arXiv subjects

François Digne

Publications and source records attributed to François Digne.

17 recordsLinked to original sources

arXiv2026

Centralisers of semi-simple elements are semidirect products

Let $\bG$ be a connected reductive algebraic group over an algebraically closed field, and let $s\in\bG$ be a semisimple element. We show that the centraliser of $s$ is the semidirect product of its identity component by its group of components. We then look at the case where $\bG$ is defined over an algebraic closure of a finite field $\Fq$, and $F$ is an endomorphism such that some power is a Frobenius endomorphism attached to an $\Fq$-structure on $\bG$. We show that if the centraliser of $s$ is $F$-stable we have a semidirect product decomposition of its $F$-fixed points.

math.GR↗

arXiv2024

Retraction to a parabolic subgroup and applications

We continue the study of the retraction from an Artin group to a standard parabolic subgroup introduced by Blufstein, Charney, Paris and the second author. Using right and left retractions we obtain new results on minimal parabolic subgroups, intersection of parabolic subgroups, double cosets with respect to parabolic subgroups and conjugacy classes in Artin groups.

math.GR↗

arXiv2023

Braid groups and symplectic Steinberg groups

We construct a homomorphism $f$ from the braid group $B_{2n+2}$ on $2n+2$ strands to the Steinberg group associated with the Lie type $C_n$ and with integer coefficients. This homomorphism lifts the well-known symplectic representation of the braid groups. We also describe the image and the kernel of $f$.

math.GR↗

arXiv2022

Ribbons in Garside monoids

We expound the properties of ribbons in a setting which is general enough to encompass spherical Artin monoids and dual braid monoids of well-generated complex reflection groups. We generalize to our setting results on parabolic subgroups of spherical Artin group of Godelle, Cumplido, and others.

math.GR↗

arXiv2021

Formulae for two-variable Green functions

Based on results of Digne-Michel-Lehrer (2003) we give two formulae for two-variable Green functions attached to Lusztig induction in a finite reductive group. We present applications to explicit computation of these Green functions, to conjectures of Malle and Rotilio, and to scalar products between Lusztig inductions of Gelfand-Graev characters.

math.GR↗

arXiv2020

Commutation of Shintani descent and Jordan decomposition

Let ${\mathbf G}^F$ be a finite group of Lie type, where ${\mathbf G}$ is a reductive group defined over ${\overline{\mathbb F}_q}$ and $F$ is a Frobenius root. Lusztig's Jordan decomposition parametrises the irreducible characters in a rational series${\mathcal E}({{\mathbf G}^F},(s)_{{\mathbf G}^{*F^*}})$ where $s\in{{\mathbf G}^{*F^*}}$ by the series ${\mathcal E}(C_{{\mathbf G}^*}(s)^{F^*},1)$.We conjecture that the Shintani twisting preserves the space of class functions generated by the union of the ${\mathcal E}({{\mathbf G}^F},(s')_{{\mathbf G}^{*F^*}})$ where$(s')_{{\mathbf G}^{*F^*}}$ runs over the semi-simple classes of ${{\mathbf G}^{*F^*}}$ geometrically conjugate to $s$;further, extending the Jordan decomposition by linearity to this space, we conjecture that there is a way to fix Jordan decomposition such that it maps the Shintani twisting to the Shintani twisting on disconnected groups defined by Deshpande, which acts on the linear span of $\coprod_{s'}{\mathcal E}(C_{{\mathbf G}^*}(s')^{F^*},1)$. We show a non-trivial case of this conjecture, the case where ${\mathbf G}$ is of type $A_{n-1}$with $n$ prime.

math.GR↗

arXiv2017

Quasi-semisimple elements

We study quasi-semisimple elements of disconnected reductive algebraic groups over an algebraically closed field. We describe their centralizers, define isolated and quasi-isolated quasi-semisimple elements and classify their conjugacy classes.

math.GR↗

arXiv2016

Dual braid monoids, Mikado braids and positivity in Hecke algebras

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical types different from $D_n$ the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type $D_n$).This property implies positivity properties of the polynomials arising in the linear expansion of their images in the Iwahori-Hecke algebra when expressed in the Kazhdan-Lusztig basis. In type $A_n$, it implies positivity properties of their images in the Temperley-Lieb algebra when expressed in the diagram basis.

