Proof of the Knop conjecture
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, p. 1105-1134

In this paper we prove the Knop conjecture asserting that two smooth affine spherical varieties with the same weight monoid are equivariantly isomorphic. We also state and prove a uniqueness property for (not necessarily smooth) affine spherical varieties.

Dans cet article nous prouvons la conjecture de Knop qui affirme que deux variétés affines sphériques lisses avec le même monoïde des poids sont isomorphes de manière équivariante. On énonce et prouve également une propriété d’unicité pour des variétés affines sphériques non nécessairement lisses.

DOI : https://doi.org/10.5802/aif.2459
Classification:  14R20,  53D20
Keywords: Spherical varieties, weight monoids, systems of spherical roots, multiplicity free Hamiltonian actions
@article{AIF_2009__59_3_1105_0,
     author = {Losev, Ivan V.},
     title = {Proof of the Knop conjecture},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     pages = {1105-1134},
     doi = {10.5802/aif.2459},
     mrnumber = {2543664},
     zbl = {1191.14075},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_3_1105_0}
}
Losev, Ivan V. Proof of the Knop conjecture. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 1105-1134. doi : 10.5802/aif.2459. http://www.numdam.org/item/AIF_2009__59_3_1105_0/

[1] Bialynicki-Birula, A. Some properties of the decompositions of algebraic varieties determined by an action of a torus, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., Tome 24 (1976), pp. 667-674 | MR 453766 | Zbl 0355.14015

[2] Brion, M. Sur l’image de l’application moment, Springer Verlag, Lect. Notes Math., Tome 1296 (1987) | MR 932055 | Zbl 0667.58012

[3] Brion, M.; Luna, D.; Vust, Th. Espaces homogènes sphériques, Invent. Math., Tome 84 (1986), pp. 617-632 | Article | MR 837530 | Zbl 0604.14047

[4] Camus, R. Variétés sphériques affines lisses, Université J. Fourier (2001) (Ph. D. Thesis)

[5] Delzant, T. Hamiltoniens périodiques et images convexes de l’application moment, Bul. Soc. Math. France, Tome 116 (1988), pp. 315-339 | Numdam | MR 984900 | Zbl 0676.58029

[6] Delzant, T. Classification des actions hamiltoniennes complètement intégrables de rang deux, Ann. Global. Anal. Geom., Tome 8 (1990), pp. 87-112 | Article | MR 1075241 | Zbl 0711.58017

[7] Grosshans, F. Constructing invariant polynomials via Tschirnhaus transformations, Springer Verlag, Lect. Notes Math., Tome 1278 (1987) | MR 924168 | Zbl 0659.14010

[8] Guillemin, V.; Jeffrey, L.; Sjamaar, R. Symplectic implosion, Transform. Groups, Tome 7 (2002), pp. 155-184 | Article | MR 1903116 | Zbl 1015.53054

[9] Kirwan, F. Convexity properties of the moment mapping III, Invent. Math., Tome 77 (1984), pp. 547-552 | Article | MR 759257 | Zbl 0561.58016

[10] Knop, F. The Luna-Vust theory of spherical embeddings, Proceedings of the Hydebarad conference on algebraic groups, Manoj Prakashan, Madras (1991), pp. 225-249 | MR 1131314 | Zbl 0812.20023

[11] Knop, F. The asymptotic behaviour of invariant collective motion, Invent. Math., Tome 114 (1994), pp. 309-328 | Article | MR 1253195 | Zbl 0802.58024

[12] Knop, F. Automorphisms, root systems and compactifications, J. Amer. Math. Soc., Tome 9 (1996), pp. 153-174 | Article | MR 1311823 | Zbl 0862.14034

[13] Knop, F.; Steirteghem, B. Van Classification of smooth affine spherical varieties, Transform. Groups, Tome 11 (2006), pp. 495-516 | Article | MR 2264463 | Zbl 1120.14042

[14] Kramer, M. Spharische Untergruppen in kompakten zusammenhangenden Liegruppen, Compos. Math., Tome 38 (1979), pp. 129-153 | Numdam | MR 528837 | Zbl 0402.22006

[15] Leahy, A.S. A classification of multiplicity free representations, J. Lie Theory, Tome 8 (1998), pp. 376-391 | MR 1650378 | Zbl 0910.22015

[16] Losev, I. Uniqueness properties for spherical homogeneous spaces (to appear in Duke Math. J.)

[17] Luna, D. Grosses cellules pour les variétés sphériques, Cambridge University Press, Austr. Math. Soc. Lect. Ser., Tome 9 (1997) | MR 1635686 | Zbl 0902.14037

[18] Luna, D. Variétés sphériques de type A, IHES Publ. Math., Tome 94 (2001), pp. 161-226 | Article | Numdam | MR 1896179 | Zbl 1085.14039

[19] Onishchik, A.L.; Vinberg, E.B. Lie groups and algebraic groups, Springer Verlag (1990) | MR 1064110 | Zbl 0722.22004

[20] Popov, V.L. Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Math. USSR-Izv, Tome 7 (1973), pp. 301-327 | Zbl 0301.14018

[21] Popov, V.L. Contractions of the actions of reductive algebraic groups, Math. USSR Sborhik, Tome 58 (1987), pp. 311-355 | Article | MR 865764 | Zbl 0627.14033

[22] Vust, Th. Plongement d’espaces symétriques algébriques: une classification, Ann. Sc. Norm. Super. Pisa, Ser. IV, Tome 17 (1990), pp. 165-194 | Numdam | MR 1076251 | Zbl 0728.14041

[23] Wasserman, B. Wonderful varieties of rank 2, Transform. Groups, Tome 1 (1996), pp. 375-403 | Article | MR 1424449 | Zbl 0921.14031

[24] Woodward, C. The classification of transversal multiplicity-free Hamiltonian actions, Ann. Global. Anal. Geom., Tome 14 (1996), pp. 3-42 | Article | MR 1375064 | Zbl 0877.58022

[25] Woodward, C. Spherical varieties and existence of invariant Kahler structure, Duke Math. J, Tome 93 (1998), pp. 345-377 | Article | MR 1625995 | Zbl 0979.53085