no. 4
Reductive group actions on affine varieties and their doubling
Annales de l'Institut Fourier, Volume 45 (1995) no. 4, pp. 929-950

We study G-actions of the form (G:X×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.

Nous étudions les actions de la forme (G:X×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.

Panyushev, Dmitri I. Reductive group actions on affine varieties and their doubling. Annales de l'Institut Fourier, Volume 45 (1995) no. 4, pp. 929-950. doi: 10.5802/aif.1479
@article{AIF_1995__45_4_929_0,
     author = {Panyushev, Dmitri I.},
     title = {Reductive group actions on affine varieties and their doubling},
     journal = {Annales de l'Institut Fourier},
     pages = {929--950},
     year = {1995},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {4},
     doi = {10.5802/aif.1479},
     zbl = {0831.14022},
     mrnumber = {96i:14039},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1479/}
}
TY  - JOUR
AU  - Panyushev, Dmitri I.
TI  - Reductive group actions on affine varieties and their doubling
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 929
EP  - 950
VL  - 45
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1479/
DO  - 10.5802/aif.1479
LA  - en
ID  - AIF_1995__45_4_929_0
ER  - 
%0 Journal Article
%A Panyushev, Dmitri I.
%T Reductive group actions on affine varieties and their doubling
%J Annales de l'Institut Fourier
%D 1995
%P 929-950
%V 45
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1479/
%R 10.5802/aif.1479
%G en
%F AIF_1995__45_4_929_0

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

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

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

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

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

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

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

[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. | Zbl | MR

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

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

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

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

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

[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

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

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

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

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

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

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

Cited by Sources: