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Equivariant $K$-theory of Springer Varieties

Vikraman Uma

TL;DR

This work presents a concrete, generators-and-relations description of the $T^l$-equivariant topological $K$-ring $K^0_{T^l}(\mathcal{F}_{\lambda})$ for type $A$ Springer varieties $\mathcal{F}_\lambda$, using $[L_i]_{T^l}$ as generators and an equivariant Tanisaki ideal $\mathcal{I}_{\lambda}$ as relations. It leverages the $T^l$-stable algebraic cell decomposition to show freeness over the representation ring $R(T^l)$ with rank ${n \choose \lambda}$ and develops an $S_n$-action compatible with the inclusion from the full flag; crucially, it constructs the equivariant analogue of Tanisaki relations via lambda-operations and a canonical splitting of tautological bundles. The main theorem asserts an isomorphism $\mathcal{R}/\mathcal{I}_{\lambda} \cong K^0_{T^l}(\mathcal{F}_{\lambda})$, where $\mathcal{R}=R(T^l)[x_1,\dots,x_n]$ and $x_i$ maps to $[L_i]_{T^l}$, and this specializes to the known ordinary K-ring presentation when $u_i=1$. By connecting the equivariant and ordinary theories through the forgetful map, the paper extends Abe–Horiguchi's cohomology description and Sankaran–Uma's $K$-theory results, providing a comprehensive computational framework for equivariant $K$-theory of Springer varieties. The results have implications for geometric representation theory and the study of equivariant invariants of Springer fibers.

Abstract

The aim of this paper is to describe the topological equivariant $K$-ring, in terms of generators and relations, of a Springer variety $\mathcal{F}_λ$ of type $A$ associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence $λ=(λ_1,\ldots, λ_l)$. This parallels the description of the equivariant cohomology ring of $\mathcal{F}_λ$ due to Abe and Horiguchi and generalizes the description of ordinary topological $K$-ring of $\mathcal{F}_λ$ due to Sankaran and Uma \cite{su}.

Equivariant $K$-theory of Springer Varieties

TL;DR

This work presents a concrete, generators-and-relations description of the -equivariant topological -ring for type Springer varieties , using as generators and an equivariant Tanisaki ideal as relations. It leverages the -stable algebraic cell decomposition to show freeness over the representation ring with rank and develops an -action compatible with the inclusion from the full flag; crucially, it constructs the equivariant analogue of Tanisaki relations via lambda-operations and a canonical splitting of tautological bundles. The main theorem asserts an isomorphism , where and maps to , and this specializes to the known ordinary K-ring presentation when . By connecting the equivariant and ordinary theories through the forgetful map, the paper extends Abe–Horiguchi's cohomology description and Sankaran–Uma's -theory results, providing a comprehensive computational framework for equivariant -theory of Springer varieties. The results have implications for geometric representation theory and the study of equivariant invariants of Springer fibers.

Abstract

The aim of this paper is to describe the topological equivariant -ring, in terms of generators and relations, of a Springer variety of type associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence . This parallels the description of the equivariant cohomology ring of due to Abe and Horiguchi and generalizes the description of ordinary topological -ring of due to Sankaran and Uma \cite{su}.
Paper Structure (14 sections, 16 theorems, 92 equations)

This paper contains 14 sections, 16 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Equivariant topological $K$-theory of cellular varieties
  3. The equivariant cohomology of $\mathcal{F}_{\lambda}$
  4. The $T^n$-equivariant cohomology of $\mathcal{F}$
  5. $T^l$-equivariant cohomology of $\mathcal{F}_{\lambda}$
  6. $T^n$-equivariant $K$-ring of the flag variety $\mathcal{F}$
  7. $T^l$-equivariant $K$-ring of ${\mathcal{F}}_{\lambda}$
  8. $R(T^l)$-module structure of $K_{T^l}(\mathcal{F}_{\lambda})$
  9. The $T^l$-fixed points of $\mathcal{F}_{\lambda}$
  10. Localization
  11. The action of the symmetric group on $K_{T^l}(\mathcal{F}_\lambda)$
  12. Sectioning canonical bundles over $\mathcal{F}_\lambda$
  13. The lambda operations in equivariant $K$-theory.
  14. Relation with the ordinary $K$-ring

Key Result

Theorem 1.1

$\!\!\!$With the above notations, let be the ring homomorphism defined by $\Psi_{\lambda}(x_j)=[L_j]_{T^l}$ for $1\leq j\leq n$. Then $\Psi_{\lambda}$ is surjective and $\hbox{ker}(\Psi_{\lambda})=\mathcal{I}_{\lambda}$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 4.1
  • Theorem 5.1
  • ...and 21 more