zbMATH — the first resource for mathematics

On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory. (English) Zbl 1035.20004
Let \(V\) be a finite-dimensional complex vector space. A nilpotent linear map \(N\colon V\to V\) is said to fix a flag \(F=\{F_0\subset F_1\subset\cdots\subset F_{n-1}\subset F_n=V\}\) if \(NF_i\subseteq F_{i-1}\) for each \(i\), where \(F_i\) is a subspace of \(V\) of dimension \(i\). The variety \({\mathcal B}_N\) of all flags in \(V\) fixed by a nilpotent map \(N\) is called a Springer fiber. The topology of the components of \({\mathcal B}_N\) and their pairwise intersections was studied by a number of people such as N. Spaltenstein, J. A. Vargas, …. The present paper extends some of their results to describe the homological structure of components of \({\mathcal B}_N\) and their pairwise intersections in the cases where the nilpotent maps \(N\) correspond to the hook and two-row shape partitions. In these cases, the components are nonsingular, and are iterated bundles of flag manifolds and Grassmannians.
The author also relates his computations to the structure of the Kazhdan-Lusztig bases of certain representations of Iwahori-Hecke algebras of type \(A\). The suitable normalized inner products of these basis vectors are polynomials in \(t\) and \(t^{-1}\) that are invariant under the map \(t\mapsto t^{-1}\). The author shows that for irreducible representations labeled by a hook or two-row shape, the suitable normalized inner products equal the intersection homology Poincaré polynomials of pairwise intersections of irreducible components of Springer fibers of the general linear group.

20C08 Hecke algebras and their representations
20G05 Representation theory for linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
Full Text: DOI arXiv
[1] A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, in: Analyse et topologie sur les espaces singuliers I (Luminy, 1981), Astérisque, Vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171.
[2] Borho, W.; Brylinski, J.-L.; MacPherson, R., Primitive ideals, nilpotent orbits, and characteristic classes, (1989), Birkhäuser Boston · Zbl 0697.17006
[3] W. Borho, R. MacPherson, Partial resolutions of nilpotent varieties, in: Analyse et topologie sur les espaces singuliers, II, III (Luminy, 1981), Astérisque, Vols. 101-102, Soc. Math. France, Paris, 1983, pp. 23-74.
[4] Chriss, N.; Ginzburg, V., Representation theory and complex geometry, (1996), Birkhäuser New York
[5] C.W. Curtis, Representations of Hecke algebras, in: Orbites unipotents et représentations, Astérisque, Vol. 168, Soc. Math. France, pp. 13-60.
[6] Curtis, C.W.; Iwahori, N.; Kilmoyer, R., Hecke algebras and characters of parabolic type of finite groups with (B,N)-pairs, Inst. hautes études sci. publ. math., 40, 81-116, (1971) · Zbl 0254.20004
[7] Dipper, R.; James, G., Representations of Hecke algebras of general linear groups, Proc. London math. soc. (3), 52, 1, 20-52, (1986) · Zbl 0587.20007
[8] J. Matthew Douglass, W-graphs and irreducible components of the flag variety, preprint. · Zbl 0895.20036
[9] Garsia, A.M.; McLarnan, T.J., Relations between Young’s natural and the Kazhdan-Lusztig representations of Sn, Adv. in math., 69, 1, 32-92, (1988) · Zbl 0657.20014
[10] Goresky, M.; MacPherson, R., Intersection homology theory, Topology, 19, 2, 135-162, (1980) · Zbl 0448.55004
[11] M. Goresky, R. MacPherson, On the topology of complex algebraic maps, Algebraic geometry (La Rábida, 1981), Lecture Notes in Mathematics, Vol. 961, Springer, Berlin-New York, 1982, pp. 119-129.
[12] Goresky, M.; MacPherson, R., Intersection homology II, Invent. math., 72, 1, 77-129, (1983) · Zbl 0529.55007
[13] Graham, J.J.; Lehrer, G.I., Cellular algebras, Invent. math., 123, 1, 1-34, (1996) · Zbl 0853.20029
[14] J.J. Güemes, On the homology classes for the components of some fibres of Springer’s resolution, in: Orbites Unipotentes et représentations III, Astérisque, Vols. 173-174, Soc. Math. France, Paris, 1989, pp. 257-269.
[15] Hotta, R., On Joseph’s construction of Weyl group representations, Tôhoku math. J. (2), 36, 1, 49-74, (1984) · Zbl 0545.20029
[16] Humphreys, J.E., Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics, Vol. 29, (1990), Cambridge University Press Cambridge · Zbl 0725.20028
[17] Kashiwara, M.; Saito, Y., Geometric construction of crystal bases, Duke math. J., 89, 1, 9-36, (1997) · Zbl 0901.17006
[18] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. math., 53, 2, 165-184, (1979) · Zbl 0499.20035
[19] D. Kazhdan, G. Lusztig, Schubert varieties and Poincaré duality, in: Geometry of the Laplace operator, Proceedings of the Symposium on Pure Mathematics, Vol. XXXVI, Amer. Math. Soc., Providence, RI, 1980, pp. 185-203.
[20] A. Lascoux, M.-P. Schützenberger, Polynômes de Kazhdan et Lusztig pour les grassmanniennes, in: Tableaux de Young et functors de Schur en algèbre et géométrie (Toruń, 1980), Astérisque, Vol. 87-88, Soc. Math. France, Paris, 1981, pp. 249-266.
[21] Lorist, P., The geometry of \(Bx\), Proc. nederl. akad. wetensch. A, 89, 4, 423-442, (1986) · Zbl 0646.14035
[22] Murphy, G.E., The representations of Hecke algebras of type An, J. algebra, 173, 97-121, (1995) · Zbl 0829.20022
[23] Ochiai, M.; Kako, F., Computational construction of W-graphs of Hecke algebras H(q,n) for n up to 15, Exp. math., 4, 1, 61-67, (1995) · Zbl 0846.20016
[24] Spaltenstein, N., The fixed point set of a unipotent transformation of the flag manifold, Proc. kon. ak. wet. Amsterdam, 79, 5, 452-456, (1976) · Zbl 0343.20029
[25] Springer, T.A., Trigonometric sums, Green functions of finite groups, and representations of Weyl groups, Invent. math., 36, 173-207, (1976) · Zbl 0374.20054
[26] T.A. Springer, Quelques applications de la cohomologie d’intersection, in: Séminaire Bourbaki, Vol. 1981/1982, Astérisque, Vols. 92-93, Soc. Math. France, Paris, 1982, pp. 249-273.
[27] T.A. Springer, On representations of Weyl groups, in: Proceedings of the Hyderabad Conference on Algebraic Groups, Manoj Prakashan Press, Madras, 1991, pp. 517-536. · Zbl 0809.20031
[28] Vargas, J.A., Fixed points under the action of unipotent elements of SLn in the flag variety, Boll. soc. math. mex., 24, 1, 1-14, (1979) · Zbl 0458.14019
[29] Westbury, B., The representation theory of the Temperley-Lieb algebras, Math. Z., 219, 4, 539-565, (1995) · Zbl 0840.16008
[30] Wolper, J., Some intersection properties of the fibres of Springer’s resolution, Proc. amer. math. soc., 91, 2, 182-188, (1984) · Zbl 0558.20026
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.