# Bruhat decomposition for G(R), R local ring or R=Z/p^r

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?

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Bruhat decomposition over $\mathbf Z/p^r\mathbf Z$ is precisely the problem we looked at in this paper. 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.
Thanks for the fast response. So, I guess the quotient $B\GL_n/B$ is pretty nasty and not an algebraic variety? – Marc Palm Mar 9 '11 at 10:02
We did $n=3$ explicitly. At least there $p$ enters the picture in a very controlled way. In general, it seems like there should some hidden structure which we do not understand. – Amritanshu Prasad Mar 9 '11 at 10:28
Despite the difficulty of the problem as posed, there is still the Iwahori version, that $G(\mathbb Z_p)=JWJ$ (disjoint union!), where $J$ is Iwahori, $W$ is the spherical Weyl group. The disjointness of the union sometimes reduces what is otherwise reasonably perceived as fairly chaotic. – paul garrett Jul 16 '11 at 17:30