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Hardcover ISBN:  9780821833568 
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Hardcover ISBN:  9780821833568 
Product Code:  CRMM/18 
List Price:  $115.00 
MAA Member Price:  $103.50 
AMS Member Price:  $92.00 
eBook ISBN:  9781470438630 
Product Code:  CRMM/18.E 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 
Hardcover ISBN:  9780821833568 
eBook ISBN:  9781470438630 
Product Code:  CRMM/18.B 
List Price:  $225.00 $170.00 
MAA Member Price:  $202.50 $153.00 
AMS Member Price:  $180.00 $136.00 

Book DetailsCRM Monograph SeriesVolume: 18; 2003; 136 ppMSC: Primary 20
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.
ReadershipGraduate students and research mathematicians interested in group theory and generalizations.

Table of Contents

Chapters

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


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

Chapters

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