(M_ n(D)\) associated to such a norm \(\alpha\) ; here, D is not assumed to be complete. - Section 2 shows that the ”enlarged building” (”immeuble élargi”) \({\mathcal I}\) of \(GL_ n(D)\) can be canonically identified with the set \({\mathcal N}\) of splittable norms and that this provides an isomorphism of the abstract simplicial complex of all facets of \({\mathcal I}\) with the set of all hereditary orders in \(M_ n(D)\), made into a simplicial complex by the inclusion relation. - To every \(x\in {\mathcal I}={\mathcal N}\), section 3 associates a smooth and connected group scheme \({\mathfrak G}_ x\) with generic fiber G, namely the multiplicative group scheme of \(M_ x\); it has a ”big cell”, and its group of integral points is the stabilizer of x in \(GL_ n(D)\). By section 4 all that behaves well under étale Galois extension. It follows that if K is Henselian and D is split by an étale extension of K, then the schemes \({\mathfrak G}_ x\) are those of the general theory of the cited papers. This provides an easy proof of a theorem of V. P. Platonov and V. I. Jančevskij [Soviet Math., Doklady 16, 424-427 (1975); translation from Dokl. Akad. Nak SSSR 221, 784-787 (1975; Zbl 0333.20035)]. - Section 5 deals with \(SL_ n(D)\); the results are similar except that the schemes are no longer always smooth if D does not split over an etable extension of K. Most results of sections 1 and 2 are extended to non- discrete valuations in an appendix.
MSC:
| 14L15 | Group schemes |
| 14L35 | Classical groups (algebro-geometric aspects) |
| 20G25 | Linear algebraic groups over local fields and their integers |
| 16Kxx | Division rings and semisimple Artin rings |
References:
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