Equivariant homology of the symplectic affine Grassmannian and dual affine Schur $P$-functions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914171345436672 |
|---|---|
| author | Ikeda, Takeshi Iwao, Shinsuke Shimozono, Mark |
| author_facet | Ikeda, Takeshi Iwao, Shinsuke Shimozono, Mark |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant homology of the symplectic affine Grassmannian and dual affine Schur $P$-functions Ikeda, Takeshi Iwao, Shinsuke Shimozono, Mark Representation Theory Combinatorics 05E05, 14N15 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. |
| title | Equivariant homology of the symplectic affine Grassmannian and dual affine Schur $P$-functions |
| topic | Representation Theory Combinatorics 05E05, 14N15 |
| url | https://arxiv.org/abs/2511.20966 |