Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one.
For each $w\in\widetilde{W}$, the affine Weyl group, one has the schubert cell $X_{w}=\overline{IwI}/I$, that is a normal projective variety.
One knows by Kumar's book on Kac-moody groups, that $Pic(X_{w})=\mathbb{Z}$, which ones are locally factorial? In other words, when do we have that $Pic(X_{w})=Cl(X_{w})$?
