Curriculum Vitæ : Jérémie Guilhot

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1 Curriculum Vitæ : Jérémie Guilhot born in Chambéry, France, the 5 th of January Personal address Professional address 106 West End Street School of Mathematics Norwich NR2 4DP University of East Anglia United Kingdom Norwich, NR4 7TJ United Kingdom Tel : Tel : jeremieguilhot@gmail.com j.guilhot@uea.ac.uk EDUCATION 2009/2011: Lecturer in Mathematics at the University of East Anglia 2008/2009: Research fellow under the supervision of Andrew Mathas and Robert Howlett (University of Sydney) 2004/2008: Co-supervised PhD in Mathematics at the University of Aberdeen and Université Claude Bernard Lyon 1 Thesis title: Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters Supervisors : M. Meinolf Geck, University of Aberdeen M. Philippe Caldéro, Université Claude Bernard Lyon 1 Examiners : M. Cédric Bonnafé, Chargé de recherche au CNRS, Université de Franche-Comté M. Georges Lusztig, Professeur au Massachusetts Institute of Technology Jury Dr. Radha Kessar, Lecturer, University of Aberdeen Dr.. Cédric Bonnafé, Chargé de Recherche au CNRS, Université de Franche-Comté Dr. Philippe Caldéro, Lecturer, Université Claude Bernard Lyon 1 Pr. Meinolf Geck, Professor, University of Aberdeen Pr. Iain Gordon (Président du Jury), Professeur, University of Edinburgh Pr. Kenji Iohara, Professor, Université Claude Bernard Lyon /2004: Master of Mathematics, Université Claude Bernard Lyon /2003: Agrégation de Mathématiques 2001/2002: Maîtrise (M1), University of Birmingham 1998/2001: Licence (L3), Université Jean Monnet, Saint-Etienne

2 TEACHING 2009/2011 Lecturer in Mathematics at the University of East Anglia Introduction to group theory, 1st year Linear Algebra (lecture), 2nd year Cryptography (lecture), 2nd/3rd year Calculus (exercise classes), 1st year Pure Mathematics (exercise classes), 1st year 2007/2008: Tutor in Mathematics, University of Aberdeen: Introduction to Analysis (exercise classes), 2nd year Calculus (exercise classes), 2nd year Linear algebra (exercise classes), 2nd year Further Real Analysis (exercise classes), 3rd year 2004/2007: Tutor in Mathematics, Université Claude Bernard Lyon 1: Analysis (exercise classes), 1st year Linear Algebra (exercise classes), 2nd year Algebraic structures and applications (exercise classes), 3rd year 1999/2001: Member of the association Groupement Etudiant National d Enseignement aux Personnes Incarcérées the aim of which was to promote the rehabilitation of prisoners. This involved giving weekly mathematics courses in prison. Published article LIST OF PUBLICATIONS (1) On the determination of Kazhdan-Lusztig cells for affine Weyl groups with unequal parameters, Journal of Algebra 318 (2007), no. 2, (2) On the lowest two-sided cell in affine Weyl groups, Representation Theory 12 (2008), (3) Generalized induction of Kazhdan-Lusztig cells, Annales de l institut Fourier, vol 59 (2009), (4) Kazhdan-Lusztig cells in affine Weyl groups of rank 2, International Mathematics Research Notices, vol (2010), (5) Affine cellularity of affine Hecke algebras of rank two, in collaboration with V. Miemietz To appear in Mathematische Zeitschrift Preprint (6) On Kazhdan-Lusztig cells in the asymptotic case, in collaboration with C. Bonnafé preprint available on arxiv

