×

zbMATH — the first resource for mathematics

Green functions of finite Chevalley groups of type \(E_ n (n=6,7,8)\). (English) Zbl 0539.20025
Let k be an algebraic closure of a finite field \(F_ q\). Let G be a connected reductive algebraic group over k defined over \(F_ q\) with Frobenius endomorphism F:\(G\to G\). In the paper Green functions [see P. Deligne and G. Lusztig, Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] are calculated for the finite group \(G^ F=\{g\in G| \quad F(g)=g\},\) when G is of the type \(E_ n\), \(n=6,7,8\). One of the applications of these calculations gives the results of K. Mizuno [see J. Fac. Sci., Univ. Tokyo, Sect. I A 24, 525-563 (1977; Zbl 0399.20044)] on the order relation for unipotent orbits in G given by inclusion of Zariski closures.
Reviewer: A.G.Elashvili

MSC:
20G40 Linear algebraic groups over finite fields
20G15 Linear algebraic groups over arbitrary fields
PDF BibTeX Cite
Full Text: DOI
References:
[1] Alekseevsky, A.V, Component groups of centralizer for unipotent elements in semisimple algebraic groups, Trudy tbiliss. mat. inst. razmadze akad. nauk gruzin. SSR, 62, 5-27, (1979)
[2] Alvis, D; Lusztig, G, On Springer’s correspondence for simple groups of type En (n = 6, 7, 8), (), 65-72
[3] Bala, P; Carter, R.W; Bala, P; Carter, R.W, Classes of unipotent elements in simple algebraic groups, (), 1-18 · Zbl 0364.22007
[4] Beynon, W.M; Lusztig, G, Some numerical results on the characters of exceptional Weyl groups, (), 417-426 · Zbl 0416.20033
[5] {\scW. M. Beynon and N. Spaltenstein}, “Computation of the Green Functions of Simple Groups of Type En (n = 6,7,8),” Computer Centre Report, no. 23, University of Warwick, England. · Zbl 0539.20025
[6] Borho, W; MacPherson, R, Représentations des groupes de Weyl et homologie d’intersection pour LES variétés de nilpotents, C. R. acad. sci. Paris Sér. I, 292, 15, 707-710, (1981) · Zbl 0467.20036
[7] Deligne, P; Lusztig, G, Representations of reductive groups over finite fields, Ann. of math., 103, 103-161, (1976) · Zbl 0336.20029
[8] Elashvili, A.G, The centralizers of nilpotent elements in the semisimple Lie algebras, Trudy tbiliss. mat. inst. razmadze akad. nauk gruzin. SSR, 46, 109-132, (1975) · Zbl 0323.17004
[9] Frame, J.S, The classes and representations of the groups of 27 lines and 28 bitangents, Ann. mat. pura. appl., 32, 83-119, (1951), (4) · Zbl 0045.00505
[10] Frame, J.S, The characters of the Weyl group E8, (), 111-130
[11] Goreski, M; MacPherson, R, Intersection homology theory, II, Invent. math., 71, 77-129, (1983)
[12] Hotta, R, On Springer’s representations, J. fac. sci. univ. Tokyo sect. IA math., 28, 863-876, (1982) · Zbl 0584.20033
[13] Hotta, R; Springer, T.A, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of the unitary groups, Invent. math., 41, 113-127, (1977) · Zbl 0389.20037
[14] Kazhdan, D, Proof of Springer’s hypothesis, Israel J. math., 28, 272-286, (1977) · Zbl 0391.22006
[15] {\scT. A. Springer}, A purity result for fixed point sets in the flag manifold, J. Fac. Sci. Univ. Tokyo, to appear. · Zbl 0581.20048
[16] Lusztig, G, Green polynomials and singularities of unipotent classes, Advan. in math., 42, 169-178, (1981) · Zbl 0473.20029
[17] Lusztig, G; Spaltenstein, N, Induced unipotent classes, J. London math. soc., 19, 41-52, (1979) · Zbl 0407.20035
[18] Mizuno, K, The conjugate classes of Chevalley groups of type E6, J. fac. sci. univ. Tokyo sect. IA math., 24, 525-563, (1977) · Zbl 0399.20044
[19] Mizuno, K, The conjugate classes of unipotent elements of the Chevalley groups E7 and E8, Tokyo J. math., 3, 391-461, (1980)
[20] Shoji, T, On the Green polynomials of a Chevalley group of type F4, Comm. algebra, 10, 505-543, (1982) · Zbl 0485.20031
[21] Spaltenstein, N, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology, 16, 203-204, (1977) · Zbl 0445.20021
[22] Spaltenstein, N, Appendix to on Springer’s correspondence for simple groups of type En (n = 6, 7, 8), (), 73-78
[23] Springer, T.A, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. math., 36, 173-207, (1976) · Zbl 0374.20054
[24] Srinivasan, B, Representations of finite Chevalley groups, () · Zbl 0434.20022
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.