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Hecke Algebras with Unequal Parameters
G. Lusztig, Massachusetts Institute of Technology, Cambridge, MA
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
2003; 136 pp; hardcover
Volume: 18
ISBN-10: 0-8218-3356-1
ISBN-13: 978-0-8218-3356-8
List Price: US$43 Member Price: US$34.40
Order Code: CRMM/18

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 KL-basis) 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 KL-basis also possess very interesting combinatorial properties.

In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called 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 co-published with the Centre de Recherches Mathématiques.

Graduate students and research mathematicians interested in group theory and generalizations.

• 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
• Two-sided cells
• Virtual cells
• Relative Coxeter groups
• Representations
• A new realization of Hecke algebras
• Bibliography
• Other titles in this series