Equivariant degenerations of spherical modules for groups of type $\mathsf {A}$

Papadakis, Stavros Argyrios; Van Steirteghem, Bart

doi:10.5802/aif.2735

Equivariant degenerations of spherical modules for groups of type

A

[Les dégénérescences équivariantes des modules sphériques de type

A

]

Papadakis, Stavros Argyrios ¹ ; Van Steirteghem, Bart ²

¹ Universidade Técnica de Lisboa Centro de Análise Matemática Geometria e Sistemas Dinâmicos Departamento de Matemática Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa (Portugal)
² Medgar Evers College Department of Mathematics City University of New York 1650 Bedford Ave. Brooklyn, NY 11225 (USA)

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1765-1809.

Résumé
Abstract

V. Alexeev et M. Brion ont introduit, pour un groupe complexe réductif donné, un schéma de modules de variétés sphériques affines ayant le même semi-groupe moment. Nous donnons de nouveaux exemples de ce schéma de modules en montrant qu’il est un espace affine lorsque le groupe donné est de type $A$ et le semi-groupe moment fixé est celui d’un module sphérique.

V. Alexeev and M. Brion introduced, for a given a complex reductive group, a moduli scheme of affine spherical varieties with prescribed weight monoid. We provide new examples of this moduli scheme by proving that it is an affine space when the given group is of type $A$ and the prescribed weight monoid is that of a spherical module.

EuDML MR Zbl

DOI : 10.5802/aif.2735

Classification : 14D22, 14C05, 14M27, 20G05
Keywords: Invariant Hilbert scheme, spherical module, spherical variety, equivariant degeneration
Mot clés : schéma de Hilbert invariant, module sphérique, variété sphérique, dégénérescence équivariante

Affiliations des auteurs :

Papadakis, Stavros Argyrios ¹ ; Van Steirteghem, Bart ²

@article{AIF_2012__62_5_1765_0,
     author = {Papadakis, Stavros Argyrios and Van Steirteghem, Bart},
     title = {Equivariant degenerations of spherical modules for groups of type $\mathsf {A}$},
     journal = {Annales de l'Institut Fourier},
     pages = {1765--1809},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {5},
     year = {2012},
     doi = {10.5802/aif.2735},
     zbl = {1267.14018},
     mrnumber = {3025153},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2735/}
}

TY  - JOUR
AU  - Papadakis, Stavros Argyrios
AU  - Van Steirteghem, Bart
TI  - Equivariant degenerations of spherical modules for groups of type $\mathsf {A}$
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 1765
EP  - 1809
VL  - 62
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2735/
DO  - 10.5802/aif.2735
LA  - en
ID  - AIF_2012__62_5_1765_0
ER  -

%0 Journal Article
%A Papadakis, Stavros Argyrios
%A Van Steirteghem, Bart
%T Equivariant degenerations of spherical modules for groups of type $\mathsf {A}$
%J Annales de l'Institut Fourier
%D 2012
%P 1765-1809
%V 62
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2735/
%R 10.5802/aif.2735
%G en
%F AIF_2012__62_5_1765_0

Papadakis, Stavros Argyrios; Van Steirteghem, Bart. Equivariant degenerations of spherical modules for groups of type $\mathsf {A}$. Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1765-1809. doi : 10.5802/aif.2735. http://archive.numdam.org/articles/10.5802/aif.2735/

Bibliographie
Cité par

[1] Alexeev, Valery; Brion, Michel Moduli of affine schemes with reductive group action, J. Algebraic Geom., Volume 14 (2005) no. 1, pp. 83-117 | DOI | MR | Zbl

[2] Benson, Chal; Ratcliff, Gail A classification of multiplicity free actions, J. Algebra, Volume 181 (1996) no. 1, pp. 152-186 | DOI | MR | Zbl

[3] Bravi, P.; Cupit-Foutou, S. Equivariant deformations of the affine multicone over a flag variety, Adv. Math., Volume 217 (2008) no. 6, pp. 2800-2821 | DOI | MR | Zbl

[4] Bravi, Paolo Classification of spherical varieties, Les cours du CIRM, Volume 1 (2010) no. 1, pp. 99-111 | DOI | EuDML | Numdam

