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Integral equivariant cohomology of affine Grassmannians

David Anderson

TL;DR

This work provides explicit integral and equivariant presentations for the cohomology rings of the affine Grassmannian and affine flag varieties in type A by realizing them as quotients of the polynomial ring $\Lambda[y]$ through natural embeddings in the Sato Grassmannian. It extends Bott’s classical approach to the equivariant setting, using a symmetric-power inverse-limit framework and a compatible coproduct to produce concrete generators and relations expressed via the symmetric-function notation $p_k(c|y)$ and their shifted variants $\widetilde{p}_k(h|y)$. A key technical advance is the introduction of double monomial symmetric functions $m_\lambda(\xi|a)$, which yield natural bases for $H_T^*(\widetilde{\mathrm{Gr}}_n)$ and clarify the kernel structure $I_n$ of the quotient, with connections to Lam–Shimozono and Molev. The results also illuminate the moduli-stack viewpoint of vector bundles on $\mathbb{P}^1$: Larson’s integral description is shown to coincide with the present presentation, linking combinatorial symmetric-function identities to geometric invariants of bundles and flag varieties, and enabling explicit calculations of the cohomology rings in terms of $\Lambda$, $y$-variables, and the shift parameter $d$.

Abstract

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on $\mathbb{P}^1$.

Integral equivariant cohomology of affine Grassmannians

TL;DR

This work provides explicit integral and equivariant presentations for the cohomology rings of the affine Grassmannian and affine flag varieties in type A by realizing them as quotients of the polynomial ring through natural embeddings in the Sato Grassmannian. It extends Bott’s classical approach to the equivariant setting, using a symmetric-power inverse-limit framework and a compatible coproduct to produce concrete generators and relations expressed via the symmetric-function notation and their shifted variants . A key technical advance is the introduction of double monomial symmetric functions , which yield natural bases for and clarify the kernel structure of the quotient, with connections to Lam–Shimozono and Molev. The results also illuminate the moduli-stack viewpoint of vector bundles on : Larson’s integral description is shown to coincide with the present presentation, linking combinatorial symmetric-function identities to geometric invariants of bundles and flag varieties, and enabling explicit calculations of the cohomology rings in terms of , -variables, and the shift parameter .

Abstract

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on .
Paper Structure (12 sections, 8 theorems, 54 equations)

This paper contains 12 sections, 8 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Infinite and affine flag varieties
  3. Definitions
  4. Chern classes and cohomology
  5. Coproduct
  6. Some algebra of symmetric functions
  7. Notation
  8. An equality of ideals
  9. Some identities in $\Lambda[y]$
  10. Proof of the main theorem
  11. Double monomial symmetric functions
  12. Moduli of vector bundles

Key Result

Theorem 1

The inclusion $\widetilde{\mathrm{Gr}}^d_n \hookrightarrow \mathrm{Gr}^d$ induces a surjection $H_{{T}}^*\mathrm{Gr}^d \twoheadrightarrow H_{{T}}^*\widetilde{\mathrm{Gr}}^d_n$, whose kernel is generated by $p_k(c|y)$ for $k> n$, together with $p_n(c|y)+d e_n(y)$. In particular, the map $c_k \mapsto where $I_n^d$ is the ideal generated by $p_k(c|y)$ for $k> n$ and $p_n(c|y)+d e_n(y)$.

Theorems & Definitions (18)

  • Theorem
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • ...and 8 more