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K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums

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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 for various symmetric pairs (G, K). We describe an interpretation of these formulas as representing the classes of particular types of degeneracy loci when evaluated at certain Chern classes. For the type A pair \((SL(p+q,{\mathbb C}),S(GL(p,{\mathbb C}) \times GL(q,{\mathbb C})))\), such degeneracy loci are described explicitly, relative to a rank \(p+q\) vector bundle V on a smooth complex variety X equipped with a flag of subbundles and a splitting of V as a direct sum of subbundles of ranks p and q. We conjecture similarly explicit descriptions of the degeneracy loci for all cases in types B and C.

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Authors and Affiliations

  1. Institut Fourier, St. Martin d’hères, France

    Benjamin J. Wyser

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  1. Benjamin J. Wyser
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Appendix: Weak order graphs and tables of formulas in examples

Appendix: Weak order graphs and tables of formulas in examples

See Figs. 1, 2, 3, 4, 5, 6, 7; Tables 5, 6, 7, 8, 9, 10.

Fig. 1
figure 1

\((GL(4,{\mathbb C}),GL(2,{\mathbb C}) \times GL(2,{\mathbb C}))\)

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Fig. 2
figure 2

\((SO(7,{\mathbb C}),S(O(4,{\mathbb C}) \times O(3,{\mathbb C})))\)

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Fig. 3
figure 3

\((Sp(6,{\mathbb C}),Sp(4,{\mathbb C}) \times Sp(2,{\mathbb C}))\)

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Fig. 4
figure 4

\((Sp(4,{\mathbb C}),GL(2,{\mathbb C}))\)

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Fig. 5
figure 5

\((SO(6,{\mathbb C}),S(O(4,{\mathbb C}) \times O(2,{\mathbb C})))\)

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Fig. 6
figure 6

\((SO(6,{\mathbb C}),GL(3,{\mathbb C}))\)

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Fig. 7
figure 7

\((SO(6,{\mathbb C}),S(O(3,{\mathbb C}) \times O(3,{\mathbb C})))\)

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Table 5 Formulas for \((SO(7,{\mathbb C}),S(O(4,{\mathbb C}) \times O(3,{\mathbb C})))\)
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Table 6 Formulas for \((Sp(6,{\mathbb C}),Sp(4,{\mathbb C}) \times Sp(2,{\mathbb C}))\)
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Table 7 Formulas for \((Sp(4,{\mathbb C}),GL(2,{\mathbb C}))\)
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Table 8 Formulas for \((SO(6,{\mathbb C}),S(O(4,{\mathbb C}) \times O(2,{\mathbb C})))\)
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Table 9 Formulas for \((SO(6,{\mathbb C}),GL(3,{\mathbb C}))\)
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Table 10 Formulas for \((SO(6,{\mathbb C}),S(O(3,{\mathbb C}) \times O(3,{\mathbb C})))\)
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Wyser, B.J. K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums. Geom Dedicata 181, 137–175 (2016). https://doi.org/10.1007/s10711-015-0117-1

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