Decomposition numbers for perverse sheaves

Juteau, Daniel

Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1177-1229.

Le but de cet article est de poser les fondations pour les nombres de décomposition des faisceaux pervers, de donner quelques méthodes pour les calculer dans des cas simples et de les déterminer explicitement dans deux situations : pour une singularité simple (kleinienne) de surface et pour l’adhérence de l’orbite nilpotente non-triviale minimale dans une algèbre de Lie simple.

Ce travail a des applications dans la théorie des représentations modulaires, pour les groupes de Weyl en utilisant le cône nilpotent de l’algèbre de Lie semi-simple correspondante, et pour les schémas en groupes réductifs en utilisant la grassmannienne affine du dual de Langlands.

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial nilpotent orbit in a simple Lie algebra.

This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group.

Classification : 55N33, 20C20
Keywords: Perverse sheaves, intersection cohomology, integral cohomology, t-structures, torsion theories, decomposition matrices, simple singularities, minimal nilpotent orbits
Mot clés : faisceaux pervers, cohomologie d’intersection, cohomologie entière, t-structures, théories de torsion, matrices de décomposition, singularités simples, orbites nilpotentes minimales

[1] Beĭlinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Soc. Math. France, Paris, 1982, pp. 5-171 | MR

[2] Borho, Walter; MacPherson, Robert Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math., Volume 292 (1981) no. 15, pp. 707-710 | MR | Zbl

[3] Borho, Walter; MacPherson, Robert Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981) (Astérisque), Volume 101, Soc. Math. France, Paris, 1983, pp. 23-74 | Numdam | MR

[4] Bourbaki, Nicolas Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 | MR | Zbl

[5] Brieskorn, Egbert Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 279-284 | MR | Zbl

[6] Deligne, Pierre La conjecture de Weil II, Publ. Math. IHES, Volume 52 (1980), pp. 137-252 | Numdam | MR | Zbl

[7] Goresky, Mark; MacPherson, Robert Intersection homology theory, Topology, Volume 19 (1980) no. 2, pp. 135-162 | DOI | MR | Zbl

[8] Goresky, Mark; MacPherson, Robert Intersection homology. II, Invent. Math., Volume 72 (1983) no. 1, pp. 77-129 | DOI | MR | Zbl

[9] Happel, Dieter; Reiten, Idun; Smalø, Sverre O. Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc., Volume 120 (1996) no. 575, pp. viii+ 88 | MR | Zbl

[10] Ito, Y.; Nakamura, I. Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996) (London Math. Soc. Lecture Note Ser.), Volume 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233 | MR | Zbl

[11] Juteau, Daniel Modular Springer correspondence and decomposition matrices (in preparation)

[12] Juteau, Daniel Modular Springer correspondence and decomposition matrices, Université Paris 7 Denis Diderot, arXiv:0901.3671 (2007) (Ph. D. Thesis arXiv:0901.3671)

[13] Juteau, Daniel Cohomology of the minimal nilpotent orbit, Transformation Groups, Volume 13 (2008) no. 2, pp. 355-387 | DOI | MR | Zbl

[14] Kashiwara, Masaki; Schapira, Pierre Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 332, Springer-Verlag, Berlin, 2006 | MR | Zbl

[15] Kazhdan, David; Lusztig, George Schubert varieties and Poincaré duality, Proc. Symposia in Pure Math., Volume 36 (1980), pp. 185-203 | MR | Zbl

[16] Letellier, Emmanuel Fourier transforms of invariant functions on finite reductive Lie algebras, Lecture Notes in Mathematics, 1859, Springer, Berlin, 2005 | MR | Zbl

[17] Lusztig, George Intersection cohomology complexes on a reductive group, Invent. Math., Volume 75 (1984), pp. 205-272 | DOI | MR | Zbl

[18] Malkin, Anton; Ostrik, Viktor; Vybornov, Maxim The minimal degeneration singularities in the affine Grassmannians, Duke Math. J., Volume 126 (2005) no. 2, pp. 233-249 | DOI | MR | Zbl

[19] Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2), Volume 166 (2007) no. 1, pp. 95-143 | DOI | MR | Zbl

[20] Slodowy, Peter Four lectures on simple groups and singularities, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, 11, Rijksuniversiteit Utrecht Mathematical Institute, Utrecht, 1980 | MR | Zbl

[21] Slodowy, Peter Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, 815, Springer, Berlin, 1980 | MR | Zbl

[22] Springer, Tonny A. Linear algebraic groups, Progress in Mathematics, 9, Birkhäuser Boston Inc., Boston, MA, 1998 | MR | Zbl

[23] Wang, Weiqiang Dimension of a minimal nilpotent orbit, Proc. Amer. Math. Soc., Volume 127 (1999) no. 3, pp. 935-936 | DOI | MR | Zbl

[24] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994 | MR | Zbl