# Workshop Programme

## for period 12 January - 17 April 2009

### Instructional workshop

12 January - 17 April 2009

Timetable

Monday 12 January | ||||

08:30-10:00 | Registration | |||

10:00-11:00 | Achar, P (Louisiana) |
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Derived categories and perverse sheaves | Sem 1 | |||

Perverse sheaves are a very important and powerful tool in representation theory. In these talks, we will begin with the formalism of derived categories and t-structures, needed to define perverse sheaves. Next, we will go through the list of remarkable properties of perverse sheaves that make them "better" than ordinary sheaves, and are the source of their usefulness. Lastly, we will look at a couple of representation theory applications. (Some basic familiarity with ordinary sheaves may be a helpful prerequisite.) |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Geck, M (Aberdeen) |
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Hecke algebras at roots of unity | Sem 1 | |||

We discuss various issues around non-semisimple Hecke algebras: cellular structure, James' conjecture, application to modular Harish-Chandra series for finite groups of Lie type. |
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12:30-13:30 | Lunch | |||

14:00-15:00 | Chuang, J (London) |
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Categorification of sl(2)-modules | Sem 1 | |||

The aim of this course is to introduce the Kazhdan-Lusztig theory, in the general framework of unequal parameters. We plan to talk about Lusztig's Conjectures, the asymptotic algebra, and to give a detailed account on some examples (symmetric group, type B,...). If time is left, we might talk about the connections with Cherednik algebras, or with representations of finite reductive groups, or with the geometry of the flag variety, or with the computation of decomposition matrices using the canonical bases of the Fock space (at most one of these four topics will be treated). |
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15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

17:30-18:30 | Wine Reception | |||

18:45-19:30 | Dinner |

Tuesday 13 January | ||||

09:00-10:00 | Ram, A (Melbourne) |
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Symmetry, polynomials and quantisation | Sem 1 | |||

10:00-11:00 | Achar, P (Louisiana) |
|||

Derived categories and perverse sheaves | Sem 1 | |||

Perverse sheaves are a very important and powerful tool in representation theory. In these talks, we will begin with the formalism of derived categories and t-structures, needed to define perverse sheaves. Next, we will go through the list of remarkable properties of perverse sheaves that make them "better" than ordinary sheaves, and are the source of their usefulness. Lastly, we will look at a couple of representation theory applications. (Some basic familiarity with ordinary sheaves may be a helpful prerequisite.) |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Geck, M (Aberdeen) |
|||

Hecke algebras at roots of unity | Sem 1 | |||

We discuss various issues around non-semisimple Hecke algebras: cellular structure, James' conjecture, application to modular Harish-Chandra series for finite groups of Lie type. |
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12:30-13:30 | Lunch | |||

14:00-15:00 | Kleshchev, A (Oregon) |
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W-algebras and Hecke algebras 1 | Sem 1 | |||

We discuss a new presentation for blocks of cyclotomic Hecke algebras which allows to grade these blocks. This opens up a way for studying graded representation theory of cyclotomic Hecke algebras. We give emphasis to the recent result of Khovanov-Lauda, Rouquier, Brundan-Kleshchev and Kleshchev-Ram. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Wednesday 14 January | ||||

09:00-10:00 | Ram, A (Melbourne) |
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Symmetry, polynomials and quantisation | Sem 1 | |||

10:00-11:00 | Achar, P (Louisiana) |
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Derived categories and perverse sheaves | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Geck, M (Aberdeen) |
|||

Hecke algebras at roots of unity | Sem 1 | |||

We discuss various issues around non-semisimple Hecke algebras: cellular structure, James' conjecture, application to modular Harish-Chandra series for finite groups of Lie type. |
||||

12:30-13:30 | Lunch | |||

18:45-19:30 | Dinner |

Thursday 15 January | ||||

09:00-10:00 | Ram, A (Melbourne) |
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Symmetry, polynomials and quantisation | Sem 1 | |||

10:00-11:00 | Achar, P (Louisiana) |
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Derived categories and perverse sheaves | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Chuang, J (London) |
|||

Categorification of sl(2)-modules | Sem 1 | |||

12:30-13:30 | Lunch | |||

14:00-15:00 | Kleshchev, A (Oregon) |
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W-algebras and Hecke algebras 2 | Sem 1 | |||

We discuss a new presentation for blocks of cyclotomic Hecke algebras which allows to grade these blocks. This opens up a way for studying graded representation theory of cyclotomic Hecke algebras. We give emphasis to the recent result of Khovanov-Lauda, Rouquier, Brundan-Kleshchev and Kleshchev-Ram. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Friday 16 January | ||||

09:00-10:00 | Ram, A (Melbourne) |
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Symmetry, polynomials and quantisation | Sem 1 | |||

10:00-11:00 | Achar, P (Louisiana) |
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Derived categories and perverse sheaves | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Chuang, J (London) |
|||

