Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T06:06:27.408Z Has data issue: false hasContentIssue false
  • English
  • Français

LECTURES ON THE GEOMETRY AND MODULAR REPRESENTATION THEORY OF ALGEBRAIC GROUPS

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  13 January 2021

JOSHUA CIAPPARA  and
GEORDIE WILLIAMSON
JOSHUA CIAPPARA*
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW2006, Australia
GEORDIE WILLIAMSON
Affiliation:
Sydney Mathematical Research Institute, The University of Sydney, NSW2006, Australia e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

Keywords

algebraic groupsmodular representation theory

MSC classification

Primary: 20G05: Representation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 1 , February 2021 , pp. 1 - 47
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Ben Martin

The first author was supported by an Australian government research training program (RTP) scholarship.

References

Achar, P. and Riche, S., ‘Reductive groups, the loop Grassmannian, and the Springer resolution’, Invent. Math. 214 (2018), 289–436.CrossRefGoogle Scholar
Andersen, H. H., ‘The Frobenius morphism on the cohomology of homogeneous vector bundles on $G/B$ ’, Ann. of Math. (2) 112(1) (1980), 113–121.CrossRefGoogle Scholar
Andersen, H. H., ‘The strong linkage principle’, J. reine angew. Math. 315 (1980), 53–59.Google Scholar
Andersen, H. H., Jantzen, J. C. and Soergel, W., Representations of Quantum Groups at a $p$ -th Root of Unity and of Semisimple Groups in Characteristic $p$ : Independence of $p$ , Astérisque, 220 (Société Mathématique de France, Paris, 1994).Google Scholar
Arkhipov, S., Bezrukavnikov, R. and Ginzburg, V., ‘Quantum groups, the loop Grassmannian, and the Springer resolution’, J. Amer. Math. Soc. 17 (2004), 595–678.CrossRefGoogle Scholar
Baumann, P. and Riche, S., ‘Notes on the geometric Satake equivalence’, in: Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms: CIRM Jean-Morlet Chair, Spring 2016 (Springer International, Switzerland, 2018), 1–134.Google Scholar
Beilinson, A. and Bernstein, J., ‘Localisation de $g$ -modules’, C. R. Math. Acad. Sci. Paris 292(1) (1981), 15–18.Google Scholar
Bernstein, J. and Lunts, V., Equivariant Sheaves and Functors, Lecture Notes in Mathematics, 1578 (Springer, Berlin, Heidelberg, 2006).Google Scholar
Bezrukavnikov, R., Riche, S. and Rider, L., ‘Modular affine Hecke category and regular unipotent centralizer, I’, Preprint, 2020.Google Scholar
Borel, A., Linear Algebraic Groups, Graduate Texts in Mathematics, 21 (Springer, New York, 2012).Google Scholar
Bourbaki, N., Lie Groups and Lie Algebras (Springer, Berlin, Heidelberg, 2002), Chs. 4--6.CrossRefGoogle Scholar
Brylinski, J. L. and Kashiwara, M., ‘Kazhdan–Lusztig conjecture and holonomic systems’, Invent. Math. 64 (1981), 387–410.CrossRefGoogle Scholar
Chriss, N. and Ginzburg, V., Representation Theory and Complex Geometry, Modern Birkhäuser Classics, 77 (Birkhäuser, Boston, MA, 2010).CrossRefGoogle Scholar
Curtis, C. W., ‘Representations of Lie algebras of classical type with application to linear groups’, J. Math. Mech. 9(2) (1960), 307–326.Google Scholar
Curtis, C. W., Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, 15 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
de Cataldo, M. A. A. and Migliorini, L., ‘The decomposition theorem, perverse sheaves and the topology of algebraic maps’, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 535–633.CrossRefGoogle Scholar
Deligne, P. and Milne, J. S., Tannakian Categories (Springer, Berlin, Heidelberg, 1982), 101–228.Google Scholar
Donkin, S., ‘The blocks of a semisimple algebraic group’, J. Algebra 67 (1980), 36–53.CrossRefGoogle Scholar
