arXiv subjects

Raphaël Rouquier

Publications and source records attributed to Raphaël Rouquier.

16 recordsLinked to original sources

arXiv2022

Cherednik algebras and Calogero-Moser cells

Using the representation theory of Cherednik algebras at $t=0$ and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-sided cells, Kazhdan-Lusztig families and Kazhdan-Lusztig cellular representations.

math.RT↗

arXiv2020

The Brauer trees of unipotent blocks

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 $\ell$-blocks of finite groups are known for $\ell>71$.

math.RT↗

arXiv2019

An asymptotic cell category for cyclic groups

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle.

math.RT↗

arXiv2018

Translation by the full twist and Deligne-Lusztig varieties

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.

math.RT↗

arXiv2016

Derived categories and Deligne-Lusztig varieties II

This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups. We also show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of [BoRo1], the finer homotopy equivalence was unknown. As a consequence, the equivalence preserves defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations.

math.RT↗

arXiv2013

Cellules de Calogero-Moser

Using the representation theory of Cherednik algebras at t=0 and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-sided cells, Kazhdan-Lusztig families and Kazhdan-Lusztig cellular representations.

math.RT↗

arXiv2012

Coxeter orbits and Brauer trees III

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 $\ell$-block of $E_7(q)$ when $\ell \mid Φ_{18}(q)$ and $E_8(q)$ when $\ell \mid Φ_{18}(q)$ or $\ell \mid Φ_{30}(q)$.

math.RT↗

arXiv2012

Khovanov-Rozansky homology and 2-braid groups

Khovanov has given a construction of the Khovanov-Rozansky link invariants (categorifying the HOMFLYPT invariant) using Hochschild cohomology of 2-braid groups. We give a direct proof that his construction does give link invariants. We show more generally that, for any finite Coxeter group, his construction provides a Markov "2-trace", and we actually show that the invariant takes value in suitable derived categories. This makes more precise a result of Trafim Lasy who has shown that, after taking the class in K_0, this provides a Markov trace.

math.RT↗

arXiv2012

Calogero-Moser versus Kazhdan-Lusztig cells

In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups. They defined in particular partitions of the group in left and two-sided cells. In 1983, Lusztig generalized this theory to Hecke algebras of Coxeter groups with unequal parameters. We propose a definition of left cells and two-sided cells for complex reflection groups, based on ramification theory for Calogero-Moser spaces. These spaces have been defined via rational Cherednik algebras by Etingof and Ginzburg. We conjecture that these coincide with Kazhdan-Lusztig cells, for real reflection groups. Counterparts of families of irreducible characters have been studied by Gordon and Martino, and we provide here a version of left cell representations. The Calogero-Moser cells will be studied in details in a forthcoming paper, providing thus several results supporting our conjecture.

math.RT↗

arXiv2010

Automizers as extended reflection groups

Broué, Malle and Michel have shown that the automizer of an abelian Sylow p-subgroup in a finite simple Chevalley group is an irreducible complex reflection group, for p not too small and different from the defining characteristic. The aim is this note is to show that a suitable version of this property holds for general finite groups. As an example, the automizer of an 11-Sylow subgroup in the Monster is the 2-dimensional complex reflection group G_{16}.

math.GR↗

arXiv2010

Perverse Equivalences and Broué's Conjecture

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué's conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups C3xC3. We also examine extensions to automorphism groups in a general setting.

math.RT↗

arXiv2008

Compactification des variétés de Deligne-Lusztig

We construct explicitly the normalisation of the Bott-Samelson-Demazure-Hansen compactification of Deligne-Lusztig varieties $X(w)$ in their covering $Y(w)$: we retrieve a result by Deligne-Lusztig about the local monodromy around the divisors of the compactification.

math.AG↗

arXiv2007

Affineness of Deligne-Lusztig varieties for minimal length elements

We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures.

math.AG↗

arXiv2006

Cohomologie des variétés de Deligne-Lusztig

We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element $w\_0$ (some indeterminacies remain for $G\_2$). We use this to construct Hecke algebra actions on the cohomology of varieties associated to $w\_0$ or its square, for groups of arbitrary rank. In a subsequent work, we construct actions associated to more general regular elements and we study their trace on cohomology.

math.RT↗

arXiv2006

On the irreducibility of Deligne-Lusztig varieties

Let $G$ be a connected reductive algebraic group defined over an algebraic closure of a finite field and let $F : G \to G$ be an endomorphism such that $F^d$ is a Frobenius endomorphism for some $d \geq 1$. Let $P$ be a parabolic subgroup of $G$ admitting an $F$-stable Levi subgroup. We prove that the Deligne-Lusztig variety $\{gP | g^{-1}F(g)\in P\cdot F(P)\}$ is irreducible if and only if $P$ is not contained in a proper $F$-stable parabolic subgroup of $G$.

math.GR↗

arXiv2006

Coxeter orbits and modular representations

We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A\_n$ and a Coxeter element. Our study is based on Lusztig's work in characteristic 0.

math.RT↗