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

TL;DR

This paper establishes a Seidel product formula within the framework of torus-equivariant quantum K-theory for generalized flag varieties, extending previous results to a broader class of varieties using advanced algebraic tools.

Contribution

It generalizes the Seidel product formula to all generalized flag varieties by employing the K-theoretic Peterson isomorphism and nil-Hecke algebra techniques.

Findings

Proves a Seidel product formula for generalized flag varieties.

Extends previous results from cominuscule to all flag varieties.

Utilizes K-theoretic Peterson isomorphism and nil-Hecke algebra methods.

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.