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Equivariant homology of the symplectic affine Grassmannian and dual affine Schur $P$-functions

Takeshi Ikeda, Shinsuke Iwao, Mark Shimozono

TL;DR

The paper develops a torus-equivariant Schubert calculus for the symplectic affine Grassmannian by introducing dual affine Schur $P$-functions that represent Schubert classes, realized through the level-zero affine nil-Hecke algebra and stabilizing to Nakagawa–Naruse’s dual factorial $P$-functions. It then establishes a precise Ginzburg–Peterson framework showing $H_*^T(\mathrm{Gr}_{\mathrm{Sp}_{2n}})$ is isomorphic to a coordinate ring of a centralizer family for the Langlands dual group $G^\vee=\mathrm{SO}_{2n+1}$, with an explicit isomorphism via symmetric functions. The work also provides a flatness argument for the centralizer coordinate ring and develops a free-fermion formulation to realize $\hat{P}_\lambda(y|a)$ and related $\hat{q}$-functions, enabling explicit Pieri-type relations and structural insights into the equivariant (co)homology. Overall, the results bridge geometric realization, representation-theoretic algebra, and integrable-system perspectives (Toda lattice) to illuminate the Schubert calculus in type C and its infinite-rank limits, with concrete algebraic models and constructive tools for further Pieri and Giambelli analyses.

Abstract

We study the torus-equivariant homology $H_*^T(\mathrm{Gr}_G)$ of the affine Grassmannian $\mathrm{Gr}_G$, where $G=\mathrm{Sp}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis, giving rise to a well-defined Schubert calculus. We realize $H_*^T(\mathrm{Gr}_G)$ in terms of symmetric functions. Our first main result introduces a new family of symmetric functions, called the \emph{dual affine Schur $P$-functions}, which represent the Schubert classes. These functions are defined through the action of the affine nil-Hecke algebra, and specialize, in the stable limit as $n\to \infty$, to the dual factorial $P$-functions of Nakagawa and Naruse. Our second main result gives a precise comparison between this symmetric function model and the geometric construction of $H_*^T (\mathrm{Gr}_G)$ due to Ginzburg and Peterson, which identifies it with a coordinate ring of a centralizer family in the Langlands dual group.

Equivariant homology of the symplectic affine Grassmannian and dual affine Schur $P$-functions

TL;DR

The paper develops a torus-equivariant Schubert calculus for the symplectic affine Grassmannian by introducing dual affine Schur -functions that represent Schubert classes, realized through the level-zero affine nil-Hecke algebra and stabilizing to Nakagawa–Naruse’s dual factorial -functions. It then establishes a precise Ginzburg–Peterson framework showing is isomorphic to a coordinate ring of a centralizer family for the Langlands dual group , with an explicit isomorphism via symmetric functions. The work also provides a flatness argument for the centralizer coordinate ring and develops a free-fermion formulation to realize and related -functions, enabling explicit Pieri-type relations and structural insights into the equivariant (co)homology. Overall, the results bridge geometric realization, representation-theoretic algebra, and integrable-system perspectives (Toda lattice) to illuminate the Schubert calculus in type C and its infinite-rank limits, with concrete algebraic models and constructive tools for further Pieri and Giambelli analyses.

Abstract

We study the torus-equivariant homology of the affine Grassmannian , where is the symplectic group. This homology admits a natural ring structure and a Schubert basis, giving rise to a well-defined Schubert calculus. We realize in terms of symmetric functions. Our first main result introduces a new family of symmetric functions, called the \emph{dual affine Schur -functions}, which represent the Schubert classes. These functions are defined through the action of the affine nil-Hecke algebra, and specialize, in the stable limit as , to the dual factorial -functions of Nakagawa and Naruse. Our second main result gives a precise comparison between this symmetric function model and the geometric construction of due to Ginzburg and Peterson, which identifies it with a coordinate ring of a centralizer family in the Langlands dual group.

Paper Structure

This paper contains 55 sections, 78 theorems, 263 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dual affine Schur $P$-functions
  3. Relation to infinite rank case
  4. Ginzburg–Peterson construction
  5. Pieri classes
  6. Dual factorial $P$-functions
  7. Root system $\mathop{\mathrm{C}}_\infty$ and strict partitions
  8. Schur $Q$-functions
  9. Factorial $Q$-functions
  10. Dual factorial $P$-functions $\hat{P}_\lambda(y|a)$
  11. Left divided difference operators
  12. $Q$-functions in time variables
  13. $\hat{q}$-functions
  14. Affine nil-Hecke algebra and equivariant homology of the affine Grassmannian
  15. Root system and Weyl group
  16. ...and 40 more sections

Key Result

Theorem 1.1

There is an isomorphism of $S$-algebras such that the Schubert class $\sigma_w$ of $H_*^T(\mathrm{Gr}_{\mathrm{Sp}_{2n}})$ corresponds to $\hat{P}_w^{(n)}(y|a).$

Figures (1)

  • Figure 1: Picture of $R$ for $n=3$

Theorems & Definitions (172)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Ivanov Iv:pri
  • proof
  • Remark 2.5
  • ...and 162 more