Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over \(p\)adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KLbasis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KLbasis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KLbasis to a more general class of Hecke algebras, the socalled algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives researchers and graduate students working in the theory of algebraic groups and their representations an invaluable insight and a wealth of new and useful information. Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students and research mathematicians interested in group theory and generalizations. Table of Contents  Introduction
 Coxeter groups
 Partial order on \(W\)
 The algebra \({\mathcal H}\)
 The bar operator
 The elements \(c_w\)
 Left or right multiplication by \(c_s\)
 Dihedral groups
 Cells
 Cosets of parabolic subgroups
 Inversion
 The longest element for a finite \(W\)
 Examples of elements \(D_w\)
 The function \(\mathbf{a}\)
 Conjectures
 Example: The split case
 Example: The quasisplit case
 Example: The infinite dihedral case
 The ring \(J\)
 Algebras with trace form
 The function \({\mathbf{a}}_E\)
 Study of a left cell
 Constructible representations
 Twosided cells
 Virtual cells
 Relative Coxeter groups
 Representations
 A new realization of Hecke algebras
 Bibliography
 Other titles in this series
