Abstract
Let G be a reductive algebraic group over a field k and let B be a Borel subgroup in G. We demonstrate how a number of results on the cohomology of line bundles on the ag manifold G/B have had interesting consequences in the representation theory for G, and vice versa. Our focus is on the case where the characteristic of k is positive. In this case, both the vanishing behavior of the cohomology modules for a line bundle on G/B and the G-structures of the nonzero cohomology modules are still very much open problems. We give an account of the developments over the years, trying to illustrate what is now known and what is still not known today.
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Achar, P., Makisumi, S., Riche, S., Williamson, G.: Koszul duality for Kac-Moody groups and characters of tilting modules. J. Amer. Math. Soc. 32, 261–310 (2019)
H. H. Andersen, Cohomology of line bundles on G/B, Ann. Scient. Éc. Norm. Sup. (4) 12 (1979), 82-100.
Andersen, H.H.: The first cohomology group of a line bundle on G/B. Invent. Math. 5, 287–296 (1979)
Andersen, H.H.: The strong linkage principle. J. reine angew. Math. 315, 53–59 (1980)
H. H. Andersen, A G-equivariant proof of the vanishing theorem for dominant line bundles on G/B, preprint Inst. Adv. Study (1979), pp.1-13.
Andersen, H.H.: The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B. Ann. of Math. 112, 113–121 (1980)
Andersen, H.H.: Vanishing theorems and induced representations. J. Alg. 62, 86–100 (1980)
Andersen, H.H.: On the structure of the cohomology of line bundles on G/B. J. Alg. 71, 242–258 (1981)
H. H. Andersen, Filtrations of cohomology modules for Chevalley groups, Ann. Scient. Éc. Norm. Sup. (4) 16 (1983), 495-528.
Andersen, H.H.: Schubert varieties and Demazure’s character formula. Invent. Math. 79, 611–618 (1985)
Andersen, H.H.: Torsion in the cohomology of line bundles on homogeneous spaces for Chevalley groups. Proc. A.M.S. 96, 537–544 (1986)
Andersen, H.H.: On the generic structure of cohomology modules for semisimple algebraic groups. Trans. Amer. Math. Soc. 295, 397–415 (1986)
Andersen, H.H.: The strong linkage principle for quantum groups at roots of 1. J. Alg. 260, 2–15 (2003)
H. H. Andersen, Cohomology of line bundles, in: Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math., Vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 13-36.
Andersen, H.H., Kaneda, M.: Cohomology of line bundles on the flag variety for type G2. Journal of Pure and Applied Algebra. 216(7), 1566–1579 (2012)
H. H. Andersen, U. Kulkarni) Sum formulas for reductive algebraic groups, Adv. in Math. 217, (2008), 419-447.
Andersen, H.H., Wen, K.: Representations of quantum algebras. The mixed case. J. Reine Angew. Math. 427, 35–50 (1992)
Bott, R.: Homogeneous vector bundles. Ann. of Math. 66, 203–248 (1957)
N. Bourbaki, Algèbre, Chapitre X, Masson 1980.
M. Brion, S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, Vol. 231, Birkhäuser Boston, Boston, MA, 2005.
Carter, R., Lusztig, G.: On the modular representation theory of the general linear and symmetric groups. Math. Z. 136, 193–242 (1974)
C. Chevalley, Théorie des Groupes de Lie, tome II, Groupes Algbriques, Actualités Sci. Ind. no. 1152, Hermann & Cie., Paris, 1951.
E. Cline, B. Parshall, L. Scott, On injective modules for infinitesimal algebraic groups, J. London Math. Soc. (2) 31 (1985), 277-291.
Curtis, C.: Representations of Lie algebras of classical types with applications to linear groups. J. Math. Mech. 9, 307–326 (1960)
Demazure, M.: Une démonstration algébrique d’un théorème de Bott. Invent. Math. 5, 349–356 (1968)
Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. 7, 53–88 (1974)
Demazure, M.: A very simple proof of Bott’s theorem. Invent. Math. 33, 271–272 (1976)
Donkin, S.: The cohomology of line bundles on the three-dimensional flag variety. J. Alg. 307, 570–613 (2007)
Doty, S.: The strong linkage principle. Amer. J. Math. 111, 135–141 (1989)
Doty, S.R., Sullivan, J.B.: On the structure of higher cohomology modules of line bundles on G/B. J. Alg. 114, 286–332 (1988)
Griffith, W.L.: Cohomology of flag varieties in characteristic p. Illinois J. Math. 24, 452–461 (1980)
Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohuku Math. J. 9, 119–221 (1957)
Haboush, W.: A short proof of the Kempf vanishing theorem. Invent. Math. 56, 109–112 (1980)
Humphreys, J.E.: Modular representations of classical Lie algebras and semisimple groups. J. Alg. 19, 51–79 (1971)
J. E. Humphreys, Ordinary and Modular Representations of Chevalley Groups, Lect. Notes in Mathematics, Vol. 528, Springer-Verlag, Berlin, 1976.
