Parity sheaves, moment graphs and the $p$-smooth locus of Schubert varieties

Fiebig, Peter; Williamson, Geordie

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 489-536.

On montre que l’algorithme de Braden-MacPherson calcule les fibres des faisceaux de parité. On en déduit que l’algorithme de Braden-MacPherson peut être utilisé pour calculer les caractères des modules basculants pour les groupes algébriques. Finalement, on montre que le lieu $p$-lisse d’une variété de Schubert coïncide avec son lieu rationnellement lisse, si le graphe de Bruhat sous-jacent satisfait une condition dite GKM.

We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the $p$-smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.

Classification : 20C20, 22E47, 55N33, 55N91, 14M15
Keywords: Modular representation theory, equivariant cohomology, moment graphs, constructible sheaves, tilting modules, Schubert varieties, $p$-smooth locus
Mot clés : théorie des représentations modulaires, cohomologie équivariante, graphes de moment, faisceaux constructibles, modules basculants, variétés de Schubert, lieu $p$-lisse

[1] Arabia, Alberto Classes d’Euler équivariantes et points rationnellement lisses, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 3, pp. 861-912 | DOI | Numdam | MR | Zbl

[2] Barthel, G. Séminaire Heidelberg-Strasbourg, 1966/1967 : Dualité de Poincaré, Publication I.R.M.A. Strasbourg, No. 3, Institut de Recherche Mathématique Avancée, Laboratoire Associé au C.N.R.S., Université de Strasbourg, Strasbourg, 1969, pp. ii+219 pp. (not consecutively paged)

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

[4] Beĭlinson, Alexandre; Bernstein, Joseph Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math., Volume 292 (1981) no. 1, pp. 15-18 | MR | Zbl

[5] Bernstein, Joseph; Lunts, Valery Equivariant sheaves and functors, Lecture Notes in Mathematics, 1578, Springer-Verlag, Berlin, 1994, pp. iv+139 | MR | Zbl

[6] Braden, Tom Hyperbolic localization of intersection cohomology, Transform. Groups, Volume 8 (2003) no. 3, pp. 209-216 | DOI | MR | Zbl

[7] Braden, Tom; MacPherson, Robert From moment graphs to intersection cohomology, Math. Ann., Volume 321 (2001) no. 3, pp. 533-551 | DOI | MR | Zbl

[8] Brion, Michel Equivariant cohomology and equivariant intersection theory, 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. 1-37 (Notes by Alvaro Rittatore) | MR | Zbl

[9] Carrell, J. B.; Goresky, R. M. A decomposition theorem for the integral homology of a variety, Invent. Math., Volume 73 (1983) no. 3, pp. 367-381 | DOI | MR | Zbl

[10] Carrell, James B. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) (Proc. Sympos. Pure Math.), Volume 56, Amer. Math. Soc., Providence, RI, 1994, pp. 53-61 | MR | Zbl

[11] Chang, Theodore; Skjelbred, Tor The topological Schur lemma and related results, Ann. of Math. (2), Volume 100 (1974), pp. 307-321 | DOI | MR | Zbl

[12] Dugundji, James Topology, Allyn and Bacon Inc., Boston, Mass., 1966, pp. xvi+447 | MR | Zbl

[13] Dyer, M. J. The nil Hecke ring and Deodhar’s conjecture on Bruhat intervals, Invent. Math., Volume 111 (1993) no. 3, pp. 571-574 | DOI | EuDML | MR | Zbl

[14] Dyer, M. J. Rank two detection of singularities of Schubert varieties (2005) (preprint http://www.nd.edu/~dyer/papers, arXiv:0906.2994)

[15] Fiebig, Peter Lusztig’s conjecture as a moment graph problem, Bull. Lond. Math. Soc., Volume 42 (2010) no. 6, pp. 957-972 | DOI | MR | Zbl