Categorification of sl(2)-modules | Sem 1 | |||

12:30-13:30 | Lunch | |||

14:00-15:00 | Kleshchev, A (Oregon) |
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W-algebras and Hecke algebras 3 | ||||

We discuss a new presentation for blocks of cyclotomic Hecke algebras which allows to grade these blocks. This opens up a way for studying graded representation theory of cyclotomic Hecke algebras. We give emphasis to the recent result of Khovanov-Lauda, Rouquier, Brundan-Kleshchev and Kleshchev-Ram. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Monday 19 January | ||||

09:00-10:00 | Broué, M (Paris) |
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Complex reflection groups and their associated braid groups and Hecke algebras | Sem 1 | |||

10:00-11:00 | Bonnafé, C (de Franche Comté) |
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Introduction to Kazhdan-Lusztig theory with unequal parameters | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Rouquier, R (Oxford) |
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Higher representations of Lie algebras | Sem 1 | |||

12:30-13:30 | Lunch | |||

14:00-15:00 | Vogan, D (Massachusetts) |
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Schubert varieties and representations of real reductive groups | Sem 1 | |||

15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Tuesday 20 January | ||||

09:00-10:00 | Broué, M (Paris) |
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Complex reflection groups and their associated braid groups and Hecke algebras | Sem 1 | |||

10:00-11:00 | Bonnafé, C (de Franche Comté) |
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Introduction to Kazhdan-Lusztig theory with unequal parameters | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Rouquier, R (Oxford) |
|||

Higher representations of Lie algebras | Sem 1 | |||

12:30-13:30 | Lunch | |||

14:00-15:00 | Vogan, D (Massachusetts) |
|||

Schubert varieties and representations of real reductive groups | Sem 1 | |||

15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Wednesday 21 January | ||||

09:00-10:00 | Broué, M (Paris) |
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Complex reflection groups and their associated braid groups and Hecke algebras | Sem 1 | |||

10:00-11:00 | Losev, I (Massachusetts) |
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Finite W-algebras and their representations | Sem 1 | |||

A (finite) W-algebra is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras. In this course I am going to explain two different definitions of W-algebras: by Hamiltonian reduction (Premet, Gan-Ginzburg) and deformation quantization (I.L). Then I am going to explain category equivalence theorems relating representations of W-algebras and universal enveloping algebras and describe relation between primitive ideals. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Rouquier, R (Oxford) |
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Higher representations of Lie algebras | Sem 1 | |||

12:30-13:30 | Lunch | |||

18:45-19:30 | Dinner |

Thursday 22 January | ||||

09:00-10:00 | Losev, I (Massachusetts) |
|||

Finite W-algebras and their representations | Sem 1 | |||

A (finite) W-algebra is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras. In this course I am going to explain two different definitions of W-algebras: by Hamiltonian reduction (Premet, Gan-Ginzburg) and deformation quantization (I.L). Then I am going to explain category equivalence theorems relating representations of W-algebras and universal enveloping algebras and describe relation between primitive ideals. |
||||

10:00-11:00 | Bonnafé, C (de Franche Comté) |
|||

Introduction to Kazhdan-Lusztig theory with unequal parameters | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Rouquier, R (Oxford) |
|||

Higher representations of Lie algebras | Sem 1 | |||

12:30-13:30 | Lunch | |||

14:00-15:00 | Vogan, D (Massachusetts) |
|||

Schubert varieties and representations of real reductive groups | Sem 1 | |||

15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Friday 23 January | ||||

09:00-10:00 | Losev, I (Massachusetts) |
|||

Finite W-algebras and their representations | Sem 1 | |||

A (finite) W-algebra is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras. In this course I am going to explain two different definitions of W-algebras: by Hamiltonian reduction (Premet, Gan-Ginzburg) and deformation quantization (I.L). Then I am going to explain category equivalence theorems relating representations of W-algebras and universal enveloping algebras and describe relation between primitive ideals. |
||||

10:00-11:00 | Bonnafé, C (de Franche Comté) |
|||

Introduction to Kazhdan-Lusztig theory with unequal parameters | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Rouquier, R (Oxford) |
|||

Higher representations of Lie algebras | Sem 1 | |||

12:30-13:30 | Lunch | |||

14:00-15:00 | Vogan, D (Massachusetts) |
|||

Schubert varieties and representations of real reductive groups | Sem 1 | |||

15:00-15:30 | Tea | |||

15:30-16:30 | Discussion | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner |

Tuesday 14 April | ||||

09:00-10:00 | Cautis, S (MSRI) |
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Geometry and categorification II Braid group actions in algebraic geometry |

Friday 17 April | ||||

16:20-17:00 | Masbaum, G (CNRS) |
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Integral lattices in TQFT and integral modular categories | ||||

The SO(3)-TQFT at an odd prime has a natural integral structure. It means that all matrix coefficients of the mapping class group representations in this theory are algebraic integers. I will discuss the structure of the resulting "integral modular functor". This is joint work with Patrick Gilmer. |