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Representations of Chevalley groups in their own characteristic. (English) Zbl 0652.20042

Representations of finite groups, Proc. Conf., Arcata/Calif. 1986, Pt. 1, Proc. Symp. Pure Math. 47, 127-146 (1987).
[For the entire collection see Zbl 0627.00009.]
The author gives a survey of the p-modular representation theory of finite Chevalley groups in characteristic p. Let G be a connected, simply connected, semisimple algebraic group defined and split over \(F_ p\). The representations that are studied are those of the finite groups \(G(p^ n)\) of \(F_{p^ n}\)-rational points of G over an algebraic closure k of \(F_ p\). Let T be a maximal torus of G defined and split over \(F_ p\) and \(X(T)_+\) the set of dominant weights with respect to T. For each \(\lambda \in (T)_+\) one has a simple G-module L(\(\lambda)\) with highest weight \(\lambda\) and character ch L(\(\lambda)\) and a “Weyl module” obtained by reduction mod p from a simple module for a corresponding group in characteristic 0, with character \(\chi\) (\(\lambda)\). One of the main open problems in the theory is to describe the multiplicities of the ch L(\(\mu)\) \((\mu \in X(T)_+)\) in the \(\chi\) (\(\lambda)\). [A solution is given by Lusztig’s conjecture, which is not stated here.]
In the first section of this paper the author describes some results, e.g. on the composition factors of the tensor product of two simple \(kG(p^ n)\)-modules, which would follow from a knowledge of these multiplicities. The principal indecomposable modules \(U_ n(\lambda)\) corresponding to the L(\(\lambda)\) form the subject of the second section. The author describes how the Cartan invariants and the decomposition matrix of \(G(p^ n)\) can be obtained, again assuming a knowledge of the multiplicities as above. The third section deals with the reduction mod p of the Deligne-Lusztig virtual characters (in characteristic 0) of \(G(p^ n)\). The author has emphasized parts of the theory which have not been treated in earlier surveys. There are useful appendices on (i) a different approach to the study of simple modules, due to Kempf, (ii) the work of Chastkofsky on the behavior of dim \(U_ n(0)\) as a function of n, and (iii) explicit calculations for \(G=Sp_ 4\).
Reviewer: B.Srinivasan

MSC:

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C20 Modular representations and characters

Citations:

Zbl 0627.00009