By R. W. Carter (auth.), A. I. Kostrikin, I. R. Shafarevich (eds.)

The finite teams of Lie kind are of primary mathematical significance and the matter of knowing their irreducible representations is of serious curiosity. The illustration conception of those teams over an algebraically closed box of attribute 0 was once built by means of P.Deligne and G.Lusztig in 1976 and as a result in a sequence of papers by way of Lusztig culminating in his publication in 1984. the aim of the 1st a part of this e-book is to offer an outline of the topic, with no together with distinct proofs. the second one half is a survey of the constitution of finite-dimensional department algebras with many define proofs, giving the fundamental idea and techniques of development after which is going directly to a deeper research of department algebras over valuated fields. An account of the multiplicative constitution and diminished K-theory offers contemporary paintings at the topic, together with that of the authors. therefore it kinds a handy and extremely readable creation to a box which within the final twenty years has obvious a lot progress.

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**Algebra IX: Finite Groups of Lie Type Finite-Dimensional Division Algebras**

The finite teams of Lie variety are of crucial mathematical value and the matter of realizing their irreducible representations is of significant curiosity. The illustration idea of those teams over an algebraically closed box of attribute 0 used to be built through P. Deligne and G. Lusztig in 1976 and consequently in a chain of papers by way of Lusztig culminating in his ebook in 1984.

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**Example text**

G(x)O in this way. When U EGis unipotent the groups ~G(u)/~G(u)O have been determined by Alexeevski [1]. If G has type Al then ~G(u)/~G(u)O = 1. If G has type BI, CI, DI then ~du)/~G(u)O is isomorphic to 'lL2 x 'lL2 X ... X 'lL2 (e factors) for some e depending on u. If G has type G2, F4, E6, E7, E8 then ~G(u)/~G(u)O is isomorphic to the symmetric group Sn for some n ::;; 5 depending on u. If the Frobenius map acts trivially on the Dynkin diagram of G then every unipotent class of G is F-stable, although this is not always true when the Frobenius map acts non-trivially on the diagram.

The details can be found in Steinberg [3]. Thus all irreducible components of the Gelfand-Graev character r occur with multiplicity 1. In order to determine how many such components there are we consider the dual E = r*. E has a very simple form. It is the generalized character of GF which takes the value IZFlql on all the regular unipotent elements of GF and value 0 on all other elements of GF • (An element is regular if and only if its centralizer has dimension equal to the rank of G). Now the number of regular unipotent elements of GF can be shown to be IGFI/IZFlql.

In the first place we have 1* = St. Thus the dual of the principal character is the Steinberg character. It follows of course that St* = 1. Secondly we take a Deligne-Lusztig generalized character R T ,6 of GF • Then we have R},6 = BGBT R T ,6' This result was proved by Deligne and Lusztig in [2]. The dual of an irreducible character ~ of GF need not be irreducible. 6 The Gelfand-Graev Character of GF We shall continue to assume that the centre Z of G is connected. Let G* be the dual group of G and (G*)' be the semisimple part of G*.