Let $G$ be a quasisplit connected reductive group over a $p$-adic field $k$. Identify $G$ with its rational points. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$, both defined over $k$. For $w \in N\_G(T)$, consider the double cosets $BwB$, which are locally closed submanifolds of $G$. Is $B$ the only closed double coset? Is there some general description of the closure of a Bruhat cell? 



Is the Borel subgroup the only closed double coset?
In general, and this holds for algebraically closed field as well as local field of charracteristic 0, at least in the split case, the description of orbit closures is as follows: 
Let $\omega \in W\_G$ be a weyl element, and write $\omega = \sigma\_{\alpha\_1} ... \sigma\_{\alpha\_k}$ as a reduced expression in simple reflections, i.e. 
$\ell(\omega) = k$. 
Let $P\_{\alpha\_i}$ denote the parabolic subgroup containing $T$ of the form 
$SL\_2^\alpha B$ where $SL\_2^\alpha$ is the $SL\_2$-triple with roots 
$\alpha, -\alpha$. Then, the orbit closure is given by  

$\overline{B \omega B} = P\_{\alpha\_1} ... P\_{\alpha\_k}B$. In particular, 
$B$ is the unique closed orbit, and the orbit closure contains exactly those orbits $B \omega' B$ such that $\omega'$ is obtained from $\omega$ by excising several reflections from the reduced expression. 


