Is the Borel subgroup the only closed double coset?

Question

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?

Answer 1

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.