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

1. Problem Statement

The paper addresses the computation of products involving Seidel classes in the $T$-equivariant quantum K-theory ($QK_T(G/B)$) of a generalized flag variety $G/B$, where $G$ is a simple, simply connected complex algebraic group and $T$ is a maximal torus.

• Context: Seidel's construction, originally developed for quantum cohomology, provides a way to understand the action of the fundamental group of the adjoint group on the quantum cohomology ring. In quantum K-theory, Buch, Chaput, and Perrin established a "Seidel product formula" specifically for cominuscule flag varieties.
• Gap: While the formula was known for cominuscule cases (and recently for Grassmannians via curve neighborhoods), a general formula for arbitrary flag varieties $G/B$ (and subsequently $G/P$) in the equivariant quantum K-theory setting was missing.
• Goal: To prove a general Seidel product formula for $QK_T(G/B)$ for any simple group $G$ and any special (cominuscule) node $i$ of the Dynkin diagram, and to extend this result to partial flag varieties $G/P$.

2. Methodology

The authors employ a "quantum equals affine" philosophy, utilizing the K-theoretic Peterson isomorphism to translate the problem from the quantum K-theory of the flag variety to the equivariant K-homology of the affine Grassmannian.

Key Tools:

1. K-theoretic Peterson Isomorphism ($\Phi$):

   • Established by Kato, this is an isomorphism of $R(T)$-algebras between the localized equivariant K-homology of the affine Grassmannian ($K_T^*(Gr_G)_{loc}$) and the localized equivariant quantum K-theory of the flag variety ($QK_T(G/B)_{loc}$).
   • It maps affine Schubert classes to quantum Schubert classes and quantum parameters to specific elements in the affine K-homology.

2. Extended K-theoretic Peterson Algebra ($\hat{L}_G$):

   • Based on the work of Ikeda, Shimozono, and Yamaguchi, the authors utilize an extended version of the K-theoretic nil-Hecke algebra associated with the extended affine Weyl group $\hat{W}_{af} = \Sigma \ltimes W_{af}$.
   • Here, $\Sigma \cong P^\vee/Q^\vee$ represents the fundamental group of the adjoint group.
   • The algebra $\hat{L}_G$ serves as an algebraic model for $K_{T_{ad}}^*(Gr_{G_{ad}})$.

3. Star Action:

   • The proof relies on the "star action" of the Weyl group $W$ on the Peterson algebra. The authors demonstrate that the left action of $W$ on $QK_T(G/B)$ corresponds to the star action on the affine side via the isomorphism $\Phi$.

4. Seidel Elements ($v_i$):

   • For a special node $i$, the Seidel element $v_i \in W$ is defined via the longest elements of $W$ and the parabolic subgroup $P_i$.
   • The authors analyze the combinatorial properties of $v_i$, specifically its interaction with the fundamental coweight $\varpi^\vee_i$ and the affine automorphism $\pi_i$.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The paper proves the Seidel product formula for the full flag variety $G/B$. Let $i$ be a special node, $v_i$ the corresponding Seidel element, and $O_w^{G/B}$ the Schubert basis elements. For any $w \in W$:
$$O_{v_i}^{G/B} \cdot v_i^L O_w^{G/B} = Q^{\varpi^\vee_i - w^{-1}(\varpi^\vee_i)} O_{v_i w}^{G/B}$$

• $v_i^L$: Denotes the left action of $v_i$ on the ring.
• $Q^\beta$: The quantum parameter associated with the coweight $\beta$.
• Significance: This generalizes the Buch-Chaput-Perrin result from cominuscule varieties to the full flag variety and incorporates the equivariant parameters naturally.

Corollary for Partial Flag Varieties (Corollary 1.2)

Using Kato's pushforward homomorphism $\pi_*: QK_T(G/B) \to QK_T(G/P)$, the authors derive the formula for any parabolic subgroup $P$:
$$O_{\lfloor v_i \rfloor_P}^{G/P} \cdot v_i^L O_w^{G/P} = Q^{\lfloor \varpi^\vee_i - w^{-1}(\varpi^\vee_i) \rfloor_P} O_{\lfloor v_i w \rfloor_P}^{G/P}$$
where $\lfloor \cdot \rfloor_P$ denotes the minimal-length coset representative. This recovers the non-equivariant formula of Buch-Chaput-Perrin when $P$ is cominuscule.

Technical Lemmas

• Lemma 3.1: Establishes a crucial identity regarding the descent sets and the element $\gamma_w$: $\gamma_{v_i w} = \gamma_w - w^{-1}(\varpi^\vee_i)$. This identity is the combinatorial engine driving the proof.
• Proposition 2.6: Proves the compatibility between the star action on the affine Peterson algebra and the left action on the quantum K-theory ring via the Peterson isomorphism.

4. Proof Strategy Overview

1. Translation: The authors translate the product $O_{v_i} \cdot (v_i^L O_w)$ in $QK_T(G/B)$ to the affine side using the isomorphism $\Phi$.
2. Affine Computation: In the affine Peterson algebra $\hat{L}_G$, they compute the product using the star action and the properties of the elements $\sigma_i = \ell_{t^{-\varpi^\vee_i}}$.
3. Key Identity: They utilize the relation $v_i t^{-\varpi^\vee_i} = \pi_i^{-1}$ and the behavior of the star action on the basis elements $\ell_x$.
4. Reconstruction: By applying Lemma 3.1, they show that the resulting affine element corresponds exactly to the quantum parameter $Q^{\varpi^\vee_i - w^{-1}(\varpi^\vee_i)}$ times the Schubert class $O_{v_i w}$.
5. Pushforward: The result for $G/P$ follows directly from the surjectivity and compatibility of Kato's pushforward map.

5. Significance

• Unification: The paper unifies the understanding of Seidel representations across different types of flag varieties, moving beyond the restrictive cominuscule case.
• Methodological Advancement: It demonstrates the power of the K-theoretic Peterson isomorphism combined with extended affine Weyl group techniques. This approach avoids the geometric complexity of curve neighborhoods used in previous K-theoretic proofs for Grassmannians.
• Hidden Symmetry: The results provide concrete evidence of the "hidden affine symmetry" in Gromov-Witten invariants and quantum K-theory, linking the topology of the flag variety to the structure of the affine Grassmannian.
• Explicit Computations: The authors provide explicit examples for types $A_{n-1}$, $C_2$, and $D_5$, verifying the formula and illustrating the combinatorial mechanics of the Seidel elements.

In summary, this paper provides a rigorous, algebraic proof of the Seidel product formula for general flag varieties in equivariant quantum K-theory, leveraging deep connections between affine Lie theory and quantum geometry.