Abstract
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of K-orbit closures on the flag variety G/B, where G = GL(n, \( \mathbb{C} \)), and where K is one of the symmetric subgroups O(n, \( \mathbb{C} \)) or Sp(n, \( \mathbb{C} \)). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.
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M. Brion, Equivariant cohomology and equivariant intersection theory, in: Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Kluwer, 1997, pp. 1–37.
M. Brion, Rational smoothness and fixed points of torus actions, Transform. Groups 4 (1999), no. 2&3, 127–156.
M. Brion, On orbit closures of spherical subgroups in flag varieties, Comm. Math. Helv. 76 (2001), no. 2, 263–299.
D. Edidin, W. Graham, Characteristic classes and quadric bundles, Duke Math. J. 78 (1995), no. 2, 277–299.
W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420.
W. Fulton, Schubert varieties in flag bundles for the classical groups, in: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan Univ., 1996, pp. 241–262.
W. Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Diff. Geom. 43 (1996), no. 2, 276–290.
W. Fulton, Intersection Theory, 2nd ed., Springer-Verlag, Berlin, 1998.
W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.
W. Graham, The class of the diagonal in flag bundles, J. Diff. Geom. 45 (1997), no. 3, 471–487.
J. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975. Russian transl.: Дж. Хамфри, Линейные алгебраические группы, Нayкa, M., 1980.
T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), no. 2, 331–357.
W. McGovern, P. Trapa, Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the fag variety, J. Algebra 322 (2009), no. 8, 2713–2730.
J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, Princeton, NJ, 1974. Russian transl.: Дж. Милнор, Дж. Cтaшeф, Характеристические классы, Мир, M., 1979.
R. W. Richardson, T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1–3, 389–436.
R. W. Richardson, T. A. Springer, Combinatorics and geometry of K-orbits on the flag manifold, in: Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., Vol. 153, Amer. Math. Soc., 1993, pp. 109–142.
R. W. Richardson, T. A. Springer, Complements to: “The Bruhat order on symmetric varieties” [Geom. Dedicata 35 (1990), no. 1–3, 389–436], Geom. Dedicata 49 (1994), no. 2, 231–238.
T. A. Springer, Some results on algebraic groups with involutions, in: Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., Vol. 6, North-Holland, 1985, pp. 525–543.
B. Wyser, Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci, PhD thesis, University of Georgia, 2012.
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Dedicated to my daughters, Avery and Carsyn
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Wyser, B.J. K-Orbit Closures on G/B as Universal Degeneracy Loci for Flagged Vector Bundles with Symmetric or Skew-Symmetric Bilinear Form. Transformation Groups 18, 557–594 (2013). https://doi.org/10.1007/s00031-013-9221-1
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DOI: https://doi.org/10.1007/s00031-013-9221-1