zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Deligne, P; Lusztig, G, Representations of reductive groups over finite fields, Ann. of math., 103, 103-161, (1976) · Zbl 0336.20029
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.