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Character sheaves. V. (English) Zbl 0602.20036
The reviewed paper is part 5 of a series [for part IV see the preceding review] devoted to the study of a class $$\hat G$$ of irreducible perverse sheaves (called character sheaves) on a connected reductive algebraic group G over an algebraically closed field k. The numbering of sections, subsections and references continues that of the earlier parts.
Most results of this paper hold under the following restrictions on the characteristic p of k: if $$p=5$$, then G has no factors of type $$E_ 8$$; if $$p=3$$, then G has no factors of type $$E_ 7$$, $$E_ 8$$, $$F_ 4$$, $$G_ 2$$; if $$p=2$$, then G has no factors of type $$E_ 6$$, $$E_ 7$$, $$E_ 8$$, $$F_ 4$$, $$G_ 2.$$
One of the main results of the paper (theorem 23.1) gives a classification of the character sheaves of G on which the group of components of the centre acts faithfully. There are four sections in the paper (from 22 to 25). Section 22 contains results for the classical groups in characteristic 2. Section 23 shows the classification of character sheaves and a multiplicity formula. Local intersection cohomology with twisted coefficients of the closure of a unipotent class are contained in Section 24. In Section 25 it is shown that in the case when G is defined over $$F_ q$$ $$(k=\bar F_ q)$$, the characteristic functions of the character sheaves A which are themselves defined over $$F_ q$$, form an orthonormal basis of the space of class functions on $$G(F_ q)$$.
Reviewer: N.I.Osetinski

##### MSC:
 20G05 Representation theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients) 20G15 Linear algebraic groups over arbitrary fields 14F30 $$p$$-adic cohomology, crystalline cohomology 14L40 Other algebraic groups (geometric aspects) 20G40 Linear algebraic groups over finite fields 20G10 Cohomology theory for linear algebraic groups
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