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

We prove a Seidel product formula for the torus-equivariant quantum $K$-theory of a generalized flag variety $G\mathrm{/}P\mathrm{.}$ 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.