SIGMA 22 (2026), 065, 39 pages      arXiv:2505.02941      https://doi.org/10.3842/SIGMA.2026.065

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

Takeshi Ikeda ^a, Shinsuke Iwao ^b, Satoshi Naito ^c and Kohei Yamaguchi ^a
a) Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
^b) Faculty of Business and Commerce, Keio University, 4-1-1 Hiyosi, Kohoku-ku, Yokohama-si, Kanagawa 223-8521, Japan
^c) Department of Mathematics, Institute of Science Tokyo, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan

Received October 26, 2025, in final form June 18, 2026; Published online July 06, 2026

Abstract
We investigate the phenomenon known as ''quantum equals affine'' in the setting of $T$-equivariant quantum $K$-theory of the flag variety $G/B$, as established by Kato for any semisimple algebraic group $G$. In particular, we focus on the $K$-Peterson isomorphism between the $T$-equivariant quantum $K$-ring $QK_T(\mathrm{SL}_n(\mathbb{C})/B)$ and the $T$-equivariant $K$-homology ring $K_*^T(\mathrm{Gr}_{\mathrm{SL}_n})$ of the affine Grassmannian, after suitable localizations on both sides. Building on an earlier work by Ikeda, Iwao, and Maeno, we present an explicit algebraic realization of the $K$-Peterson map via a rational substitution that sends the generators of the quantum $K$-theory ring to explicit rational expressions in the fundamental generators of $K_*^T(\mathrm{Gr}_{\mathrm{SL}_n})$, thereby matching the Schubert bases on both sides. Our approach builds on recent developments in the theory of $QK_T(\mathrm{SL}_n(\mathbb{C})/B)$ by Maeno, Naito, and Sagaki, as well as the theory of $K$-theoretic double $k$-Schur functions introduced by Ikeda, Shimozono, and Yamaguchi. This concrete formulation provides new insight into the combinatorial structure of the $K$-Peterson isomorphism in the equivariant setting. As an application, we establish a factorization formula for the $K$-theoretic double $k$-Schur function associated with the maximal $k$-irreducible $k$-bounded partition.

Key words: equivariant quantum $K$-theory; affine Grassmannian; Peterson isomorphism; relativistic Toda lattice; $k$-Schur functions.

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