Grothendieck polynomials were firstly defined in
Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux. C. R. Acad. Sci. Paris S´er. I Math., 295(11):629–633, 1982.
to be a computation of K-theory of flag manifold. It is in French and hard to get access. And as far as I saw, I did not find out any reference explaining the geometric meaning in a precise way.
For this I have a series of questions. Let me use $G/B$ to stand for the flag manifolds.
Firstly, how to define the Schubert cells in $K(G/B)$? Are the following two definitions the same?
If one prefers to work in algebraic K-theory, then we can simply define it to be the push forward of $\mathcal{O}_{Bw_0wB/B}$ over $G/B$.
If one insists to use the topological one, then I believe the only way to define it is by the Atiyah–Hirzebruch spectral sequence. Since $E^{pq}_2 = H^p(G/B;K^q(pt))$ has only even dimensions, so it collapses, and we can define the Schubert cells to be the corresponding Schubert cells in $H^{2k}(G/B;K^{-2k}(pt))$.
Due to some big theorem (for example, projective bundle theorem), the algebraic K-group coincides with topological K-group in our case. Are these two definitions the same?
Secondly, what is the meaning for the indeterminants $x_1,\ldots,x_n$?
Actually, in Lascoux and Sch¨utzenberger exposition
Symmetry and flag manifolds, Invariant theory, Springer, 1983, pp. 118–144.
The $x_1,\ldots,x_n$ are exchanged from $L_1,\ldots,L_n$ by $x_i=1-L_i^{-1}$. I guess this $L_i$ is the line bundle $G\times_B \mathbb{C}$ with $B$ acts over $\mathbb{C}$ by the $i$-th index over diagonal.
Thirdly, how to compute the isobaric Demazure operators?
Firstly, how to define push forward in topological sense is a little hard. But one still can do so by Atiyah–Hirzebruch spectral sequence or some variantion of Thom isomorphism.
I think one can compute the isobaric Demazure operators $\pi_i$ for general $\mathbb{C}P^1$ bundle. This may not be very hard. But the problem is why $\pi_i \Sigma_w=\begin{cases}\Sigma_{s_iw},&\ell(w)=\ell(s_iw)+1\\0,& \ell(w)=\ell(s_iw)-1. \end{cases}$
Lastly, why the topmost element is presented by $x_1^{n-1}\cdots x_{n-1}$?
Definitely, there are a lot of choices of the presentitve, but why it is $\bmod \mathbb{Z}[x]^W_+$? In cohomology case, we can compute taking advantage of dimension, but in K-theory case $\pi_i$ does not reduce dimensions.
By the way, I believe that one can get this from the equivariant case, by the localization theorem. But I am not sure that the equivariant topology was wildly used that time. Especially, the algebraic equivairant K-theory is established in 1983 by Thomason.
Generally, I suspect that the Chern character is mainly used, but I did not figure out how Chern character reflects the cell structures.
