  We give a new proof of the generalized Cauchy identity for double
Grothendieck polynomials, a combinatorial interpretation of the stable double
Grothendieck polynomials in terms of triples of tableaux, and an interpolation
between the stable double Grothendieck polynomial and the weak stable double
Grothendieck polynomial. This so-called half weak stable double Grothendieck
polynomial evaluated at $x=y$ generalizes the type $B$ Stanley symmetric
function of Billey and Haiman and is $Q$-Schur positive by degree
