J'ai donné un cours de l'ED d'introduction à la théorie de Deligne--Lusztig. Les notes se trouvent ici (attention, beaucoup de coquilles encore !)
- Decomposition matrices for groups of Lie type in non-defining characteristic, to appear in the Memoirs of the AMS, 2023 (with G. Malle).
- We determine approximations to the decomposition matrices for unipotent
l-blocks of several series of finite reductive groups of classical and
exceptional type over F_{q} of low rank in non-defining good
characteristic l.
- Decomposition numbers for the principal Φ_{2n}-block of Sp_{4n}(q) and SO_{4n+1}(q), accepted in Annales de l'Institut Fourier, 2022 (with E. Norton).
- We compute the decomposition numbers of the unipotent characters lying in the principal l-block of a finite group of Lie type of type B_{2n}(q) or C_{2n}(q) when q is an odd prime power and l is an odd prime number such that the order of q mod l is 2n. Along the way, we extend to these finite reductive groups the results of Dudas--Varagnolo--Vasserot on the branching graph for Harish-Chandra induction and restriction.
- The Ext-algebra of the Brauer tree algebra associated to a line, accepted in Revista de la Unión Matemática Argentina, 2021.
- We compute the Ext-algebra of the Brauer tree algebra associated to a line with no exceptional vertex.
- Unitriangular shape of decomposition matrices of unipotent blocks, accepted in Ann. of Math., 2020 (with O. Brunat and J. Taylor).
- We show that the decomposition matrix of unipotent l-blocks of a finite reductive group G(F_{q}) has a unitriangular shape, assuming q is a power of a good prime and l is very good for G. This was conjectured by Geck in 1990. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.
- Translation by the full twist and Deligne--Lusztig varieties, J. Algebra 558 (2020), 129--145. (with C. Bonnafé and R. Rouquier).
- We prove several conjectures about the cohomology of Deligne--Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broué--Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectured for finite groups of Lie type.
- Categorical actions on unipotent representations of finite unitary groups, Publications Mathématiques de l'IHES 129 (2019), 129--197. (with M. Varagnolo and É. Vasserot).
- Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincide with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broué's abelian defect group conjecture for unipotent l-blocks at linear primes l.
- Non-uniqueness of supercuspidal support for finite reductive groups. Appendix to the article Simple subquotients of big parabolically induced representations of p-adic groups by J.-F. Dat, J. Algebra 510 (2018), 508--512.
- We exhibit a simple representation of the finite symplectic group of Sp_{8}(q) in characteristic l | q^{2}+1 having two non-conjugate supercuspidal supports. This answers a question of M.-F. Vignéras.
- Bounding Harish-Chandra series, Transactions of the AMS 371 (2019), 6511--6530.(with G. Malle).
- We use the progenerator constructed in [15] to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.
- Brauer trees of unipotent blocks, JEMS 22 (2020), no. 9, 2821--2877. (with D. Craven and R. Rouquier).
- In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal l-blocks of finite groups
are known for l>71.
- Alvis-Curtis duality for finite general linear groups and a generalized Mullineux involution, SIGMA 14 (2018) (with N. Jacon).
- We study the effect of Alvis-Curtis duality on the unipotent representations of GL_{n}(q) in non-defining characteristic l. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both l and the order of q modulo l.
- Lectures on modular Deligne-Lusztig theory, in Local representation theory and simple groups, 107--177, EMS Ser. Lect. Math., Eur. Math. Soc., Zürich, 2018.
- These notes are based on a series of lectures given by the author at the Centre Bernoulli (EPFL) in July 2016. They aim at illustrating the importance of the mod-l cohomology of Deligne--Lusztig varieties in the modular representation theory of finite reductive groups.
- Splendeur des variétés de Deligne-Lusztig, d'après Deligne-Lusztig, Broué, Rickard, Bonnafé-Dat-Rouquier, Séminaire Bourbaki, Exp. 1137, to appear in Astérisque, 2017. [You can watch the talk here]
- Les travaux fondateurs de Deligne et Lusztig en 1976 ont permis la
construction et l'étude des représentations complexes des groupes
réductifs finis (tels que GL_{n}(q) et Sp_{2n}(q)),
à partir de la cohomologie de certaines variétés algébriques désormais
connues sous le nom de "variétés de Deligne-Lusztig". Dans cet exposé
nous t&ahat;cherons d'expliquer comment ces constructions s'adaptent
parfaitement au cas des représentations dites modulaires (à
coefficients dans un corps de caractéristique positive). Nous
l'illustrerons en détaillant les travaux récents de Bonnafé-Rouquier
(2003) et Bonnafé-Dat-Rouquier (2017) sur la constructions
d'équivalences splendides entre blocs de représentations, équivalences
prédites par Broué 25 ans auparavant.
- Modular irreducibility of cuspidal unipotent characters, Invent. Math. 211 (2018), 579--589 (with G. Malle).
- We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after l-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand--Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand--Graev representations attached to cuspidal unipotent classes, as defined and studied by Geck--Malle.
- Categorical actions from Lusztig induction and restriction on finite general linear groups in Representation theory - current trends and perspectives, 59--74, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2017 (with M. Varagnolo and É. Vasserot).
- In this note we explain how Lusztig's induction and restriction functors
yield categorical actions of Kac-Moody algebras on the derived category of unipotent
representations. We focus on the example of finite general linear groups and induction/restriction
associated with split Levi subgroups, providing a derived analogue of Harish-Chandra
induction/restriction as studied by Chuang-Rouquier.