[16] Fiebig, Peter The multiplicity one case of Lusztig’s conjecture, Duke Math. J., Volume 153 (2010) no. 3, pp. 551-571 | DOI | MR | Zbl

[17] Fiebig, Peter Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture, J. Amer. Math. Soc., Volume 24 (2011) no. 1, pp. 133-181 | DOI | MR | Zbl

[18] Fulton, William Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, pp. x+260 (With applications to representation theory and geometry) | MR | Zbl

[19] Goresky, Mark; Kottwitz, Robert; MacPherson, Robert Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math., Volume 131 (1998) no. 1, pp. 25-83 | DOI | MR | Zbl

[20] Jantzen, Jens Carsten Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003, pp. xiv+576 | MR | Zbl

[21] Jantzen, Jens Carsten Moment graphs and representations, Séminaires et Congrès, Volume 24 (2010), pp. 227-318

[22] Juteau, D.; Mautner, C.; Williamson, G. Parity sheaves and tilting modules (In preparation, see http://arxiv.org/abs/0906.2994v1) | MR

[23] Juteau, D.; Mautner, C.; Williamson, G. Parity sheaves (2009) (preprint http://arxiv.org/abs/0906.2994) | MR

[24] Juteau, D.; Mautner, C.; Williamson, G. Perverse sheaves and modular representation theory, Séminaires et Congrès, Volume 24 (2012) no. II, pp. 313-350 | MR | Zbl

[25] Juteau, D.; Williamson, G. Kumar’s criterion modulo $p$ (2011) (preprint http://arxiv.org/abs/1201.5341) | MR

[26] Juteau, Daniel Decomposition numbers for perverse sheaves, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 3, pp. 1177-1229 | DOI | EuDML | Numdam | MR | Zbl

[27] Kashiwara, Masaki; Schapira, Pierre Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 292, Springer-Verlag, Berlin, 1994, pp. x+512 (With a chapter in French by Christian Houzel, Corrected reprint of the 1990 original) | MR | Zbl

[28] Knop, Friedrich; Kraft, Hanspeter; Luna, Domingo; Vust, Thierry Local properties of algebraic group actions, Algebraische Transformationsgruppen und Invariantentheorie (DMV Sem.), Volume 13, Birkhäuser, Basel, 1989, pp. 63-75 | MR | Zbl

[29] Kumar, Shrawan The nil Hecke ring and singularity of Schubert varieties, Invent. Math., Volume 123 (1996) no. 3, pp. 471-506 | DOI | EuDML | MR | Zbl

[30] Kumar, Shrawan Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser Boston Inc., Boston, MA, 2002, pp. xvi+606 | MR | Zbl

[31] Lusztig, George Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) (Astérisque), Volume 101, Soc. Math. France, Paris, 1983, pp. 208-229 | Numdam | MR | Zbl

[32] Milnor, John W.; Stasheff, James D. Characteristic classes, Princeton University Press, Princeton, N. J., 1974, pp. vii+331 (Annals of Mathematics Studies, No. 76) | MR | Zbl

[33] 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

[34] Soergel, Wolfgang On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra, Volume 152 (2000) no. 1-3, pp. 311-335 Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998) | DOI | MR | Zbl

[35] Spaltenstein, N. Resolutions of unbounded complexes, Compositio Math., Volume 65 (1988) no. 2, pp. 121-154 | EuDML | Numdam | MR | Zbl

[36] Springer, T. A. Quelques applications de la cohomologie d’intersection, Bourbaki Seminar, Vol. 1981/1982 (Astérisque), Volume 92, Soc. Math. France, Paris, 1982, pp. 249-273 | EuDML | Numdam | MR | Zbl

[37] Springer, T. A. A purity result for fixed point varieties in flag manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 31 (1984) no. 2, pp. 271-282 | MR | Zbl

[38] Sumihiro, Hideyasu Equivariant completion, J. Math. Kyoto Univ., Volume 14 (1974), pp. 1-28 | MR | Zbl