arXiv subjects

Jean Michel

Publications and source records attributed to Jean Michel.

At least 19 recordsLinked to original sources

arXiv2025

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

Tower equivalence and Lusztig's truncated Fourier transform

We give a proof of the results of Chapuy and Douvropoulos [3] for irreducible spetsial reflection groups based on Deligne-Lusztig combinatorics. In particular, if f denotes the truncated Lusztig Fourier transform, we show that the image by f of the normalized characteristic function of a Coxeter element is the alternate sum of the exterior powers of the reflection representation, and that a class function is tower equivalent to its image by f .

math.RT↗

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↗

arXiv2017

Cyclotomic root systems and bad primes

We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes.

math.RT↗

arXiv2016

The Sylow Subgroups of a Finite Reductive Group

We describe the structure of Sylow {\ell}-subgroups of a finite reduc-tive group G(Fq) when q $\not\equiv$ 0 (mod {\ell}) that we find governed by a complex reflection group attached to G and {\ell}, which depends on {\ell} only through the set of cyclotomic factors of the generic order of G(Fq) whose value at q is divisible by {\ell}. We also tackle the more general case G F where F is an isogeny whose a power is a Frobenius morphism.

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

Addenda to "Foundations of Garside Theory"

This text consists of additions to the book "Foundations of Garside Theory", EMS Tracts in Mathematics, vol. 22 (2015) -- see introduction and table of contents in arXiv:1309.0796 -- namely skipped proofs and solutions to selected exercises.

math.GR↗

arXiv2014

"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element

Chapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.

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↗

arXiv2013

The development version of the CHEVIE package of GAP3

I describe the current state of the development version of the CHEVIE package, which deals with Coxeter groups, reductive algebraic groups, complex reflection groups, Hecke algebras, braid monoids, etc... Examples are given, showing the code to check some results of Lusztig.

math.RT↗

arXiv2013

Foundations of Garside Theory

This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context. The complete text can be found at http://www.math.unicaen.fr/~garside/Garside.pdf.

math.GR↗

arXiv2013

Garside families and Garside germs

Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations of Garside families, that is, equivalently, various criteria for establishing the existence of normal decompositions of a certain type.

math.GR↗

arXiv2012

Split Spetses for primitive reflection groups

Let $(V,W)$ be an exceptional spetsial irreducible reflection group $W$ on a complex vector space $V$, that is a group $G_n$ for $n \in \{4, 6, 8, 14, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37\}$ in the Shephard-Todd notation. We describe how to determine some data associated to the corresponding (split) "spets", given complete knowledge of the same data for all proper subspetses (the method is thus inductive). The data determined here is the set Uch$(\mathbb G)$ of "unipotent characters" of $\mathbb G$ and the associated set of Frobenius eigenvalues, and its repartition into families. The determination of the Fourier matrices linking unipotent characters and "unipotent character sheaves" will be given in another paper. The approach works for all split reflection cosets for primitive irreducible reflection groups. The result is that all the above data exist and are unique (note that the cuspidal unipotent degrees are only determined up to sign).

math.RT↗

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↗