Is there an invariant, which encodes the failure of the Bruhat decomposition to hold for a reductive group with coefficients in a local ring like the p-adic integers or the ring $\mathbb{Z}/\mathfrak{p}^r$?


Example $G =GL\_2$: Fix a Borel subgroup $B$, e.g. the upper diagonal matrixes. The coset space $B \backslash G$ (or $G/B$) are isomorphic to the projective line. However, the Bruhat decomposition $G = BWB$, where $W$ is the Weyl group, does not hold for the group of $R$ points, where $R$ is not a field. Can we describe $B\backslash G/B$ as a variety over $\mathbb{Z}$ here?



Bruhat decomposition for G(R), R local ring or R=Z/p^r
Bruhat decomposition over $\mathbf Z/p^r\mathbf Z$ is precisely the problem we looked at in [this paper](http://dx.doi.org/10.1080/00927870600876250). We defined several invariants of double cosets, and classified the pairs $(n,k)$ for which, when $G=GL\_n(\mathbf Z/p^k\mathbf Z)$, the cardinality of $B\backslash G/B$ does not depend on $p$. Unfortunately, the general question seems to involve wild classification problems.


