Computation of Weyl groups of 𝐺-varieties

Losev, Ivan

doi:10.1090/S1088-4165-10-00365-1

Computation of Weyl groups of $G$-varieties
HTML articles powered by AMS MathViewer

by Ivan V. Losev

Represent. Theory 14 (2010), 9-69

DOI: https://doi.org/10.1090/S1088-4165-10-00365-1

Published electronically: January 5, 2010

PDF | Request permission

Abstract:

Let $G$ be a connected reductive group. To any irreducible $G$-variety one assigns a certain linear group generated by reflections called the Weyl group. Weyl groups play an important role in the study of embeddings of homogeneous spaces. We establish algorithms for computing Weyl groups for homogeneous spaces and affine homogeneous vector bundles. For some special classes of $G$-varieties (affine homogeneous vector bundles of maximal rank, affine homogeneous spaces, homogeneous spaces of maximal rank with a discrete group of central automorphisms) we compute Weyl groups more or less explicitly.

References

E. M. Andreev, È. B. Vinberg, and A. G. Èlašvili, Orbits of highest dimension of semisimple linear Lie groups, Funkcional. Anal. i Priložen. 1 (1967), no. 4, 3–7 (Russian). MR 0267040
N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1364, Hermann, Paris, 1975 (French). MR 0453824
Michel Brion and Franz Pauer, Valuations des espaces homogènes sphériques, Comment. Math. Helv. 62 (1987), no. 2, 265–285 (French). MR 896097, DOI 10.1007/BF02564447
P. Bravi. Wonderful varieties of type D. Ph. D. thesis, Università di Roma “La Sapienza”, 2003.
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
Elie Cartan, Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple, Ann. Sci. École Norm. Sup. (3) 44 (1927), 345–467 (French). MR 1509283, DOI 10.24033/asens.781
S. Cupit-Foutou. Wonderful varieties: A geometrical realization. Preprint (2009), arXiv:0907.2852.
E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
A. G. Èlašvili, Canonical form and stationary subalgebras of points in general position for simple linear Lie groups, Funkcional. Anal. i Priložen. 6 (1972), no. 1, 51–62 (Russian). MR 0304554
A. G. Èlašvili, Stationary subalgebras of points of general position for irreducible linear Lie groups, Funkcional. Anal. i Priložen. 6 (1972), no. 2, 65–78 (Russian). MR 0304555
Friedrich Knop, Weylgruppe und Momentabbildung, Invent. Math. 99 (1990), no. 1, 1–23 (German, with English summary). MR 1029388, DOI 10.1007/BF01234409
Friedrich Knop, 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
Friedrich Knop, Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind, Math. Ann. 295 (1993), no. 2, 333–363 (German, with English summary). MR 1202396, DOI 10.1007/BF01444891
Friedrich Knop, The asymptotic behavior of invariant collective motion, Invent. Math. 116 (1994), no. 1-3, 309–328. MR 1253195, DOI 10.1007/BF01231563
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
Friedrich Knop, Some remarks on multiplicity free spaces, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 301–317. MR 1653036
Friedrich Knop, A connectedness property of algebraic moment maps, J. Algebra 258 (2002), no. 1, 122–136. Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958900, DOI 10.1016/S0021-8693(02)00509-4
Andrew S. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), no. 2, 367–391. MR 1650378
I. V. Losev, Computation of Cartan spaces for affine homogeneous spaces, Mat. Sb. 198 (2007), no. 10, 31–56 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 9-10, 1407–1431. MR 2362821, DOI 10.1070/SM2007v198n10ABEH003889
Ivan Losev, Combinatorial invariants of algebraic Hamiltonian actions, Mosc. Math. J. 8 (2008), no. 3, 493–519, 616 (English, with English and Russian summaries). MR 2483222, DOI 10.17323/1609-4514-2008-8-3-493-519
I.V. Losev. On fibers of algebraic invariant moment maps. Preprint (2007), arXiv:math.AG/0703296, 37 pages. To appear in Transform. Groups.
Ivan V. Losev, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147 (2009), no. 2, 315–343. MR 2495078, DOI 10.1215/00127094-2009-013
