Summary
LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toK U. Then we study the set of invariant valuations having some fixed restrictionv 0, toK B. Ifv 0 is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupW X. Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofK B.
Similar content being viewed by others
Literatur
[Br] Brion, M.: Vers une généralisation des espaces symétriques. J. Algebra134, 115–143 (1990)
[BLV] Brion, M., Luna, D., Vust, Th.: Espaces homogènes sphériques. Invent. Math.84, 617–632 (1986)
[BP] Brion, P., Pauer, F.: Valuation des espaces homogènes sphériques. Comment. Math. Helv.62, 265–285 (1987)
[Cox] Coxeter, H.S.M.: Discrete groups generated by reflections. Ann. Math.35, 588–621 (1934)
[CP] De Concini, C., Procesi, C.: Complete symmetric varieties. In: Gherardelli, F. (ed.) Invariant theory. Proceedings, Montecatini. (Lect. Notes Math., vol. 996, pp. 1–44) Berlin Heidelberg New York: Springer 1983
[De] Demazure, M.: Sous-groups algébriques de rang maximum du group de Cremona. Ann. Sci. Éc. Norm. Supér.3, 507–588 (1970)
[Gr1] Grosshans, F.D.: The invariants of unipotent radicals of parabolic subgroups. Invent. math.73, 1–9 (1983)
[Gr2] Grosshans, F.D.: Contractions of the actions of reductive algebraic groups in arbitrary characteristic. Invent. Math.107, 127–133 (1992)
[EGA] Dieudonné, J., Grothendieck, A.: Eléments de géométrie algébrique. IV. Publ. Math. Inst. Hautes Étud. Sci.28 (1966)
[Ha] Hartshorne, R.: Algebraic geometry. (Graduate Texts Math., vol. 52). Berlin Heidelberg New York: Springer 1977
[Hu] Humphreys, J.E.: Linear algebraic groups. (Graduate Texts Math., vol. 21) Berlin Heidelberg New York: Springer 1981
[KKMS] Kempf, G., Knudson, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings, I. (Lect. Notes Math., vol. 339) Berlin Heidelberg New York: Springer 1973
[Kn1] Knop, F.: Weylgruppe und Momentabbildung. Invent. Math.99 1–23 (1990)
[Kn2] Knop, F.: The Luna-Vust theory of spherical embeddings. In: Ramanan, S. (ed.): Proceedings of the Hyderabad conference on algebraic groups. Madras: Manoj Prakashan 1991
[Kn3] Knop, F.: Weylgruppe und äquivariante Einbettungen (Preprint 1990)
[KKV] Knop, F., Kraft, H., Vust, Th.: The Picard group of aG-variety. In: Kraft, H., Slodowy, P., Springer, T. (eds.) Algebraische Transformationsgrupen und Invariantentheorie. (DMV Semin., vol. 13, pp. 77–88) Basel Boston Berlin: Birkhäuser 1989
[KKLV] Knop, F., Kraft, H., Luna, D., Vust, Th.: Local properties of algebraic group actions. In: Kraft, H., Slodowy, P., Springer, T. (eds.) Algebraische Transformationsgruppen und Invariantentheorie. (DMV Semin., vol. 13, pp. 63–76) Basel Boston Berlin: Birkhäuser 1989
[LV] Luna, D., Vust, Th.: Plongements d'espaces homogènes Comment. Math. Helv.58, 186–245 (1983)
[MO] Miyake K., Oda, T.: Almost homogeneous algebraic varieties under algebraic torus actions. In: Hattori, A. (ed.) Manifolds-Tokyo, 1973, pp. 373–381 Tokyo: University of Tokyo Press 1975
[Pan] Panyushev, D.: Complexity and rank of homogeneous spaces. Geom. Dedicata34, 249–269 (1990)
[Pau] Pauer, F.: “Caracterisation valuative” d'une classe de sous-groupes d'un groupe algébrique. C.R. 109e Congrès Nat. Soc. Sav.3, 159–166 (1984)
[Sa] Satake, I.: On representations and compactifications of symmetric Riemannian spaces. Ann. Math.71, 77–110 (1960)
[Sl] Slodowy, P.: Simple singularities and simple algebraic groups. (Lect. Notes Math., vol. 815) Berlin Heidelberg New York: Springer 1980
[Su] Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ.14, 19–28 (1974)
[ZS] Zariski, O., Samuel, P.: Commutative algebra, vol. II. Princeton: Van Nostrand 1960
Author information
Authors and Affiliations
Additional information
Unterstützt durch den Schweizerischen Nationalfonds zur Förderung der wissenschaftlichen Forschung.