Relativistic Toda Lattice and Equivariant $K$-Homology of Affine Grassmannian

Technical Summary: Relativistic Toda Lattice and Equivariant K-Homology of Affine Grassmannian

Problem Statement
This paper investigates the "quantum equals affine" phenomenon within the context of $T$-equivariant quantum $K$-theory of the flag variety $G/B$ (specifically for $G=SL_n(\mathbb{C})$) and its relationship to the $T$-equivariant $K$-homology of the affine Grassmannian $Gr_{SL_n}$. While the non-equivariant version of this correspondence was previously explored using the relativistic Toda lattice, the equivariant case required a more rigorous algebraic realization. The central challenge addressed is the explicit construction of the $K$-Peterson isomorphism—a map connecting the localized quantum $K$-ring $QK_T(SL_n(\mathbb{C})/B)$ and the equivariant $K$-homology ring $K_T^*(Gr_{SL_n})$—specifically establishing a correspondence between their respective Schubert bases.

Methodology
The authors employ an integrable systems approach, utilizing the relativistic Toda lattice introduced by Ruijsenaars. The methodology proceeds through the following steps:

1. Geometric Setup: The authors define the phase space $Y$ of the relativistic Toda lattice and a centralizer family $Z$ associated with $PGL_n(\mathbb{C})$. They establish that the coordinate rings of these varieties correspond to the polynomial versions of the quantum $K$-ring and the equivariant $K$-homology ring, respectively:
   $$\mathcal{O}(Y) \cong QK_T^{pol}(SL_n(\mathbb{C})/B), \quad \mathcal{O}(Z) \cong K_T^*(Gr_{SL_n}).$$
2. Birational Map Construction: By solving the relativistic Toda lattice equations, the authors construct a birational map $\Phi_n$ between the open dense subsets $Y^\circ$ and $Z^\circ$. This map is defined via a rational substitution sending the dynamical variables $(z_i, Q_i)$ of the Toda lattice to specific rational expressions involving $\tau$-functions ($\tau_i$) and $\sigma$-functions ($\sigma_i$) derived from the centralizer family.
3. Algebraic Realization: The paper utilizes recent developments in the presentation of the quantum $K$-ring via quantum double Grothendieck polynomials ($G_w^Q$) and the realization of the affine $K$-homology via $K$-theoretic double $k$-Schur functions ($\tilde{g}_x^{(k)}$).
4. Commutative Diagram: The authors demonstrate that the map $\Phi_n$, when twisted by a specific algebra automorphism $\sigma$ (denoted as $\tilde{\Phi}_n = \sigma \circ \Phi_n$), fits into a commutative diagram with Kato's abstract $K$-Peterson map.

Key Contributions and Results

• Explicit Algebraic Realization of the $K$-Peterson Map: The primary result (Theorem 1.1) provides an explicit formula for the $K$-Peterson isomorphism. The map $\tilde{\Phi}_n$ sends the quantum Schubert class $Q^\xi G_w^Q(z|\eta)$ to the affine Schubert class $\tilde{g}_{x\omega_k}^{(k)}(y|b)$, where $x = w t_\xi$ and $\omega_k$ is a specific involution on the affine Weyl group. This establishes a concrete combinatorial correspondence between the two bases.
• Intertwining of Demazure Operators: The authors prove that the constructed map intertwines the actions of the quantum Demazure operators ($D_i^Q$) on the quantum side with the affine Demazure operators ($D_i$) on the affine side (up to the involution $\omega_k$). This property is crucial for extending the correspondence from the longest element to all Schubert classes.
• Determinantal Formulas: For $k$-small partitions (partitions $\lambda$ satisfying $\lambda_1 + \ell(\lambda) \leq n$), the paper derives explicit determinantal formulas for the $K$-theoretic double $k$-Schur functions $\tilde{g}_\lambda^{(k)}(y|b)$ and their non-closed counterparts $g_\lambda^{(k)}(y|b)$. These formulas express the functions as determinants of matrices involving the coordinate functions of the centralizer family.
• Factorization Formula: As a direct application of the main theorem, the authors derive a factorization formula for the $K$-theoretic double $k$-Schur function associated with the unique maximal $k$-irreducible $k$-bounded partition $\nu_n$ (Theorem 1.2). This formula expresses $\tilde{g}_{\nu_n}^{(k)}$ as a product of simpler functions and $\tau$-function related terms, confirming a conjecture from the non-equivariant setting.

Significance
The paper claims significance in providing a concrete algebraic formulation of the $K$-Peterson isomorphism in the equivariant setting. By grounding the abstract geometric correspondence in the theory of the relativistic Toda lattice, the authors offer new insights into the combinatorial structure of the isomorphism. Specifically, the work bridges the gap between the quantum double Grothendieck polynomials and the $K$-theoretic double $k$-Schur functions, allowing for the translation of results between the quantum and affine geometries. The derivation of the factorization formula for the maximal $k$-irreducible partition serves as a non-trivial verification of the theory's consistency and utility. The results rely on and extend prior work by Kato, Maeno, Naito, Sagaki, Ikeda, Shimozono, and Yamaguchi, solidifying the connection between integrable systems and equivariant quantum $K$-theory.