[5] Bravi, Paolo; Luna, Domingo An introduction to wonderful varieties with many examples of type $F_{4}$ , J. Algebra, Volume 329 (2011) no. 1, pp. 4-51 | DOI | MR | Zbl

[6] Brion, Michel Variétés sphériques, notes de la session de la Société Mathématique de France “Opérations hamiltoniennes et opérations de groupes algébriques,” Grenoble, http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html, 1997

[7] Brion, Michel Introduction to actions of algebraic groups, Les cours du CIRM, Volume 1 (2010) no. 1, pp. 1-22 | DOI | EuDML | Numdam | MR

[8] Brion, Michel Invariant Hilbert schemes, arXiv:1102.0198v2 [math.AG], 2011

[9] Camus, Romain Variétés sphériques affines lisses, Institut Fourier, Grenoble (2001) (Ph. D. Thesis)

[10] Cupit-Foutou, S. Invariant Hilbert schemes and wonderful varieties, arXiv: 0811.1567v2 [math.AG], 2009

[11] Cupit-Foutou, S. Wonderful varieties: a geometrical realization, arXiv:0907.2852v3 [math.AG], 2010

[12] Delzant, Thomas Classification des actions hamiltoniennes complètement intégrables de rang deux, Ann. Global Anal. Geom., Volume 8 (1990) no. 1, pp. 87-112 | DOI | MR | Zbl

[13] Grayson, Daniel R.; Stillman, Michael E. Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/

[14] Howe, Roger; Umeda, Tōru The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann., Volume 290 (1991) no. 3, pp. 565-619 | DOI | MR | Zbl

[15] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York, 1975 (Graduate Texts in Mathematics, No. 21) | MR | Zbl

[16] Jansou, Sébastien Déformations des cônes de vecteurs primitifs, Math. Ann., Volume 338 (2007) no. 3, pp. 627-667 | DOI | MR | Zbl

[17] Kac, V. G. Some remarks on nilpotent orbits, J. Algebra, Volume 64 (1980) no. 1, pp. 190-213 | DOI | MR | Zbl

[18] Knop, Friedrich Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc., Volume 9 (1996) no. 1, pp. 153-174 | DOI | MR | Zbl

[19] Knop, Friedrich Some remarks on multiplicity free spaces, Representation theories and algebraic geometry (Montreal, PQ, 1997) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 301-317 | MR | Zbl

[20] Knop, Friedrich Automorphisms of multiplicity free Hamiltonian manifolds, J. Amer. Math. Soc., Volume 24 (2011) no. 2, pp. 567-601 | DOI | MR | Zbl

[21] Leahy, Andrew S. A classification of multiplicity free representations, J. Lie Theory, Volume 8 (1998) no. 2, pp. 367-391 | MR | Zbl

[22] Losev, Ivan V. Proof of the Knop conjecture, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 3, pp. 1105-1134 | DOI | Numdam | MR | Zbl

[23] Losev, Ivan V. Uniqueness property for spherical homogeneous spaces, Duke Math. J., Volume 147 (2009) no. 2, pp. 315-343 | DOI | MR | Zbl

[24] Northcott, D. G. Syzygies and specializations, Proc. London Math. Soc. (3), Volume 15 (1965), pp. 1-25 | DOI | MR | Zbl

[25] Panyushev, Dmitri On deformation method in invariant theory, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 4, pp. 985-1012 | DOI | Numdam | MR | Zbl

[26] Papadakis, Stavros; Van Steirteghem, Bart Equivariant degenerations of spherical modules for groups of type $A$ , arXiv:1008.0911v3 [math.AG], 2011

[27] Pezzini, Guido Lectures on spherical and wonderful varieties, Les cours du CIRM, Volume 1 (2010) no. 1, pp. 33-53 | DOI | Numdam

[28] Popov, V. L. Contractions of actions of reductive algebraic groups, Mat. Sb. (N.S.), Volume 130(172) (1986) no. 3, p. 310-334, 431 English translation in Math. USSR-Sb. 58 (1987), no. 2, 311–335 | MR | Zbl

[29] Vinberg, È. B.; Popov, V. L. A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat., Volume 36 (1972), pp. 749-764 English translation in Math. USSR Izv. 6 (1972), 743–758 | MR | Zbl

Cité par Sources :