J. E. Humphreys, Weyl modules and Bott's theorem in characteristic p, in: Lie Theories and their Applications, Queen’s Papers in Pure & Appl. Math., Vol. 48, Kingston, Ont., 1978, pp. 474-483.
Humphreys, J.E.: Cohomology of G/B in characteristic p. Adv. in Math. 59, 170–183 (1986)
Humphreys, J.E.: Cohomology of line bundles on G/B for the exceptional group G2. J. Pure Appl. Math. 44, 227–239 (1987)
J. E. Humphreys, Cohomology of line bundles on flag varieties in prime characteristic, in: Proc. Hyderabad Conference on Algebraic Groups, Manoj Prakashan, Madras, 1991, pp. 193-204.
J. E. Humphreys, Cohomology of line bundles on flag varieties, Note posted on homepage, May 2018.
Jantzen, J.C.: Sur Characterformel gewisser Darstellungen halbeinfacher Gruppen und Lie Algebren. Math. Z. 140, 127–149 (1974)
Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und kontravariante Formen. J. reine angew. Math. 290, 117–141 (1977)
J. C. Jantzen, Representations of Algebraic Groups, 2nd edition, Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.
Kac, V., Weisfeiler, B.: Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p. Indag. Math. 38, 136–151 (1976)
Kempf, G.: Vanishing theorem for flag manifolds. Amer. J. Math. 98, 325–331 (1976)
Kempf, G.: Linear systems on homogeneous spaces. Ann. of Math. 103, 557–591 (1976)
Lakshmibai, V., Musili, C., Seshadri, C.S.: Cohomology of line bundles on G/B. Ann. Sci. École Norm. Sup. 7, 88–132 (1974)
Lin, Z.: The structure of cohomology of line bundles on G/B for semisimple algebraic groups. J. Alg. bf. 134, 225–256 (1990)
Lin, Z.: Socle series of cohomology groups of line bundles on G/B. J. Pure Appl. Algebra. 72, 275–294 (1991)
Liu, L.: On the cohomology of line bundles over certain flag schemes. J. Combinatorial Theory, Ser. A. 182, 105448 (2021)
Liu, L., Polo, P.: On the cohomology of line bundles over certain flag schemes II. J. Combinatorial Theory, Ser. A. 178, 105352 (2021)
Mehta, V., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. of Math. 122, 27–45 (1985)
S. Riche, G. Williamson, A simple character formula, Annales Henri Lebesgue, UFR de Mathmatiques-IRMAR (2020), 503-535.
J.-P. Serre, Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts, Séminaire N. Bourbaki, 1954, exp. no. 100, 447-454.
C. S. Seshadri, Line bundles on Schubert varieties, in: Vector Bundles on Algebraic Varieties (Bombay, 1984), Tata Inst. Fund. Res., Bombay, 1987, pp. 499-528.
Sobaje, P.: On character formulas for simple and tilting modules. Adv. in Math. 369, 107172 (2020)
Steinberg, R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1963)
D.-N. Verma, Rôle of affine Weyl groups in representation theory of algebraic Chevalley groups and their Lie algebras, in: Lie Groups and Their Lie Algebras, Proc. Summer School, Bolyai János Math. Soc., Budapest 1971, Halsted, New York, 1975, pp. 653-705.
Wong, W.: Very strong linkage principle for cohomology of line bundles on G/B. J. Alg. 113, 71–80 (1988)
Wong, W.: Weyl modules for p-singular weights. J. Alg. 114, 357–368 (1988)
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Dedicated to the memory of Jim Humphreys
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ANDERSEN, H.H. REPRESENTATION THEORY VIA COHOMOLOGY OF LINE BUNDLES. Transformation Groups 28, 1033–1058 (2023). https://doi.org/10.1007/s00031-022-09769-x
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DOI: https://doi.org/10.1007/s00031-022-09769-x