zbMATH — the first resource for mathematics

Intersection cohomology complexes on a reductive group. (English) Zbl 0547.20032
Let G be a reductive connected algebraic group over an algebraically closed field, and let u be a unipotent element of G. Let \(A_ G(u)\) be the group of components of the centralizer \(Z_ G(u)\). \(A_ G(u)\) acts naturally by permutations on the set of irreducible components of the variety of Borel subgroups containing u and Springer has shown that (with some restrictions on the characteristic) the irreducible representations of \(A_ G(u)\) appearing in this permutation representation for various u (up to conjugacy) are in 1-1 correspondence with the irreducible representations of the Weyl group. However, in general, not all irreducible representations of \(A_ G(u)\) appear in this permutation representation. In this paper, the author investigates the missing representations.
Let P be a parabolic subgroup of G with Levi decomposition \(P=LU_ P\), and let v be a unipotent element in L. Let \(Y_{u,v}=\{gZ^ 0_ L(v)U_ p|\quad g\in G,\quad g^{-1}ug\in vU_ P\}.\) Then \(\dim Y_{u,v}\leq d=1/2(\dim Z_ G(u)-\dim Z_ L(v)).\) The group \(Z_ G(u)\) acts naturally on \(Y_{u,v}\) by left translation. This induces an action of the finite group \(A_ G(u)\) on the finite set \(S_{u,v}\) of irreducible components of dimension d of \(Y_{u,v}\). When P is a Borel subgroup and \(v=1\), this is just the action considered by Springer. An irreducible representation of \(A_ G(u)\) is said to be cuspidal if it does not appear in the permutation representation on \(S_{u,v}\) for any \(P\), \(v\) as above with \(P\neq G\). The author shows that very few representations of \(A_ G(u)\) are cuspidal. More precisely, for a fixed character \(\chi\) of the group \(\Gamma\) of components of the center of G, and for a field k of good characteristic, there is at most one pair \((u,\rho)\) with u unipotent in G (up to conjugacy) such that \(\rho\) is an irreducible cuspidal representation of \(A_ G(u)\) on which \(\Gamma\) acts according to \(\chi\). Given a pair \((u,\rho)\), the author defines a triple \((L,v,\rho')\) up to conjugacy, where \(L\) is the Levi subgroup of a parabolic subgroup of \(G\), \(v\) a unipotent element in \(L\), and \(\rho'\) is a cuspidal representation of \(A_ L(v)\), and he shows that the set of pairs \((u,\rho)\) giving rise to a fixed triple \((L,v,\rho')\) as above may be naturally put into 1-1 correspondence with the set of irreducible representations of the group of components of the normalizer of \(L\) which is shown to be a Coxeter group. It reduces to the correspondence described originally by Springer, in the case where \(L\) is a maximal torus, and it is called generalized Springer correspondence. The author determines in a combinatorial way this correspondence in the case of symplectic and special orthogonal groups in odd characteristic. This generalizes the main result of Shoji on the usual Springer correspondence for these groups. Throughout the paper, the intersection cohomology theory of Deligne-Goresky-MacPherson is used extensively. Using the result of this paper, by Spaltenstein, the correspondence for exceptional groups in arbitrary characteristic has also been determined explicitly in almost all cases.
Reviewer: E.Abe

20G05 Representation theory for linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F99 (Co)homology theory in algebraic geometry
Full Text: DOI EuDML
[1] Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analyse et topologie sur les espaces singuliers, I. Astérisque, vol. 100 (1982), Société Mathématique de France · Zbl 0536.14011
[2] Borho, W., MacPherson, R.: Representations des groupes de Weyl et homologie d’intersection pour les varietes nilpotentes, Comptes rendus. Acad. Sci. Paris t292, 707-710 (1981) · Zbl 0467.20036
[3] Borho, W., MacPherson, R.: Partial resolutions of nilpotent varieties. In: Analyse et topologie sur les espaces singuliers, II. Astérisque, vol. 101-102 (1983), Société Mathématique de France · Zbl 0576.14046
[4] Goresky, M., MacPherson, R.: Intersection homology, II, Invent. Math.72, 77-129 (1983) · Zbl 0529.55007
[5] Hardy, G.H., Wright, E.M.: The theory of numbers. Oxford: Clarendon Press 1971 · Zbl 0020.29201
[6] Lusztig, G.: Irreducible representations of finite classical groups. Invent. Math.43, 125-175 (1977) · Zbl 0372.20033
[7] Lusztig, G.: Coxeter orbits and eigensspaces of Frobenius. Invent Math.38, 101-159 (1976) · Zbl 0366.20031
[8] Lusztig, G.: On the finiteness of the number of unipotent classes. Inv. Math.34, 201-213 (1976) · Zbl 0371.20039
[9] Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. in Math.42, 169-178 (1981) · Zbl 0473.20029
[10] Lusztig, G.: Characters of reductive groups over a finite field. Ann. of Math. Studies, to appear · Zbl 0556.20033
[11] Lusztig, G., Spaltenstein, N.: Induced unipotent classes. J. London. Math. Soc.19, 41-52 (1979) · Zbl 0407.20035
[12] Mizuno, K.: The conjugate classes of Chevalley groups of typeE 6. J. Fac. Sci. Univ. Tokyo24, 525-563 (1977) · Zbl 0399.20044
[13] Mizuno, K.: The conjugate classes of unipotent elements of the Chevalley groups,E 7 andE 8. Tokyo J. Math.3, 391-461 (1980) · Zbl 0454.20046
[14] Shinoda, K.: The conjugacy classes of Chevalley groups of typeF 4 over finite fields of characteristic 2. J. Fac. Sci. Univ. Tokyo21, 133-159 (1974) · Zbl 0306.20013
[15] Shoji, T.: On the Springer representations of the Weyl groups of classical algebraic groups. Comm. in Alg.7, 1713-1745, 2027-2033 (1979) · Zbl 0423.20042
[16] Shoji, T.: On the Springer representations of Chevalley groups of typeF 4. Comm. in Alg.8, 409-440 (1980) · Zbl 0434.20026
[17] Spaltenstein, N.: Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic, preprint · Zbl 0575.17007
[18] Spaltenstein, N.: Classes unipotentes et sous-groupes de Borel. Lecture Notes in Mathematics, vol. 946. Berlin-Heidelberg-New York: Springer · Zbl 0486.20025
[19] Spaltenstein, N.: Appendix. Math. Proc. Camb. Phil. Soc.92, 73-78 (1982)
[20] Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math.36, 173-207 (1976) · Zbl 0374.20054
[21] Springer, T.A.: A construction of representations of Weyl groups. Invent. Math.44, 279-293 (1978) · Zbl 0376.17002
[22] Springer, T.A., Steinberg, R.: Conjugacy classes. In: Borel, A., et al.: Seminar on algebraic groups and related finite groups. Lecture Notes in Mathematics, Vol. 131. Berlin-Heidelberg-New York: Springer
[23] Steinberg, R.: Regular elements of semisimple algebraic groups. Publ. Math., I.H.E.S., no.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.