Modular Deligne--Lusztig learning seminar
Friday 15:00--17:00 during the acadamic year 2017--2018
(The graduate course (this year) on modular rep thy of finite groups (by Fei Xu) should be helpful for the seminar audiences.)
Contents
(I will roughly follow the order of contents of Bonnafé's book.)
- Lecture 1 (2017-Sep-1). Intro
- Lecture 2 (2017-Sep-8). Basic properties of SL_2(F_q) - cardinals, subgroups, comparison with SL_2(R), relations with modular forms etc
- Lecture 3 (2017-Sep-15). Basic properties of the Drinfeld curve and some elementary alg geom revisited - connectedness, smoothness, compactification, gp actions, quotients etc
- Lecture 4 (2017-Sep-22). Basic properties of the Drinfeld curve II - Frobenius action, point counting, exercises etc
- Lecture 5 (2017-Oct-12). Basic rep thy of finite groups revisited - definition, induction, restriction, Frobenius reciprocity, Mackey intertwining formula etc
- Lecture 6 (2017-Oct-20). Harish-Chandra theory (the SL_2(F_q) case) - ``cusp forms'', the intertwining formula, some dimension computations etc
- Lecture 7 (2017-Nov-3). Deligne--Lusztig theory (the SL_2(F_q) case) I - some generalities on étale cohomology, cohomology of the Drinfeld curve, Deligne--Lusztig induction, the Mackey formula etc
- Lecture 8 (2017-Nov-17). Deligne--Lusztig theory (the SL_2(F_q) case) II - representation counting, dimensions, conjugacy classes, Frobenius actions (1) etc
- Lecture 9 (2017-Nov-24). Deligne--Lusztig theory (the SL_2(F_q) case) III - Frobenius actions (2), point counting, zeta function, dual representations, exercises etc
- Lecture 10 (2017-Dec-8). Basics of modular representations - the gp alg and idempotents, the cde triangle, blocks and defect groups, some examples etc
- Lecture 11 (2017-Dec-29). The McKay conjecture - some recaps, Brauer correspondents and Broué's abelian defect gp conj, the McKay conj, verification for SL_2(F_q) etc
- Lecture 12 (2018-Jan-5). Modular Deligne--Lusztig construction I (mainly the case of SL_2(F_q)) - cohomology complexes and ``projectivisation'', some criteria of equivalences etc
- Lecture 13 (2018-Jan-19). Modular Deligne--Lusztig construction II (mainly the case of SL_2(F_q)) - endomorphism algebras, blocks, Morita and derived equivalences etc
- (Winter vacation and Chineses new year)
- Lecture 14 (2018-Mar-9). (rough plan) Modular Deligne--Lusztig construction III - an overview
- Lecture 15 (2018-Mar-16). (rough plan) ``An addendum'' - the definining characteristic case
References
Bonnafé - Representations of SL_2(F_q)
Bonnafé, Dat, and Rouquier - Derived categories and Deligne--Lusztig varieties II
Bonnafé et Rouquier - Catégories dérivées et variétés de Deligne--Lusztig
Borel/Humphreys/Springer - Linear algebraic groups
Broué - Isométries de caractères et équivalences de Morita ou dérivées
Broué und Malle - Zyklotomische Heckealgebren
Broué et Michel - Blocs et séries de Lusztig dans un groupe réductif fini
Cabanes and Enguehard - Representation theory of finite reductive groups
Carter - Finite groups of Lie type
Deligne and Lusztig - Representations of reductive groups over finite fields
Digne and Michel - Representations of finite groups of Lie type
Dudas - Coxeter orbits and Brauer trees
Dudas - Cohomology of Deligne--Lusztig varieties for unipotent blocks of GL_n(F_q)
Dudas - Lectures on modular Deligne--Lusztig theory (arXiv 1705.08234)
Geck - A first guide to the character theory of finite groups of Lie type (arXiv 1705.05083)
Lusztig - Coxeter orbits and eigenspaces of Frobenius
Lusztig - CBMS lectures
Lusztig - Corvallis paper
Lusztig - orange book
Rickard - Finite group actions and étale cohomology
Rouquier - Complexes de chaînes étales et courbes de Deligne--Lusztig
Serre - Linear representations of finite groups
SGA 3, SGA 4, SGA 4 1/2