3 TALKS Affine cellularity of affine Hecke algebras of rank 2 Algebra seminar, Leicester, December 2010 Séminaire Chevalley, Paris, December 2010 On the lowest two-sided cell in the asymptotic case Nikolaus Conference on computational aspects of representation theory, Aachen, December 2009 Cells in affine Weyl groups Algebra seminar of the program Algebraic Lie Theory, Cambridge, April 2009 Semicontinuity properties of Kazhdan-Lusztig cells in affine Weyl groups Algebra seminar, Sydney, March 2009 Algebra seminar, Melbourne, March 2009 Cellules de Kazhdan-Lusztig dans les groupes de Weyl affines Séminaire d Algèbre, Lyon, January 2008 Séminaire d Algèbre, Amiens, January 2008 Séminaire Algèbres enveloppantes, Paris, January 2008 Séminaire d Algèbre, Besançon, June 2008 On the lowest two-sided cell in affine Weyl groups Algebra Seminar, Syndey, September 2009 The Maxwell Institute Algebra Seminar, Edinburgh, November 2007 Algebra Seminar, Manchester, October 2007 Representation Theory Seminar, Oxford, October 2007 Algebra Seminar, Leeds, October 2007 VISITS April/May 2009, Isaac Newton Institute Invited for 3 weeks to the research program Algebraic Lie theory Bursary awarded by the council of the Cambridge Philosophical Society. June 2008, Université de Besançon. Invited by C. Bonnafé for two weeks. December 2007, Université de Besançon. Invited by C. Bonnafé for two weeks.

4 Organisation of Conference MISCENALLEOUS In collaboration with M.Geck, I have organised a one day meeting in Representation theory at the University of Aberdeen (Scotland, UK) on the 5th of June in the framework of the program Representation theory across the channel. Attended Conferences August 2010 : XIV International Conferences on Representations of Algebras, Tokyo, Japon August 2010 : Representation Theory of Algebraic Groups and Quantum Groups 10, Nagoya, Japon June 2010 : Group Representation Theory and Related Topics, in honour of Jacques Thevenaz 60 th birthday, Lausanne, Switzerland December 2009 : Nikolaus Conference on computational aspects of representation theory, Lehrstuhl D für Mathematik, Aachen, Germany April-May 2009: Algebraic Lie Theory, Isaac Newton Institute for Mathematical Sciences, Cambridge, Royaume-Uni. March 2008 : Lie Theory, Mathematical Sciences Research Institute, Berkeley, United-State September 2007 : Conference in memoriam of Fokko du Cloux, Ile de Berder, France April 2007 : Algèbres de Cherednik, Centre International de Rencontres Mathématiques, Luminy, France October 2006 : Conférence en l honneur du 60ème anniversaire de Michel Broué, Ecole normale supérieure, Paris, France August 2006 : Representation Theory of Algebraic Groups and Quantum Groups 06, Nagoya, Japon October 2005 : Sixth Century Conference in Representation Theory, University of Aberdeen, Scotland June 2005 : Conference on algebraic groups and finite reductive groups, Ecole Polytechnique Fédérale de Lausanne, Switzerland Administrative activities Member of the school board of the school of mathematics at the University of East Anglia Participation of the open-day at the University of Norwich. In collaboration with another lecturer, I have organised a presentation for the visiting student during the open day at the university.

5 RESEARCH STATEMENT General interest. Hecke algebras with unequal parameters naturally arise in the representation theory of reductive groups over finite or p-adic fields as endomorphism algebras of certain induced representations. The overall philosophy is that a significant amount of the representation theory of the group is controlled by the representation theory of those endomorphism algebras and this is why understanding their representations is a question of great interest. These algebras are specializations of Iwahori-Hecke algebras which can be defined without reference to algebraic groups, namely by explicit generators and relations in terms of a Coxeter group and a weight function. A major breakthrough in the study of representations of Hecke algebras was achieved in 1979 when Kazhdan and Lusztig first introduced the notion of cells for an arbitrary Coxeter group. Each cell defines a representation of the Iwahori-Hecke algebra together with a canonical basis (the so-called Kazhdan-Lusztig basis) indexed by the elements of the cell. In the case where the weight function is constant on the generators of the corresponding Coxeter group (known as the equal parameter case), the Kazhdan-Lusztig basis can be interpreted in terms of intersection cohomology. Lustzig also proposed a conjectural geometric interpretation (extremely complex) for some specific choices of parameters. Inspired by some results obtained in those cases, Lusztig has formulated 15 conjectures which capture essential properties of cells for all choices of parameters. These properties can be used as an axiomatic basis for studying the structure and representations of Hecke algebras. In particular, a proof of these conjectures will bring the knowledge of the Kazhdan-Lusztig theory in the general case in the same level of completeness as in the equal parameter case. My Research. I am especially interested in Kazhdan-Lusztig theory in affine Weyl groups for an arbitrary weight function. A special feature of affine Weyl groups is that there is a distinguished two-sided cell, the so called lowest two-sided cell, which contains, roughly speaking, most of the elements of the group. This two-sided cell has been thoroughly studied in the equal parameter case (Shi, 1988) using the geometric interpretation of Kazhdan-Lusztig polynomials, which is not available in the general case of unequal parameters. In that case, some partial results have been obtained by Bremke and Xi. The main result of my paper On the lowest two-sided cell in affine Weyl groups brings the knowledge of this cell in the general case to the same level of completeness as in the equal parameter case. In particular, it provides alternative proofs of some results by Shi. In order to determine all cells, the first step is to find a decomposition into pieces such that each of them is included in a cell. To this end, in On the determination of Kazhdan-Lusztig cells for affine Weyl groups with unequal parameters, I proved that the Kazhdan-Lusztig polynomials are invariant under long enough translations. The second step is then to determine how the pieces of the above partition are grouped together into cells. Methods for dealing with this problem are developed in Generalized induction of Kazhdan-Lusztig cells. Under specific technical conditions on the parameters, I proved that the cells in a certain parabolic subgroup are in fact cells in the whole group. The efficiency of these methods is illustrated by the fact that I completely determine the partition into cells of all affine Weyl groups of rank 2 (see Kazhdan-Lusztig cells in