D. Luna and R. W. Richardson, A generalization of the Chevalley restriction theorem, Duke Math. J. 46 (1979), no. 3, 487–496. MR 544240, DOI 10.1215/S0012-7094-79-04623-4
Domingo Luna, Sur les orbites fermées des groupes algébriques réductifs, Invent. Math. 16 (1972), 1–5 (French). MR 294351, DOI 10.1007/BF01391210
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 and Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245 (French). MR 705534, DOI 10.1007/BF02564633
William M. McGovern, The adjoint representation and the adjoint action, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia Math. Sci., vol. 131, Springer, Berlin, 2002, pp. 159–238. MR 1925831, DOI 10.1007/978-3-662-05071-2_{3}
A. L. Onishchik and È. B. Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. MR 1064110, DOI 10.1007/978-3-642-74334-4
D. I. Panyushev, Orbit spaces of finite and connected linear groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 1, 95–99, 191 (Russian). MR 643895
Dmitrii I. Panyushev, Complexity and nilpotent orbits, Manuscripta Math. 83 (1994), no. 3-4, 223–237. MR 1277527, DOI 10.1007/BF02567611
Dmitri I. Panyushev, A restriction theorem and the Poincaré series for $U$-invariants, Math. Ann. 301 (1995), no. 4, 655–675. MR 1326762, DOI 10.1007/BF01446653
Dmitri Panyushev, On homogeneous spaces of rank one, Indag. Math. (N.S.) 6 (1995), no. 3, 315–323. MR 1351150, DOI 10.1016/0019-3577(95)93199-K
D. I. Panyushev, Complexity and rank of actions in invariant theory, J. Math. Sci. (New York) 95 (1999), no. 1, 1925–1985. Algebraic geometry, 8. MR 1708594, DOI 10.1007/BF02169155
G. Pezzini. Wonderful varieties of type C. Ph. D. thesis, Università di Roma “La Sapienza”, 2003.
V. L. Popov, Criteria for the stability of the action of a semisimple group on the factorial of a manifold, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 523–531 (Russian). MR 0262416
È. B. Vinberg and V. L. Popov, Invariant theory, Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 137–314, 315 (Russian). MR 1100485
Gerald W. Schwarz, Representations of simple Lie groups with a free module of covariants, Invent. Math. 50 (1978/79), no. 1, 1–12. MR 516601, DOI 10.1007/BF01406465
Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37–135. MR 573821, DOI 10.1007/BF02684776
A. V. Smirnov, Quasiclosed orbits in projective representation of semisimple complex Lie groups, Tr. Mosk. Mat. Obs. 64 (2003), 213–270 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2003), 193–247. MR 2030190
A. A. Sukhanov, The description of observable subgroups of linear algebraic groups, Mat. Sb. (N.S.) 137(179) (1988), no. 1, 90–102, 144 (Russian); English transl., Math. USSR-Sb. 65 (1990), no. 1, 97–108. MR 965881, DOI 10.1070/SM1990v065n01ABEH001141
D. A. Timashëv, Classification of $G$-manifolds of complexity $1$, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 2, 127–162 (Russian, with Russian summary); English transl., Izv. Math. 61 (1997), no. 2, 363–397. MR 1470147, DOI 10.1070/im1997v061n02ABEH000119
D.A. Timashev, Homogeneous spaces and equivariant embeddings. Preprint (2006), arXiv:math.AG/0602228.
È. B. Vinberg, Invariant linear connections in a homogeneous space, Trudy Moskov. Mat. Obšč. 9 (1960), 191–210 (Russian). MR 0176418
Thierry Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317–333 (French). MR 366941, DOI 10.24033/bsmf.1782
Thierry Vust, Plongements d’espaces symétriques algébriques: une classification, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 2, 165–195 (French). MR 1076251
B. Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), no. 4, 375–403. MR 1424449, DOI 10.1007/BF02549213
B.Yu. Weisfeller. On a class of unipotent subgroups of semisimple algebraic groups. Usp. Mat. Nauk 22(1966), no. 2, 222-223 (in Russian).
H. Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I, Math. Z. 23 (1925), no. 1, 271–309 (German). MR 1544744, DOI 10.1007/BF01506234