Wonderful varieties of type 𝐷

Bravi, Paolo; Pezzini, Guido

doi:10.1090/S1088-4165-05-00260-8

Wonderful varieties of type $D$
HTML articles powered by AMS MathViewer

by Paolo Bravi and Guido Pezzini PDF

Represent. Theory 9 (2005), 578-637 Request permission

Abstract:

Let $G$ be a connected semisimple group over $\mathbb C$, whose simple components have type $\mathsf A$ or $\mathsf D$. We prove that wonderful $G$-varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known result of Luna for groups of type $\mathsf A$; thanks to another result of Luna, this implies also the classification of all spherical $G$-varieties for the groups $G$ we are considering. For these $G$ we also prove the smoothness of the embedding of Demazure.

References

Dmitry Ahiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), no. 1, 49–78. MR 739893, DOI 10.1007/BF02329739
Valery Alexeev and Michel Brion, Moduli of affine schemes with reductive group action, J. Algebraic Geom. 14 (2005), no. 1, 83–117. MR 2092127, DOI 10.1090/S1056-3911-04-00377-7
Shôrô Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. MR 153782
M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), no. 2, 189–208 (French). MR 906369
Michel Brion, On spherical varieties of rank one (after D. Ahiezer, A. Huckleberry, D. Snow), Group actions and invariant theory (Montreal, PQ, 1988) CMS Conf. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 1989, pp. 31–41. MR 1021273
Michel Brion, Vers une généralisation des espaces symétriques, J. Algebra 134 (1990), no. 1, 115–143 (French). MR 1068418, DOI 10.1016/0021-8693(90)90214-9

Variétés sphériques affines lisses

C. De Concini and C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1–44. MR 718125, DOI 10.1007/BFb0063234
Thomas Delzant, Classification des actions hamiltoniennes complètement intégrables de rang deux, Ann. Global Anal. Geom. 8 (1990), no. 1, 87–112 (French). MR 1075241, DOI 10.1007/BF00055020
Friedrich Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153–174. MR 1311823, DOI 10.1090/S0894-0347-96-00179-8
Manfred Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), no. 2, 129–153 (German). MR 528837
D. Luna, Toute variété magnifique est sphérique, Transform. Groups 1 (1996), no. 3, 249–258 (French, with English summary). MR 1417712, DOI 10.1007/BF02549208
D. Luna, Grosses cellules pour les variétés sphériques, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 267–280 (French). MR 1635686
D. Luna, Variétés sphériques de type $A$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226 (French). MR 1896179, DOI 10.1007/s10240-001-8194-0
D. Luna, Sur les plongements de Demazure, J. Algebra 258 (2002), no. 1, 205–215 (French). Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958903, DOI 10.1016/S0021-8693(02)00507-0
D. Luna and Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245 (French). MR 705534, DOI 10.1007/BF02564633

Integrability of invariant Hamiltonian systems with homogeneous configuration spaces

57

B. Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), no. 4, 375–403. MR 1424449, DOI 10.1007/BF02549213