Elias, B., Makisumi, S., Thiel, U. and Williamson, G., An Introduction to Soergel Bimodules (Springer International, Switzerland, 2020).Google Scholar
Elias, B. and Williamson, G., ‘Soergel calculus’, Represent. Theory 20 (2016), 295–374.Google Scholar
Fiebig, P., ‘Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture’, J. Amer. Math. Soc. 24 (2011), 133–181.CrossRefGoogle Scholar
Fiebig, P., ‘An upper bound on the exceptional characteristics for Lusztig's character formula’, J. reine angew. Math. 673 (2012), 1–31.Google Scholar
Geck, M. and Malle, G., The Character Theory of Finite Groups of Lie Type: A Guided Tour (Cambridge University Press, Cambridge, 2020).CrossRefGoogle Scholar
Görtz, U., ‘Affine Springer fibers and affine Deligne–Lusztig varieties’, in: Affine Flag Manifolds and Principal Bundles (ed. Schmitt, A.) (Springer, Basel, 2010), 1–50.Google Scholar
Haboush, W. J., ‘A short proof of the Kempf vanishing theorem’, Invent. Math. 56(2) (1980), 109–112.Google Scholar
Hall, B. C.. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (Springer International, Switzerland, 2015).Google Scholar
Hartshorne, R.. Algebraic Geometry, Graduate Texts in Mathematics, 52 (Springer, New York, 1977).CrossRefGoogle Scholar
He, X. and Williamson, G., ‘Soergel calculus and Schubert calculus’, Bull. Inst. Math. Acad. Sin. (N.S.) 13 (2015), 317–350.Google Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T., D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, 236 (Birkhäuser, Boston, MA, 2007).Google Scholar
Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9 (Springer, New York, 1972).CrossRefGoogle Scholar
Humphreys, J. E., Linear Algebraic Groups, Graduate Texts in Mathematics, 21 (Springer, New York, 1975).Google Scholar
Humphreys, J. E.. Representations of Semisimple Lie Algebras in the BGG Category ${\mathcal{O}}$ (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
Iwahori, N. and Matsumoto, H., ‘On some Bruhat decomposition and the structure of the Hecke rings of $p$ -adic Chevalley groups’, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5–48.Google Scholar
James, G., ‘The decomposition matrices of $G{L}_n(q)$ for $n\le 10$ ’, Proc. Lond. Math. Soc. (3) 60(2) (1990), 225–265.CrossRefGoogle Scholar
Jantzen, J. C., Representations of Algebraic Groups, 2nd edn, Mathematical Surveys and Monographs, 107 (American Mathematical Society, Providence, RI, 2003).Google Scholar
Juteau, D., ‘Decomposition numbers for perverse sheaves’, Ann. Inst. Fourier (Grenoble) 59(3) (2009), 1177–1229.CrossRefGoogle Scholar
Juteau, D., Mautner, C., and Williamson, G., ‘Perverse sheaves and modular representation theory’, Sémin. Congr. 24(II) (2012), 313–350.Google Scholar
Juteau, D., Mautner, C. and Williamson, G., ‘Parity sheaves’, J. Amer. Math. Soc. 27(4) (2014), 1169–1212.CrossRefGoogle Scholar
Kashiwara, M. and Tanisaki, T., ‘Kazhdan–Lusztig conjecture for affine Lie algebras with negative level’, Duke Math. J. 77 (1995), 21–62.Google Scholar
Kashiwara, M. and Tanisaki, T., ‘Kazdhan–Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case’, Duke Math. J. 84 (1996), 771–813.CrossRefGoogle Scholar
Kazhdan, D. and Lusztig, G., ‘Schubert varieties and Poincaré duality’, in: Geometry of the Laplace Operator, Proceedings of Symposia in Pure Mathematics, 36 (American Mathematical Society, Providence, RI, 1980), 185–203.Google Scholar
Kazhdan, D. and Lusztig, G., ‘Tensor structures arising from affine Lie algebras I, II’, J. Amer. Math. Soc. 6 (1993), 905–947, 949–1011.Google Scholar
Kazhdan, D. and Lusztig, G., ‘Tensor structures arising from affine Lie algebras III’, J. Amer. Math. Soc. 7 (1994), 335–381.CrossRefGoogle Scholar
