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  • Print publication year: 2020
  • Online publication date: February 2020

1 - Reductive Groups and Steinberg Maps

Summary

The purpose of Chapter 1 is to collect some basic results about algebraic groups (with proofs where appropriate) which will be needed for the discussion of characters and applications in later chapters. In particular, one of our aims is to arrive at the point where we can give a precise definition of a `series of finite groups of Lie type' ${G(q)}$, Indexed by a parameter $q$. We also introduce a number of tools which will be helpful in the discussion of examples. For a reader familiar with the basic notions about algebraic groups, root data and Frobenius maps, it might just be sufficient to browse through this chapter on a first reading, in order to see some of our notation. A central role is played by the 'isogeny theorem' which is illustrated with numerous examples, including a quite thorough discussion of Frobenius and Steinberg maps. The final section discusses in some detail the first applications to the character theory of finite groups of Lie type: the `Multiplicity--Freeness' Theorem.

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