Character sheaves. V.

*(English)*Zbl 0602.20036The 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 \(G(F_ q)\).

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 |

##### Keywords:

irreducible perverse sheaves; character sheaves; connected reductive algebraic group; multiplicity formula; intersection cohomology; unipotent class; characteristic functions; class functions
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##### References:

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