Reductive group actions on affine varieties and their doubling

Panyushev, Dmitri I.

Annales de l'Institut Fourier, Tome 45 (1995), p. 929-950 / Harvested from Numdam

Résumé

Nous étudions les actions de la forme $(G:X\times X^{*})$ où $X^*$ est la $G$-variété duale de $X$. Ces actions sont appelées les doubles. Nous donnons une interprétation géométrique de la complexité de l’action $(G:X)$. Nous montrons que les actions doublées ont un certain nombre de bonnes propriétés, lorsque $X$ est sphérique ou de complexité un.

Abstract

We study $G$-actions of the form $(G:X\times X^{*})$, where $X^{*}$ is the dual (to $X$) $G$-variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action $(G:X)$ is given. It is shown that the doubled actions have a number of nice properties, if $X$ is spherical or of complexity one.

Détail

Publié le : 1995-01-01

MR 96i:14039 | Zbl 0831.14022 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1479

Bibliographie

[B1] M. Brion, Invariants d'un sous-groupe unipotent maximal d'un groupe semi-simple, Ann. Inst. Fourier, 33-1 (1983), 1-27. | Numdam | MR 85a:14031 | Zbl 0475.14038

[B2] M. Brion, Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J., 58 (1989), 397-424. | MR 90i:14048 | Zbl 0701.14052

[HH] R. Howe, R. Huang, Projective invariants of four subspaces, Preprint. | Zbl 0852.15021

[KR] B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971), 753-809. | MR 47 #399 | Zbl 0224.22013

[Li] P. Littelmann, On spherical double cones, J. Algebra, 166 (1994), 142-157. | MR 95c:14066 | Zbl 0823.20040

[Lu] D. Luna, Adhérences d'orbite et invariants, Invent. Math., 29 (1975), 231-238. | MR 51 #12879 | Zbl 0315.14018

[LR] D. Luna, R.W. Richardson, A generalization of the Chevalley restriction theorem, Duke Math. J., 46 (1979), 487-496. | MR 80k:14049 | Zbl 0444.14010

[P1] D. Panyushev, Orbits of maximal dimension of solvable subgroups of reductive algebraic groups and reduction for U-invariants, Math. USSR-Sb., 60 (1988), 365-375. | MR 88h:14047 | Zbl 0663.20044

[P2] D. Panyushev, Complexity and rank of homogeneous spaces, Geom. Dedicata, 34 (1990), 249-269. | MR 92e:14046 | Zbl 0706.14032

[P3] D. Panyushev, Complexity and rank of double cones and tensor product decompositions, Comment. Math. Helv., 68 (1993), 455-468. | MR 94g:14025 | Zbl 0804.14024

[P4] D. Panyushev, Complexity and nilpotent orbits, Manuscripta Math., 83 (1994), 223-237. | MR 95e:14039 | Zbl 0822.14024

[P5] D. Panyushev, A restriction theorem and the Poincaré series for U-invariants, Math. Annalen, 301 (1995), 655-675. | MR 96d:13005 | Zbl 0820.14033

[P6] D. Panyushev, Good properties of algebras of invariants and defect of linear representations, J. Lie Theory, 5 (1995). | MR 96j:14034 | Zbl 0845.14008

[Po1] V.L. Popov, A stability criterion for an action of a semisimple group on a factorial variety, Math. USSR-Izv., 4 (1971), 527-535. | Zbl 0261.14011

[Po2] V.L. Popov, Contractions of the actions of reductive algebraic groups, Math. USSR-Sbornik, 58 (1987), 311-335. | Zbl 0627.14033

[Ri] R.W. Richardson, On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc., 25 (1982), 1-28. | MR 83i:14041 | Zbl 0467.14008

[Sch1] G. Schwarz, Representations of simple Lie groups with a free module of covariants, Invent. Math., 50 (1978), 1-12. | MR 80c:14008 | Zbl 0391.20033

[Sch2] G. Schwarz, Lifting smooth homotopies of orbit spaces, Publ. Math. I.H.E.S., 51 (1980), 37-135. | Numdam | MR 81h:57024 | Zbl 0449.57009

[VP1] E.B. Vinberg, V.L. Popov, On a class of quasihomogeneous affine varieties, Math. USSR-Izv., 6 (1972), 743-758. | MR 47 #1815 | Zbl 0255.14016

[VP2] V.L. Popov, E.B. Vinberg, Invariant theory, in : “Encyclopaedia Math. Sci.” 55, Berlin, Springer, 1994, 123-284. | Zbl 0789.14008