Greeting, in this paper, the local model of a ramified extension is defined as follow
Such a functor is representable by a closed subscheme of a product of affine grassmanians. However i don't get the following part of the paper
Recall that given a scheme over DVR with residue field $k$, $\pi:X\rightarrow \mathrm{spec}R$, the special fiber is $X\times_{\mathrm{spec}R}\mathrm{spec}k$. I don't get how the writer represents the point in the special fiber with a Jordan matrix $\overline{M}$, neither how he represents the subspace of $R$-valued points with a matrice $M$ whose reduction $mod \pi$ is $\overline{M}$. I'll appreciate any help. Thanks in advance.