math.GR↗

arXiv2016

Présentation des groupes de tresses purs et de certaines de leurs extensions

This paper dates back to 1999 but was never published. The major part of it was included in the joint paper [Digne-Gomi, Presentation of pure braid groups, J. Knot Theory and its Ramifications 10 (2001) 609--623]. Sections 2 and 6 were not included there. They are independent from the rest of the paper. In section 2 we define a morphism from a braid group B associated to a Coxeter Group W to the semi-direct product of ZT (where T is the set of reflections) by W. This morphism lifts the map N from W to (Z/2Z)T defined by Dyer in [Reflection subgroups of Coxeter systems, J. Algebra 135 (1990) 57--73] and its restriction to the pure braid group P identifies with the abelianization morphism. In section 6 we prove that our morphism gives the cocycle associated to the extension B/(P,P) of W by P/(P,P). This extension has been studied recently by Beck [Abelianization of subgroups of reflection groups and their braid groups: an application to cohomology. Manuscripta Math. 136 (2011) 273--293] and by Marin [Reflection groups acting on their hyperplanes J. Algebra 322 (2009) 2848--2860] and [Crystallographic groups and flat manifolds from complex reflection groups, arXiv 1509.06213].

math.GR↗

arXiv2015

On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front set are determined by such data. In some exceptional groups we show that the projection of a generalised Gelfand-Graev character to a family with the same wave front set is (up to sign) the dual of a Mellin transform. Using these results, in certain cases we are able to determine roots of unity which relate almost characters to the characteristic functions. In particular we show how to compute the values of all unipotent characters at all unipotent classes for the exceptional adjoint groups of type G2, F4, E6, E7 and E8. We also pro- vide an appendix which gives a complete list of the cuspidal character sheaves on all quasi-simple groups.

math.RT↗

arXiv2014

Parabolic Deligne-Lusztig varieties

Motivated by the Broué conjecture on blocks with abelian defect groups for finite reductive groups, we study "parabolic" Deligne-Lusztig varieties and construct on those which occur in the Broué conjecture an action of a braid monoid, whose action on their $\ell$-adic cohomology will conjecturally factor trough a cyclotomic Hecke algebra. In order to construct this action, we need to enlarge the set of varieties we consider to varieties attached to a "ribbon category"; this category has a {\em Garside family}, which plays an important role in our constructions, so we devote the first part of our paper to the necessary background on categories with Garside families.

math.GR↗

arXiv2011

A Garside presentation for Artin-Tits groups of type $\tilde C_n$

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde C$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type $\tilde A$ under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.

math.GR↗

arXiv2011

The center of pure complex braid groups

Broué, Malle and Rouquier conjectured in that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for the full braid group due to Bessis, and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center.

math.GR↗

arXiv2007

Garside and locally Garside categories

We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.

math.GR↗

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↗

arXiv2005

Présentations duales des groupes de tresses de type affine $\tilde A$

In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type $\tilde A_n$ we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin-Tits group. More precisely, if $W$ is a Coxeter group, one can consider the length $l_R$ on $W$ with respect to the generating set $R$ consisting of all reflections. Let $c$ be a Coxeter element in $W$ and let $P_c$ be the set of elements $p\in W$ such that $c$ can be written $c=pp'$ with $l_R(c)=l_R(p)+l_R(p')$. We define the monoid $M(P_c)$ to be the monoid generated by a set $\underline P_c$ in one-to-one correspondence, $p\mapsto \underline p$, with $P_c$ with only relations $\underline{pp'}=\underline p.\underline p'$ whenever $p$, $p'$ and $pp'$ are in $P_c$ and $l_R(pp')=l_R(p)+l_R(p')$. We conjecture that the group of quotients of $M(P_c)$ is the Artin-Tits group associated to $W$ and that it has a simple presentation (see \ref{conjecture} (ii)). These conjectures are known to be true for spherical type Artin-Tits groups. Here we prove them for Artin-Tits groups of type $\tilde A$. Moreover, we show that for exactly one choice of the Coxeter element (up to diagram automorphism) we obtain a (quasi-) Garside monoid. The proof makes use of non-crossing paths in an annulus which are the counterpart in this context of the non-crossing partitions used for type $A$.

math.GR↗

arXiv2005

Endomorphisms of Deligne-Lusztig varieties

This paper is a following to math.RT/0410454. For a finite group of Lie type we study the endomorphisms, commuting with the group action, of a Deligne-Lusztig variety associated to a regular element of the Weyl group. We state some general conjectures, in particular that the endomorphism algebra induced on the cohomology is a cyclotomic Hecke algebra, and prove them in some cases. On the way we also prove results on centralizers in braid groups.

math.RT↗