Smooth components of Springer fibers

Graham, William; Zierau, R.

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2139-2182.

Cet article étudie les composantes des fibres de Springer pour $\mathrm{𝔤𝔩}\left(n\right)$ qui sont associées à des orbites fermées de $GL\left(p\right)\times GL\left(q\right)$ dans la variété de drapeaux de $GL\left(n\right),n=p+q$. Ces composantes apparaîssent dans toute fibre de Springer. En contraste avec le cas de composantes arbitraires, ces composantes sont des variétés lisses. En utilisant des résultats de Barchini et Zierau, nous montrons que ces composantes sont des fibrés itérés et sont stables sous l’action d’un tore maximal de $GL\left(n\right)$. Nous démontrons que si $ℒ$ est un fibré en droites sur la variété de drapeaux associée à un poids dominant, alors les groupes de cohomologie de degré supérieur de la restriction de $ℒ$ à ces composantes s’annulent. Nous déduisons quelques conséquences des théorèmes de localisation en cohomologie équivariante et $K$-théorie, appliqués à ces composantes. Dans l’appendice, nous indentifions les tableaux correspondants à ces composantes, via la correspondance bijective entre les composantes des fibres de Springer pour $GL\left(n\right)$ et les tableaux standard.

This article studies components of Springer fibers for $\mathrm{𝔤𝔩}\left(n\right)$ that are associated to closed orbits of $GL\left(p\right)\times GL\left(q\right)$ on the flag variety of $GL\left(n\right),n=p+q$. These components occur in any Springer fiber. In contrast to the case of arbitrary components, these components are smooth varieties. Using results of Barchini and Zierau we show these components are iterated bundles and are stable under the action of a maximal torus of $GL\left(n\right)$. We prove that if $ℒ$ is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of $ℒ$ to these components vanish. We derive some consequences of localization theorems in equivariant cohomology and $K$-theory, applied to these components. In the appendix we identify the tableaux corresponding to these components, under the bijective correspondence between components of Springer fibers for $GL\left(n\right)$ and standard tableaux.

[1] Atiyah, M. F.; Bott, R. The moment map and equivariant cohomology, Topology, Volume 23 (1984) no. 1, pp. 1-28 | DOI | MR | Zbl

[2] Barchini, L.; Zierau, R. Certain components of Springer fibers and associated cycles for discrete series representations of $\mathrm{SU}(p,q)$, Represent. Theory, Volume 12 (2008), pp. 403-434 (With an appendix by Peter E. Trapa) | DOI | MR | Zbl

[3] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991 | MR | Zbl

[4] Brion, Michel Equivariant cohomology and equivariant intersection theory, Representation theories and algebraic geometry (Montreal, PQ, 1997) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 1-37 (Notes by Alvaro Rittatore) | MR | Zbl

[5] Chang, Jen-Tseh Characteristic cycles of discrete series for $\mathbf{R}$-rank one groups, Trans. Amer. Math. Soc., Volume 341 (1994) no. 2, pp. 603-622 | DOI | MR | Zbl

[6] Chriss, Neil; Ginzburg, Victor Representation theory and complex geometry, Birkhäuser Boston Inc., Boston, MA, 1997 | MR | Zbl

[7] De Concini, C.; Lusztig, G.; Procesi, C. Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc., Volume 1 (1988) no. 1, pp. 15-34 | DOI | MR | Zbl

[8] Edidin, Dan; Graham, William Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math., Volume 120 (1998) no. 3, pp. 619-636 | DOI | MR | Zbl

[9] Fresse, Lucas Betti numbers of Springer fibers in type $A$, J. Algebra, Volume 322 (2009) no. 7, pp. 2566-2579 | DOI | MR | Zbl

[10] Fresse, Lucas Singular components of Springer fibers in the two-column case, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 6, pp. 2429-2444 | DOI | Numdam | MR | Zbl

[11] Fresse, Lucas A unified approach on Springer fibers in the hook, two-row and two-column cases, Transform. Groups, Volume 15 (2010) no. 2, pp. 285-331 | DOI | MR

[12] Fung, Francis Y. C. On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory, Adv. Math., Volume 178 (2003) no. 2, pp. 244-276 | DOI | MR | Zbl

[13] Garfinkle, Devra The annihilators of irreducible Harish-Chandra modules for $\mathrm{SU}(p,q)$ and other type $A_{n-1}$ groups, Amer. J. Math., Volume 115 (1993) no. 2, pp. 305-369 | DOI | MR | Zbl

[14] Graham, William Equivariant $K$-theory and Schubert varieties (preprint)

[15] Graham, William Positivity in equivariant Schubert calculus, Duke Math. J., Volume 109 (2001) no. 3, pp. 599-614 | DOI | MR | Zbl

[16] Graham, William; Kumar, Shrawan On positivity in $T$-equivariant $K$-theory of flag varieties, Int. Math. Res. Not. IMRN (2008), pp. Art. ID rnn 093, 43 | MR | Zbl

[17] Grothendieck, A. Séminaire de géométrie algébrique. Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, 224, Springer-Verlag, Heidelberg, 1971 | MR

[18] Güemes, J. J. On the homology classes for the components of some fibres of Springer’s resolution, Astérisque (1989) no. 173-174, pp. 257-269 | Numdam | MR | Zbl

[19] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[20] Iversen, Birger A fixed point formula for action of tori on algebraic varieties, Invent. Math., Volume 16 (1972), pp. 229-236 | DOI | MR | Zbl

[21] Knutson, Allen Schubert patches degenerate to subword complexes, Transform. Groups, Volume 13 (2008) no. 3-4, pp. 715-726 | DOI | MR | Zbl

[22] Kumar, Shrawan Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser Boston Inc., Boston, MA, 2002 | MR | Zbl

[23] Lusztig, G. Green polynomials and singularities of unipotent classes, Adv. in Math., Volume 42 (1981) no. 2, pp. 169-178 | DOI | MR | Zbl

[24] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 | MR | Zbl

[25] Pagnon, N. G. J.; Ressayre, N. Adjacency of Young tableaux and the Springer fibers, Selecta Math. (N.S.), Volume 12 (2006) no. 3-4, pp. 517-540 | MR | Zbl

[26] Spaltenstein, Nicolas Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, 946, Springer-Verlag, Berlin, 1982 | MR | Zbl

[27] Springer, T. A. Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., Volume 36 (1976), pp. 173-207 | DOI | MR | Zbl

[28] Springer, T. A. A construction of representations of Weyl groups, Invent. Math., Volume 44 (1978) no. 3, pp. 279-293 | DOI | MR | Zbl

[29] Springer, T. A. Contribution to Open problems in algebraic groups (1983) (Taniguchi Foundation, Katata)

[30] Steinberg, Robert An occurrence of the Robinson-Schensted correspondence, J. Algebra, Volume 113 (1988) no. 2, pp. 523-528 | DOI | MR | Zbl

[31] Trapa, Peter E. Generalized Robinson-Schensted algorithms for real groups, Internat. Math. Res. Notices (1999) no. 15, pp. 803-834 | DOI | MR | Zbl

[32] Vargas, J. A. Fixed points under the action of unipotent elements of ${\mathrm{SL}}_n$ in the flag variety, Bol. Soc. Mat. Mexicana (2), Volume 24 (1979) no. 1, pp. 1-14 | MR | Zbl

[33] Willems, Matthieu Cohomologie et $K$-théorie équivariantes des variétés de Bott-Samelson et des variétés de drapeaux, Bull. Soc. Math. France, Volume 132 (2004) no. 4, pp. 569-589 | Numdam | MR | Zbl