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Character sheaves. IV. (English) Zbl 0602.20035
The paper under review is part of a series [see Adv. Math. 56, 193-237 (1985; Zbl 0586.20018); 57, 226-265, 266-315 (1985; Zbl 0586.20019 and Zbl 0594.20031)] devoted to the study of a class \(\hat G\) of irreducible perverse sheaves (called character sheaves) on a connected reductive algebraic group G. The numbering of chapters, sections and references continues that of the previous parts.
This paper contains a classification of character sheaves of G assuming that G is almost simple of type A (ch. 18), or some classical group of low rank (ch. 19), or a group of type \(E_ 6\), \(E_ 7\), \(G_ 2\) at least under certain restrictions on char(k) of the ground field k (ch. 20), or a group of type \(E_ 8\), \(F_ 4\) and char(k) is good (ch. 21). It is proved that such G are clean and satisfy the parity condition [see part III Points (13.9.2) and (15.3)] and that the class of character sheaves coincides with the class of admissible complexes [see the author, Invent. Math. 75, 205-272 (1984; Zbl 0547.20032)] and that a multiplicity formula holds rather analogous to the main theorem (Point (4.23)) of the book of the author [Characters of reductive groups over a finite field (Ann. Math. Stud. 107) (1984; Zbl 0556.20033)].
Reviewer: N.I.Osetinski

MSC:
20G05 Representation theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
14L30 Group actions on varieties or schemes (quotients)
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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