Normality and non-normality of group compactifications in simple projective spaces  [ Normalité et non-normalité des plongements d’un groupe dans un espace projectif simple ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 6, p. 2435-2461
Étant donné une représentation irreductible V d’un groupe complexe semi-simple et simplement connexe G, nous considérons l’adhérence X de l’image de G dans (End(V)). Nous déterminons les représentations V pour lesquelles X est normale, respectivement lisse.
Given an irreducible representation V of a complex simply connected semisimple algebraic group G we consider the closure X of the image of G in (End(V)). We determine for which V the variety X is normal and for which V is smooth.
DOI : https://doi.org/10.5802/aif.2679
Classification:  14L30,  14M17
Mots clés: groupes algébriques semi-simples, plongements projectifs d’un groupe
@article{AIF_2011__61_6_2435_0,
     author = {Bravi, Paolo and Gandini, Jacopo and Maffei, Andrea and Ruzzi, Alessandro},
     title = {Normality and non-normality of group compactifications in simple projective spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     pages = {2435-2461},
     doi = {10.5802/aif.2679},
     mrnumber = {2976317},
     zbl = {pre06043084},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_6_2435_0}
}
Bravi, Paolo; Gandini, Jacopo; Maffei, Andrea; Ruzzi, Alessandro. Normality and non-normality of group compactifications in simple projective spaces. Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2435-2461. doi : 10.5802/aif.2679. https://www.numdam.org/item/AIF_2011__61_6_2435_0/

[1] Bourbaki, N. Éléments de mathématique, Hermann Paris, Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV, V, VI, Actualités Scientifiques et Industrielles, Tome 1337 (1968) | MR 453824 | Zbl 0186.33001

[2] Brion, M. Variétés sphériques et théorie de Mori, Duke Math. J., Tome 72 (1993) no. 2, pp. 369-404 | Article | MR 1248677 | Zbl 0821.14029

[3] Chirivì, R.; De Concini, C.; Maffei, A. On normality of cones over symmetric varieties, Tohoku Math. J. (2), Tome 58 (2006) no. 4, pp. 599-616 | Article | MR 2297202 | Zbl 1141.14033

[4] Chirivì, R.; Maffei, A. Projective normality of complete symmetric varieties, Duke Math. J., Tome 122 (2004), pp. 93-123 | Article | MR 2046808 | Zbl 1064.14058

[5] De Concini, C. Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties, Springer, Berlin (Encyclopaedia Math. Sci.) Tome 132 (2004), pp. 15-35 | MR 2090668 | Zbl 1058.22013

[6] De Concini, C.; Procesi, C. Complete symmetric varieties, Invariant Theory, Springer, Berlin (Lecture Notes in Math.) Tome 996 (1983), pp. 1-44 | Zbl 0581.14041

[7] Kannan, S.S. Projective normality of the wonderful compactification of semisimple adjoint groups, Math. Z., Tome 239 (2002), pp. 673-682 | Article | MR 1902056 | Zbl 0997.14012

[8] Knop, F. The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras (1991), pp. 225-249 | MR 1131314 | Zbl 0812.20023

[9] Knop, F.; Kraft, H.; Luna, D.; Vust, T. Local properties of algebraic group actions, DMV Sem., Tome 13 (1989), pp. 63-75 | MR 1044585 | Zbl 0722.14032

[10] Ruzzi, A. Smooth projective symmetric varieties with Picard number equal to one (To appear in Internat. J. Math)

[11] Stembridge, J.R. The partial order of dominant weights, Adv. Math., Tome 136 (1998) no. 2, pp. 340-364 | Article | MR 1626860 | Zbl 0916.06001

[12] Timashev, D.A. Equivariant compactifications of reductive groups, Sb. Math., Tome 194 (2003) no. 3-4, pp. 589-616 | Article | MR 1992080 | Zbl 1074.14043

[13] Vust, Th. Plongements d’espaces symétriques algébriques: une classification, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Tome 17 (1990) no. 2, pp. 165-195 | Numdam | MR 1076251 | Zbl 0728.14041