Character sheaves. IV.

*(English)*Zbl 0602.20035The 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)].

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 |

##### Keywords:

irreducible perverse sheaves; character sheaves; connected reductive algebraic group; admissible complexes; multiplicity formula##### References:

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[2] | Lusztig, G, Intersection cohomology complexes on a reductive group, Invent. math., 75, 205-272, (1984) · Zbl 0547.20032 |

[3] | Lusztig, G, Character sheaves, I, Adv. in math., 56, 193-237, (1985) · Zbl 0586.20018 |

[4] | Lusztig, G, Characters of reductive groups over a finite field, () · Zbl 0930.20041 |

[5] | \scG. Lusztig, Character sheaves, II, Adv. in Math., in press. · Zbl 0586.20019 |

[6] | Lusztig, G, Coxeter orbits and eigenspaces of Frobenius, Invent. math., 38, 101-159, (1976) · Zbl 0366.20031 |

[7] | \scG. Lusztig, Character sheaves, III, Adv. in Math., in press. · Zbl 1229.20041 |

[8] | Steinberg, R, Regular elements in semisimple algebraic groups, Publ. math. IHES, 25, 49-80, (1965) · Zbl 0136.30002 |

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