We construct double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by A.~N. Kirillov in arXiv:1504.01469 and identify them with the polynomials defined by T.~Ikeda and H.~Naruse in Adv. Math. (2013) for the case of maximal Grassmannian permutations. We also give geometric interpretation of them in terms of algebraic localization map and give explicit combinatorial formulas.
"Construction of Double Grothendieck Polynomials of Classical Types using IdCoxeter Algebras." Tokyo J. Math. 39 (3) 695 - 728, March 2017. https://doi.org/10.3836/tjm/1491465733