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Representations of \(\mathsf{SL}_2(\mathbb F_q)\). (English) Zbl 1203.22001

Algebra and Application 13. London: Springer (ISBN 978-0-85729-156-1/hbk; 978-1-4471-2599-0/pbk; 978-0-85729-157-8/ebook). xxii, 186 p. (2011).
The Deligne-Lusztig theory aims to study representations of finite reductive groups by geometric methods, particularly, using \(\ell\)-adic cohomology. Many excellent texts present, with different perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely, the group \(\mathsf{SL}_2(\mathbb F_q)\) of 2 by 2 matrices with determinant 1 over a field of \(q\) elements. The Deligne-Lusztig theory was inspired by Drinfeld’s example of 1974, and the book is based upon this example and extends it to modular representations. In order to efficiently use \(\ell\)-adic cohomology, a precise study of the geometric properties of the action of \(\mathsf{SL}_2(\mathbb F_q)\) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.

MSC:

22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
22E50 Representations of Lie and linear algebraic groups over local fields
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