Kazhdan-Lusztig Cells with Unequal Parameters

This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case.

Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig's seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group.

Kazhdan-Lusztig Cells with Unequal Parameters will appeal to graduate students and researchers interested in related subject areas, such as Lie theory, representation theory, and combinatorics of Coxeter groups. Useful examples and various exercises make this book suitable for self-study and use alongside lecture courses.

Information for readers: The character {\mathbb{Z}} has been corrupted in the print edition of this book and appears incorrectly with a diagonal line running through the symbol.

Provides a self-contained introduction to Kazhdan-Lusztig cells Includes figures of the partition into cells for small finite, affine, or hyperbolic Coxeter groups Explains Geck and Guilhot induction results, as well as the action of the cactus group Reviews and adds substantial results to an active field of research

Auteur
Cédric Bonnafé is an expert in representation theory of finite reductive groups and related objects (such as Hecke algebras or rational Cherednik algebras). He is the author of several papers on the Kazhdan-Lusztig theory of cells.

Contenu
Part I Preliminaries.- 1 Preorders on Bases of Algebras.- 2 Lusztig's Lemma.- Part II Coxeter Systems, Hecke Algebras.- 3 Coxeter Systems.- 4 Hecke Algebras.- Part III KazhdanLusztig Cells.- 5 The KazhdanLusztig Basis.- 6 KazhdanLusztig Cells.- 7 Semicontinuity.- Part IV General Properties of Cells.- 8 Cells and Parabolic Subgroups.- 9 Descent Sets, Knuth Relations and Vogan Classes.- 10 The Longest Element and Duality in Finite Coxeter Groups.- 11 The Guilhot Induction Process.- 12 Submaximal Cells (Split Case).- 13 Submaximal Cells (General Case).- Part V Lusztig's a-Function.- 14 Lusztig's Conjectures.- 15 Split and quasi-split cases.- Part VI Applications of Lusztig's Conjectures.- 16 Miscellanea.- 17 Multiplication by Tw0.- 18 Action of the Cactus Group.- 19 Asymptotic Algebra.- 20 Automorphisms.- Part VII Examples.- 21 Finite Dihedral Groups.- 22 The Symmetric Group.- 23 Affine Weyl Groups of Type A2.- 24 Free Coxeter Groups.- 25 Rank 3.- 26 Some Bibliographical Comments.- Appendices.- A Symmetric Algebras.- B Reflection Subgroups of Coxeter Groups.- References.- Index.