Below are provisional synopses for the courses along with some possible
prereading. If you have any questions, please email me
goodwin(at)maths.bham.ac.uk.
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Introduction to algebraic groups
The course will cover the basics of the theory of linear algebraic groups,
hopefully ending with an outline of the classification of
reductive linear algebraic groups. Topics that are likely to be covered include:
- Actions and representations of algebraic groups;
- Jordan decomposition;
- Lie algebra of an algebraic group;
- Solvable groups and Borel subgroups;
- Parabolic subgroups;
- Maximal tori and root datum;
- Reductive groups.
Suggested prereading:
Basic knowledge from affine algebraic geometry will be assumed.
This can be found in:
- Reid, Undergraduate algebraic geometry, Chapter II.
- Geck, An introduction to algebraic geometry and algebraic groups, 1.1–1.3, 2.1.
The material to be covered in the course will mostly be contained in:
- Geck, An introduction to algebraic geometry and algebraic groups.
- Humphreys, Linear algebraic groups.
- Springer, Linear algebraic groups.
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Finite groups of Lie type and Hecke algebras
A preliminary synopsis is as follows:
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Frobenius maps on algebraic groups, examples
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Harish-Chandra series of representations
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Hecke algebras
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Kazhdan-Lusztig basis and cellular structure
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Examples, open problems
Suggested prereading:
Basic knowledge of representation theory will be assumed, as would be contained in
first course. This is covered in for example:
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James and Liebeck, Representations and Characters of Groups.
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Serre, Linear Representations of Finite Groups, Chapter 1.
A single reference to give an idea of what will be covered in the course is:
- Geck, Modular representations of Hecke algebras.
http://arxiv.org/abs/math/0511548
in
Group representation theory (EPFL, 2005; eds. M. Geck, D. Testerman and
J. Th\'evenaz), pp.~301--353, Presses Polytechniques et Universitaires
Romandes, EPFL-Press, Lausanne, 2007.
Prereading for Frobenius morphisms and representations of finite groups of Lie type:
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Geck, An introduction to algebraic geometry and algebraic groups.
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Digne and Michel, Representations of Finite Groups of Lie Type.
Prereading for Hecke algebras and cellular structures:
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Humphreys, Reflection groups and Coxeter groups.
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Mathas, Iwahori–Hecke algebras and Schur algebras of the
symmetric group.
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Modular representations of Lie algebras
A provisional synopsis is as follows:
- Simple Lie algebras over C: roots
and Weyl group, Dynkin diagrams;
highest weight modules; nilpotent
orbits.
- Reduced enveloping algebras; problem
primes; Kac–Weisfeiler Conjecture;
Verma modules, simple modules,
blocks; special cases.
- Nilpotent variety: Springer resolution
and fibers; BMR theorems;
subregular case.
- Cells in affine Weyl group, Lusztig bijection;
examples; dimension formulas?
- Lusztig’s program: assign modules
to left cells? conjectures and
examples
Suggested prereading:
Familiarity with Lie algebras, root systems and Weyl groups will
be assumed. This material is covered in for example:
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Erdmann and Wildon, Introduction to Lie algebras.
- Humphreys, Introduction to Lie algebras and representation theory.
- Humphreys, Reflection groups and Coxeter groups.
For general information and references on representations of Lie algebras
in positive characteristic, see for example
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Jantzen, Representations of Lie algebras in prime characteristic,
in Representation theories and algebraic geometry, Kluwer (1998).
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Jantzen, Survey on representations of Lie algebras in prime characteristic,
available at http://home.imf.au.dk/jantzen/.
For information on nilpotent orbits see
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Jantzen,
Nilpotent orbits in representation theory, in Lie Theory,
Progress in Math., vol. 228, Birkh\"auser, 2004.
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