Kazhdan, D. and Lusztig, G., ‘Tensor structures arising from affine Lie algebras IV’, J. Amer. Math. Soc. 7 (1994), 383–453.CrossRefGoogle Scholar
Kempf, G. R., ‘Linear systems on homogeneous spaces’, Ann. of Math. (2) 103(3) (1976), 557–591.CrossRefGoogle Scholar
Kempf, G. R., ‘Representations of algebraic groups in prime characteristics’, Ann. Sci. Éc. Norm. Supér. (4) 14(1) (1981), 61–76.CrossRefGoogle Scholar
Kumar, S., Kac–Moody Groups, their Flag Varieties and Representation Theory, Progress in Mathematics, 204 (Birkhäuser, Basel, 2002).CrossRefGoogle Scholar
Lusztig, G., ‘Some problems in the representation theory of finite Chevalley groups’, in: The Santa Cruz Conference on Finite Groups (University of California, Santa Cruz, CA, 1979) (American Mathematical Society, Providence, RI, 1980), 313–317.Google Scholar
Lusztig, G., ‘Green polynomials and singularities of unipotent classes’, Adv. Math. 42 (1981), 169–178.CrossRefGoogle Scholar
Lusztig, G., ‘Monodromic systems on affine flag manifolds’, Proc. R. Soc. Lond. A 445(1923) (1994), 231–246.Google Scholar
Lusztig, G., ‘Errata: Monodromic systems on affine flag manifolds’, Proc. R. Soc. Lond. A 450(1940) (1995), 731–732.Google Scholar
McConell, J. and Robson, J., Noncommutative Noetherian Rings (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
Milne, J. S., Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge Studies in Advanced Mathematics, 170 (Cambridge University Press, Cambridge, 2017).CrossRefGoogle Scholar
Mirković, I. and Vilonen, K., ‘Geometric Langlands duality and representations of algebraic groups over commutative rings’, Ann. of Math. (2) 166 (2007), 95–143.CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, Modern Surveys in Mathematics, 34 (Springer, Berlin, Heidelberg, 1994).CrossRefGoogle Scholar
Rietsch, K., ‘An introduction to perverse sheaves’, Fields Inst. Commun. 40 (2004), 391–429.Google Scholar
Soergel, W., ‘Kazdhan–Lusztig polynomials and a combinatoric[s] for tilting modules’, Represent. Theory 1 (1997), 83–114.CrossRefGoogle Scholar
Soergel, W., ‘On the relation between intersection cohomology and representation theory in positive characteristic’, J. Pure Appl. Algebra 152 (2000), 311–335.Google Scholar
Springer, T. A., Linear Algebraic Groups, Progress in Mathematics, 9 (Birkhäuser, Boston, MA, 1981).Google Scholar
Steinberg, R., Conjugacy Classes in Algebraic Groups, Lecture Notes in Mathematics, 366 (Springer, Berlin, Heidelberg, 1974).CrossRefGoogle Scholar
Sullivan, J., ‘Simply connected groups, the hyperalgebra, and Verma's conjecture’, Amer. J. Math. 100(5) (1978), 1015–1019.Google Scholar
The Stacks project authors, ‘The Stacks project’, https://stacks.math.columbia.edu, 2020.Google Scholar
Waterhouse, W. C., Introduction to Affine Group Schemes (Springer, New York, 1979).CrossRefGoogle Scholar
Webb, P., A Course in Finite Group Representation Theory, Cambridge Studies in Advanced Mathematics, 161 (Cambridge University Press, Cambridge, 2016).Google Scholar
Williamson, G., ‘Algebraic representations and constructible sheaves’, Jpn. J. Math. 12(2) (2017), 211–259.Google Scholar
Williamson, G., ‘On torsion in the intersection cohomology of Schubert varieties’, J. Algebra 475 (2017), 207–228.Google Scholar
Williamson, G., ‘Schubert calculus and torsion explosion’, J. Amer. Math. Soc. 30 (2017), 1023–1046.Google Scholar
Zhu, X., ‘An introduction to affine Grassmannians and the geometric Satake equivalence’, in: Geometry of Moduli Spaces and Representation Theory (American Mathematical Society, Providence, RI, 2017).Google Scholar