  We study the generalized double $\beta$-Grothendieck polynomials for all
types. We study the Cauchy formulas for them. Using this, we deduce the
K-theoretic version of the comodule structure map $\alpha^*: K(G/B)\to
K(G)\otimes K(G/B)$ induced by the group action map for reductive group $G$ and
its flag variety $G/B$. Furthermore, we give a combinatorial formula to compute
the localization of Schubert classes as a generalization of Billey's formula.