- Categorical actions on unipotent representations of finite classical groups in Categorification and higher representation theory, 41--104, Contemp. Math. (683), Amer. Math. Soc., Providence, RI, 2017 (with M. Varagnolo and É. Vasserot).
- We review the categorical representation of a Kac-Moody algebra
on unipotent representations of finite unitary groups in non-defining characteristic
given in [12], using Harish-Chandra induction and restriction.
Then, we extend this construction to finite reductive groups of types B or C in
non-defining characteristic. We show that the decategorified
representation is isomorphic to a direct sum of level 2 Fock spaces.
We deduce that the Harish-Chandra branching graph coincides with the crystal graph of
these Fock spaces. We also obtain derived equivalences between blocks, yielding
Broué's abelian defect group conjecture for unipotent l-blocks at linear primes
l.
- Decomposition matrices for exceptional groups at d=4, J. Pure Appl. Algebra 220 (2016), 1096--1121 (with G. Malle).
- We determine the decomposition matrices of unipotent l-blocks of defect Φ_{4}^{2} for
exceptional groups of Lie type up to a few unknowns. For this we employ the new cohomological methods of the first
author, together with properties of generalized Gelfand-Graev characters which were recently shown to hold whenever
the underlying characteristic is good.
- Decomposition matrices for low rank unitary groups, Proc. London Math. Soc. 110 (2015), 1517--1557 (with G. Malle).
- We study the decomposition matrices of the unipotent l-blocks of finite special unitary groups
SU_{n}(q) for unitary primes l larger than n. Up to few unknown entries,
we give a complete solution for n=2,...,10. We also prove a general result for two-column partitions
when l divides q+1. This is achieved using projective modules coming from the l-adic cohomology of
Deligne--Lusztig varieties.
- Projective modules in the intersection cohomology of Deligne--Lusztig varieties, C. R. Acad. Sci. (2014), 467--471. (with G. Malle).
[I gave a talk about this here]
- We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual
of the intersection cohomology of Deligne--Lusztig varieties. This conjecture provides a powerful tool
to determine decomposition numbers of unipotent l-blocks of finite reductive groups.
- Cohomology of Deligne-Lusztig varieties for unipotent blocks of GL_{n}(q), Represent. Th. 17 (2013), 647--662.
- We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GL_{n}(q). We show that the geometric version of Broué's conjecture over Q_{l}, together with Craven's formula, holds for any unipotent block whenever it holds for the principal Φ_{1}-block, that is for the variety X(π).
- A note on decomposition numbers for groups of Lie type of small rank , J. Algebra 388 (2013), 364--373.
- Using virtual projective characters coming from the l-adic cohomology of Deligne-Lusztig varieties, we determine new decomposition numbers for principal blocks of some groups of Lie type of small rank, including G_{2}(q) and Steinberg's triality groups ^{3}D_{4}(q).
- Coxeter orbits and Brauer trees III, J. Amer. Math. Soc. 27 (2014), 1117--1145 (with R. Rouquier).
- This article is the final one of a series of articles on certain blocks of modular representations of finite groups of Lie type and the associated geometry.We prove the conjecture of Broué on derived equivalences induced by the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter elements whenever the defining characteristic is good. We also prove a conjecture of Hiss, Lübeck and Malle on the Brauer trees of the corresponding blocks. As a consequence, we determine the Brauer trees (in particular, the decomposition matrix) of the principal l-block of E_{7}(q) when l divides Φ_{18}(q) and E_{8}(q) when l divides Φ_{18}(q) or Φ_{30}(q).
- Cohomology of Deligne-Lusztig varieties for short-length regular elements in exceptional groups, J. Algebra 392 (2013), 276--298.
- We determine the cohomology of Deligne-Lusztig varieties associated to some short-length regular elements for split groups of type F_{4} and E_{n}. As a byproduct, we obtain conjectural Brauer trees for the principal Φ_{14}-block of E_{7} and the principal Φ_{24}-block of E_{8}.
- Quotient of Deligne-Lusztig varieties, J. Algebra 381 (2013), 1--20.
- We study the quotient of parabolic Deligne-Lusztig varieties by a finite unipotent group U^{F} where U is the unipotent radical of a rational parabolic subgroup P = L U. We show that in some particular cases the cohomology of this quotient can be expressed in terms of "smaller" parabolic Deligne-Lusztig varieties associated to the Levi subgroup L.
- Coxeter orbits and Brauer trees II, Int. Math. Res. Not. 15 (2014), 4100--4123.
- The purpose of this paper is to discuss the validity of the assumptions (W) and (S) stated in a previous work, about the torsion in the modular l-adic cohomology of Deligne-Lusztig varieties associated to Coxeter elements. We prove that both (W) and (S) hold except for groups of type E_{7} or E_{8}.
- Coxeter orbits and Brauer trees, Adv. Math. 229 (2012), 3398--3435.
- We study the cohomology with modular coefficients of Deligne-Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal l-block of a finite reductive group G(F_{q}) when the order of q modulo l is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, Lübeck and Malle) and the derived equivalence predicted by the geometric version of Broué's conjecture.
- Deligne-Lusztig restriction of a Gelfand-Graev module, Ann. Sci. Ec. Norm. Sup. 42 (2009), 653--674.
- Using Deodhar's decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module.