**CRM Monograph Series**

Volume: 18;
2003;
136 pp;
Hardcover

MSC: Primary 20;

Print ISBN: 978-0-8218-3356-8

Product Code: CRMM/18

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

**Electronic ISBN: 978-1-4704-3863-0
Product Code: CRMM/18.E**

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

# Hecke Algebras with Unequal Parameters

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*G. Lusztig*

A co-publication of the AMS and Centre de Recherches Mathématiques

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.

#### Readership

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

#### Table of Contents

# Table of Contents

## Hecke Algebras with Unequal Parameters

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction 18
- Coxeter groups 916
- Partial order on ๐ 1926
- The algebra โ 2330
- The bar operator 2734
- The elements ๐_{๐ค} 3138
- Left or right multiplication by ๐_{๐ } 3542
- Dihedral groups 3744
- Cells 4350
- Cosets of parabolic subgroups 4754
- Inversion 5360
- The longest element for a finite ๐ 5764
- Examples of elements ๐ท_{๐ค} 6168
- The function ๐ 6370
- Conjectures 6774
- Example: The split case 7380
- Example: The quasisplit case 7784
- Example: The infinite dihedral case 8188
- The ring ๐ฝ 8592
- Algebras with trace form 9198
- The function ๐_{๐} 93100
- Study of a left cell 101108
- Constructible representations 105112
- Two-sided cells 117124
- Virtual cells 121128
- Relative Coxeter groups 125132
- Representations 127134
- A new realization of Hecke algebras 129136
- Bibliography 135142
- Back Cover Back Cover1145