On deformation method in invariant theory
Annales de l'Institut Fourier, Volume 47 (1997) no. 4, pp. 985-1012.

In this paper we relate the deformation method in invariant theory to spherical subgroups. Let G be a reductive group, Z an affine G-variety and HG a spherical subgroup. We show that whenever G/H is affine and its semigroup of weights is saturated, the algebra of H-invariant regular functions on Z has a G-invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of G. The deformation method in its usual form, as developed by Luna et al., is a particular case of this construction. Our result also applies to the description of invariants of some reducible representations of reductive groups.

New applications of the deformation method are given which concern the property of being complete intersection for algebras of invariants. We also give some applications of the deformation method to doubled actions.

Dans cet article nous relions la méthode de déformation en théorie des invariants aux sous-groupes sphériques. Soient G un groupe réductif, Z une G-variété affine et HG un sous-groupe sphérique. Lorsque G/H est quasi-affine et que son semi-groupe des poids est saturé, nous montrons que l’algèbre des fonctions régulières H-invariantes sur Z a une filtration stable par G telle que l’algèbre graduée associée est l’algèbre des invariants d’un sous-groupe horosphérique explicite de G. La méthode de déformation sous sa forme habituelle, développée par Luna et d’autres auteurs, est un cas particulier de cette construction. Notre résultat s’applique aussi à la description des invariants de certaines représentations réductibles des groupes réductifs.

Nous donnons ensuite de nouvelles applications de la méthode de déformation ; elles concernent les algèbres d’invariants qui sont intersection complète et aussi les actions doublées.

@article{AIF_1997__47_4_985_0,
     author = {Panyushev, Dmitri},
     title = {On deformation method in invariant theory},
     journal = {Annales de l'Institut Fourier},
     pages = {985--1012},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {4},
     year = {1997},
     doi = {10.5802/aif.1589},
     zbl = {0878.14008},
     mrnumber = {99a:13001},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1589/}
}
TY  - JOUR
AU  - Panyushev, Dmitri
TI  - On deformation method in invariant theory
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 985
EP  - 1012
VL  - 47
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1589/
DO  - 10.5802/aif.1589
LA  - en
ID  - AIF_1997__47_4_985_0
ER  - 
%0 Journal Article
%A Panyushev, Dmitri
%T On deformation method in invariant theory
%J Annales de l'Institut Fourier
%D 1997
%P 985-1012
%V 47
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1589/
%R 10.5802/aif.1589
%G en
%F AIF_1997__47_4_985_0
Panyushev, Dmitri. On deformation method in invariant theory. Annales de l'Institut Fourier, Volume 47 (1997) no. 4, pp. 985-1012. doi : 10.5802/aif.1589. https://aif.centre-mersenne.org/articles/10.5802/aif.1589/

[1] N. Andruskiewitsch, H. Tirao, A restriction theorem for modules having a spherical submodule, Trans. Amer. Math. Soc., 331 (1992), 705-725. | MR | Zbl

[2] N. Beklemishev, Algebras invariants of forms that are complete intersections, Math. USSR Izv., 23 (1984), 423-429. | Zbl

[3] W. Borho, H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv., 54 (1979), 61-104. | EuDML | MR | Zbl

[4] M. Brion, Sur la théorie des invariants, Publ. Math. Univ. Pierre et Marie Curie, 45 (1981), 1-92. | Zbl

[5] M. Brion, Représentations exceptionnelles des groupes semi-simples. Ann. Sci. Éc. Norm. Sup., IV Sér., 18 (1985), 345-387. | EuDML | Numdam | MR | Zbl

[6] M. Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math., 55 (1986), 191-198. | EuDML | MR | Zbl

[7] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge University Press, 1993. | MR | Zbl

[8] F.D. Grosshans, The invariants of unipotent radicals of parabolic subgroups, Invent. Math., 73 (1983), 1-9. | EuDML | MR | Zbl

[9] F.D. Grosshans, Contractions of the actions of reductive groups in arbitrary characteristic, Invent. Math., 107 (1992), 127-133. | EuDML | MR | Zbl

[10] R. Howe, R. Huang, Projective invariants of four subspaces, Adv. in Math., 118 (1996), 295-336. | MR | Zbl

[11] R. Howe, T. Umeda, The Capelly identity, the double commutant theorem, and multiplicity-free actions, Math. Ann., 290 (1991), 565-619. | EuDML | Zbl

[12] F. Knop, Über die Glattheit von Quotientenabbildungen, Manuscripta Math., 56 (1986), 419-427. | EuDML | MR | Zbl

[13] F. Knop, Der kanonische Modul eines Invariantenrings, J. Algebra, 127 (1989), 40-54. | MR | Zbl

[14] F. Knop, Weylgruppe und Momentabbildung, Invent. Math., 99 (1990), 1-23. | EuDML | MR | Zbl

[15] F. Knop, Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins, Math. Z., 213 (1993), 33-36. | EuDML | MR | Zbl

[16] M. Krämer, Sphärische Untergruppen in kompakten zusammenhängender Lie Gruppen, Compositio Math., 38 (1979), 129-153. | EuDML | Numdam | Zbl

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

[18] N. Mohan Kumar, Complete intersections, J. Math. Kyoto Univ., 17 (1977), 533-538. | MR | Zbl

[19] H. Nakajima, Representations of a reductive algebraic group whose algebras of invariants are complete intersections, J. reine angew. Math., 367 (1986), 115-138. | EuDML | MR | Zbl

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

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

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

[23] D. Panyushev, Reductive group actions on affine varieties and their doubling, Ann. Inst. Fourier, 45-4 (1995), 929-950. | EuDML | Numdam | MR | Zbl

[24] D. Panyushev, Good properties of algebras of invariants and defect of linear representations, Journal of Lie Theory, 5 (1995), 81-99. | EuDML | MR | Zbl

[25] F. Pauer, Sur les espaces homogènes de complication nulle, Bull. Soc. Math. France, 112 (1984), 377-385. | EuDML | Numdam | MR | Zbl

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

[27] R. Stanley, Hilbert functions of graded algebras, Adv. Math., 28 (1978), 57-83. | MR | Zbl

[28] E.B. Vinberg, Complexity of actions of reductive groups, Funct. Anal. Appl., 20 (1986), 1-11. | MR | Zbl

[29] E.B. Vinberg, B.N. Kimel'Fel'D, Homogeneous domains on flag varieties and spherical subgroups of semisimple Lie groups, Funct. Anal. Appl., 12 (1978), 168-174. | MR | Zbl

[30] A.L. Onishchik, E.B. Vinberg, Lie groups and algebraic groups, Berlin Heidelberg New York, Springer, 1990. | MR | Zbl

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

[32] V.L. Popov, E.B. Vinberg, Invariant theory, In: Parshin, A.N., Shafarevich, I.R. (Eds.): Algebraic Geometry IV (Encyclopaedia Math. Sci., vol. 55, pp. 123-284) Berlin Heidelberg New York: Springer, 1994. | Zbl

[33] Th. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France, 102 (1974), 317-333. | EuDML | Numdam | MR | Zbl

Cited by Sources: