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Seidel product formula in equivariant quantum $K$-theory of flag varieties

Takeshi Ikeda, Takafumi Kouno, Satoshi Naito

Abstract

We prove a Seidel product formula for the torus-equivariant quantum $K$-theory of a generalized flag variety $G/P.$ This is a natural generalization of the corresponding results by Buch, Chaput, and Perrin for the cominuscule flag varieties. Our proof is based on the $K$-theoretic Peterson isomorphism, due to Kato. We also use a version of the $K$-theoretic nil-Hecke algebra associated with the extended affine Weyl group, which was studied by Ikeda, Shimozono, and Yamaguchi.

Seidel product formula in equivariant quantum $K$-theory of flag varieties

Abstract

We prove a Seidel product formula for the torus-equivariant quantum -theory of a generalized flag variety This is a natural generalization of the corresponding results by Buch, Chaput, and Perrin for the cominuscule flag varieties. Our proof is based on the -theoretic Peterson isomorphism, due to Kato. We also use a version of the -theoretic nil-Hecke algebra associated with the extended affine Weyl group, which was studied by Ikeda, Shimozono, and Yamaguchi.
Paper Structure (25 sections, 19 theorems, 59 equations)

This paper contains 25 sections, 19 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Seidel classes
  3. Seidel product formula for $QK_T(G/B)$
  4. Parabolic case
  5. $K$-theoretic Peterson isomorphism
  6. Extended $K$-theoretic Peterson algebra
  7. Related works
  8. Extended K-theoretic Peterson algebra: Preliminaries
  9. Root systems
  10. Extended affine Weyl group $\hat{W}_\mathrm{af}$
  11. Extended $K$-theoretic Peterson algebra
  12. $K$-theoretic Peterson isomorphism
  13. Seidel classes on the affine side
  14. Some properties of $v_{i}$
  15. Proof of the Seidel product theorem
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $i$ be a special (cominuscule) node of the Dynkin diagram of $G,$ and $v_{i}$ the Seidel element corresponding to $i$. Then, for any $w\in W,$ we have in torus-equivatiant quantum $K$-theory $QK_T(G/B)$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: ISY, cf. IIS
  • Proposition 2.2: ISY
  • Lemma 2.3: ISY
  • Proposition 2.4
  • proof
  • Theorem 2.5: Kato Kato, see also CL, LLMS
  • Proposition 2.6
  • Proposition 2.7
  • ...and 24 more