×

Decomposition numbers for perverse sheaves. (English) Zbl 1187.14022

The author presents the problem for the decomposition numbers for perverse sheaves, obtains some methods for the their computation in some simple cases and computes them explicitly for a simple surface singularity and for the closure of the minimal non-trivial nilpotent orbit in a simple Lie algebra. After recalling the definition of perverse sheaves over \(\mathbb{K}, \mathbb{Q}, \mathbb{F}\) he considers a \(t\)-category having the heart endowed with a torsion theory. He studies the interaction between torsion theories and \(t\)-structures and obtains the failure of commutativity between truncations and modular reduction. The decomposition numbers are obtained in the setting of recollements, they are defined for perverse shaves then it is studied the equivariance. Then the author presents some techniques for computation of the decomposition numbers. In characteristic zero, there is obtained some information from the study of semi-small and small proper separable morphisms. The results are important for the modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra and for reductive algebraic groups schemes using the affine Grassmannian of the Langlands dual group.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
55N33 Intersection homology and cohomology in algebraic topology
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
20C20 Modular representations and characters
PDFBibTeX XMLCite
Full Text: DOI arXiv Numdam EuDML

References:

[1] Beĭlinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre, Analysis and topology on singular spaces, I (Luminy, 1981), 100, 5-171 (1982)
[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., 292, 15, 707-710 (1981) · Zbl 0467.20036
[3] Borho, Walter; MacPherson, Robert, Analysis and topology on singular spaces, II, III (Luminy, 1981), 101, 23-74 (1983)
[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 (1968) · Zbl 0186.33001
[5] Brieskorn, Egbert, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, 279-284 (1971) · Zbl 0223.22012
[6] Deligne, Pierre, La conjecture de Weil II, Publ. Math. IHES, 52, 137-252 (1980) · Zbl 0456.14014
[7] Goresky, Mark; MacPherson, Robert, Intersection homology theory, Topology, 19, 2, 135-162 (1980) · Zbl 0448.55004 · doi:10.1016/0040-9383(80)90003-8
[8] Goresky, Mark; MacPherson, Robert, Intersection homology. II, Invent. Math., 72, 1, 77-129 (1983) · Zbl 0529.55007 · doi:10.1007/BF01389130
[9] Happel, Dieter; Reiten, Idun; Smalø, Sverre O., Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc., 120, 575 (1996) · Zbl 0849.16011
[10] Ito, Y.; Nakamura, I., New trends in algebraic geometry (Warwick, 1996), 264, 151-233 (1999) · Zbl 0954.14001
[11] Juteau, Daniel, Modular Springer correspondence and decomposition matrices · Zbl 1187.14022
[12] Juteau, Daniel, Modular Springer correspondence and decomposition matrices (2007)
[13] Juteau, Daniel, Cohomology of the minimal nilpotent orbit, Transformation Groups, 13, 2, 355-387 (2008) · Zbl 1152.22007 · doi:10.1007/s00031-008-9009-x
[14] Kashiwara, Masaki; Schapira, Pierre, Categories and sheaves, 332 (2006) · Zbl 1118.18001
[15] Kazhdan, David; Lusztig, George, Schubert varieties and Poincaré duality, Proc. Symposia in Pure Math., 36, 185-203 (1980) · Zbl 0461.14015
[16] Letellier, Emmanuel, Fourier transforms of invariant functions on finite reductive Lie algebras, 1859 (2005) · Zbl 1076.43001
[17] Lusztig, George, Intersection cohomology complexes on a reductive group, Invent. Math., 75, 205-272 (1984) · Zbl 0547.20032 · doi:10.1007/BF01388564
[18] Malkin, Anton; Ostrik, Viktor; Vybornov, Maxim, The minimal degeneration singularities in the affine Grassmannians, Duke Math. J., 126, 2, 233-249 (2005) · Zbl 1078.14016 · doi:10.1215/S0012-7094-04-12622-3
[19] Mirković, I.; Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2), 166, 1, 95-143 (2007) · Zbl 1138.22013 · doi:10.4007/annals.2007.166.95
[20] Slodowy, Peter, Four lectures on simple groups and singularities, 11 (1980) · Zbl 0425.22020
[21] Slodowy, Peter, Simple singularities and simple algebraic groups, 815 (1980) · Zbl 0441.14002
[22] Springer, Tonny A., Linear algebraic groups, 9 (1998) · Zbl 0927.20024
[23] Wang, Weiqiang, Dimension of a minimal nilpotent orbit, Proc. Amer. Math. Soc., 127, 3, 935-936 (1999) · Zbl 0909.22009 · doi:10.1090/S0002-9939-99-04946-1
[24] Weibel, Charles A., An introduction to homological algebra, 38 (1994) · Zbl 0797.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.