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

Benjamin J. Wyser|arXiv (Cornell University)|Jan 8, 2013

Advanced Algebra and Geometry28 references4 citations

TL;DR

This paper computes torus-equivariant cohomology classes of K-orbit closures in the flag variety G/B for classical symmetric pairs (G,K), using equivariant localization and divided difference operators. It establishes that these classes correspond to universal degeneracy loci defined by relative position conditions between flags and direct sum decompositions of vector bundles, with explicit formulas provided for type A and conjectures for types B/C.

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 $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\\C) \ imes GL(q,\\C)$-orbits on $GL(p+q,\\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic description of $K$-orbit closures, which we provide via a detailed combinatorial analysis of the poset of "$(p,q)$-clans," which parametrize the orbit closures. We describe precisely how our formulas for the equivariant classes of $K$-orbit closures can be interpreted as formulas for the classes of such degeneracy loci in the Chern classes of the bundles involved. In the cases outside of type $A$, we suggest that the orbit closures should parametrize degeneracy loci involving a vector bundle equipped with a non-degenerate symmetric or skew-symmetric bilinear form, a single flag of subbundles which are isotropic or Lagrangian with respect to the form, and a splitting as a direct sum of subbundles with each summand satisfying some property (depending on $K$) with respect to the form. The precise description of such a degeneracy locus is conjectured for all cases in types $B$ and $C$.

Motivation & Objective

• To determine torus-equivariant fundamental cohomology classes of K-orbit closures in G/B for symmetric pairs (G,K) with G classical.
• To interpret these cohomology classes as representing degeneracy loci in flagged vector bundles equipped with a direct sum splitting.
• To provide explicit formulas for such degeneracy loci in type A, and conjecture similar descriptions for types B and C.
• To extend the framework of universal degeneracy loci—previously used for Schubert varieties—to K-orbit closures as generalized Schubert varieties.
• To establish a geometric realization of K-orbit closures as degeneracy loci defined by relative position constraints between flag subbundles and summands of a split vector bundle.

Proposed method

• Apply equivariant localization and the self-intersection formula to compute the S-equivariant cohomology classes of closed K-orbits.
• Use divided difference operators to derive formulas for the equivariant classes of all K-orbit closures, building from closed orbit classes.
• Employ combinatorial models of K\G/B and weak order Hasse diagrams, distinguishing solid and dashed edges via divided difference rules.
• Define degeneracy loci via 'clan' parameters encoding relative positions between flag fibers and direct summands V' and V'' of a vector bundle V.
• Leverage explicit set-theoretic descriptions of K-orbit closures (e.g., Theorem 3.3 for type A) to construct degeneracy locus conditions.
• Conjecture that similar degeneracy locus descriptions hold for types B and C based on structural analogies, though they fail in type D.

Experimental results

Research questions

• RQ1 How can the equivariant cohomology classes of K-orbit closures in G/B be computed for symmetric pairs (G,K) with G classical?
• RQ2 Can K-orbit closures be realized as universal degeneracy loci in the same way Schubert varieties are, but for vector bundles with direct sum splittings?
• RQ3 What are the precise conditions on the relative position of flag subbundles and direct summands that define such degeneracy loci?
• RQ4 Why do naive generalizations of the type A degeneracy locus description fail in type D, despite structural similarities?
• RQ5 To what extent can the degeneracy locus framework be extended from Schubert varieties to K-orbit closures in types B, C, and D?

Key findings

• The paper provides explicit formulas for the torus-equivariant cohomology classes of K-orbit closures in G/B for all classical symmetric pairs (G,K), as stated in Theorem 2.15.
• For the type A pair (SL(p+q,C), S(GL(p,C)×GL(q,C))), the degeneracy loci are described as the set of flags where the relative position between the flag and the direct sum decomposition V=V'⊕V'' satisfies conditions encoded by (p,q)-clans.
• The formulas for the equivariant classes are given explicitly in Tables 1–10, such as [Y_γ] = (x₁−y₃)(x₁+y₃)(x₂−y₃)(x₂+y₃) for the (Sp(6,C),Sp(4,C)×Sp(2,C)) pair with clan ++--++.
• The paper conjectures that similar degeneracy locus descriptions hold for types B and C, based on structural parallels, though a naive extension fails in type D as shown in Fact 1.
• The method successfully generalizes Fulton’s universal degeneracy locus framework from Schubert varieties to K-orbit closures by incorporating direct sum splittings as additional structure.
• The work completes the program initiated in [Wys13a] by extending the equivariant cohomology and degeneracy locus description to the remaining symmetric pairs in classical types.