6 affine Weyl groups of rank 2 ) for any parameters. I also proposed a conjecture on the decomposition of an affine Weyl group into cells. The underlying idea of this conjecture is that, from the knowledge of Kazhdan Lusztig cells in the finite parabolic subgroups, one can recover the partition of the whole group using the process of induction of Kazhdan Lusztig cells. Recently S.König and C. Xi have introduced the notion of affine cellularity, generalising the notion of cellularity for finite dimensional algebra in the sense of Graham and Lehrer to infinite dimensional algebras. Together with V.Miemietz, we have proved that affine Hecke algebras of rank 2 are cellular for all generic choices of parameters (see Affine Hecke algebras of rank 2 are affine cellular ) using the knowledge of Kazhdan-Lusztig cells in this case. Previously, only the extended affine Hecke algebras of type A were known to be affine cellular (Konig and Xi) and our result provides the first example of a multi-parameter cellular affine Hecke algebra. Research project. As a natural continuation of my current research, I intend to keep working on the Kazhdan-Lusztig cells in affine Weyl groups but I also would like to to spend most of my time pursuing new directions. For instance, here are some questions I am working on: (1) The explicit partition of the affine Weyl groups of rank 2 into cells allows us to verify some of Lusztig s conjectures. I am undertaking a thorough study of these conjectures in this case. (2) The question of whether all affine Hecke algebras are affine cellular is wide open: as mentioned earlier König and C. Xi only established the cellular structure for affine Hecke algebras of type A. Further, their proofs relies on the use of the asymptotic algebra defined by Lusztig and the fact that we are in the equal parameter case plays a key role. Hence it would be very interesting to generalise the methods develop by V.Miemietz and myself in rank 2 to study this problem. (3) The past few years have seen the emergence of a new phenomenon in the study of Hecke algebras with unequal parameters: the semicontinuity phenomenon. This concept first appeared in 2007 in the work of M. Chlouveraki on Rouquier blocks in cyclotomic Hecke algebras. It also appeared in 2008 in the work of Opdam and Solleveld on representations of affine Hecke algebras where they introduced a continuous space of parameters with respect to which the harmonic analysis naturally deforms. Finally, also in 2008, C. Bonnafé conjectured that the partition into cells of an arbitrary Coxeter group changes in a semicontinuous manner as the parameters vary. In the work of Opdam and Solvelled some critical values of the parameters arise for each type of affine Hecke algebras. It turns out that in type G 2 these parameters are exactly those for which we see a change in the cell partition, that is the critical parameters according to Bonnafé s semicontinuity conjecture. This leads to the following exciting and fundamental problem: find a representation-theoretic interpretation, in terms of harmonic analysis, of the semicontinuity conjecture for Kazhdan Lusztig cells.