Ebook: Representations of SL2(Fq)
Author: Cédric Bonnafé (auth.)
- Tags: Algebra, Algebraic Geometry, Group Theory and Generalizations
- Series: Algebra and Applications 13
- Year: 2011
- Publisher: Springer-Verlag London
- Edition: 1
- Language: English
- pdf
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provide the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.
The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.
At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provide the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.
The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.
At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provide the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.
The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.
At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.
Content:
Front Matter....Pages I-XXII
Front Matter....Pages 1-2
Structure of $mathrm{SL}_{2}({mathbb{F}_{!q}})$ ....Pages 3-14
The Geometry of the Drinfeld Curve....Pages 15-25
Front Matter....Pages 27-28
Harish-Chandra Induction....Pages 29-35
Deligne-Lusztig Induction....Pages 37-50
The Character Table....Pages 51-58
Front Matter....Pages 59-61
More about Characters of G and of its Sylow Subgroups....Pages 63-69
Unequal Characteristic: Generalities....Pages 71-84
Unequal Characteristic: Equivalences of Categories....Pages 85-96
Unequal Characteristic: Simple Modules, Decomposition Matrices....Pages 97-108
Equal Characteristic....Pages 109-126
Front Matter....Pages 127-128
Special Cases....Pages 129-148
Deligne-Lusztig Theory: an Overview*....Pages 149-157
Back Matter....Pages 159-186
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provide the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.
The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.
At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.
Content:
Front Matter....Pages I-XXII
Front Matter....Pages 1-2
Structure of $mathrm{SL}_{2}({mathbb{F}_{!q}})$ ....Pages 3-14
The Geometry of the Drinfeld Curve....Pages 15-25
Front Matter....Pages 27-28
Harish-Chandra Induction....Pages 29-35
Deligne-Lusztig Induction....Pages 37-50
The Character Table....Pages 51-58
Front Matter....Pages 59-61
More about Characters of G and of its Sylow Subgroups....Pages 63-69
Unequal Characteristic: Generalities....Pages 71-84
Unequal Characteristic: Equivalences of Categories....Pages 85-96
Unequal Characteristic: Simple Modules, Decomposition Matrices....Pages 97-108
Equal Characteristic....Pages 109-126
Front Matter....Pages 127-128
Special Cases....Pages 129-148
Deligne-Lusztig Theory: an Overview*....Pages 149-157
Back Matter....Pages 159-186
....