Let $G$ be a quasisplit connected, reductive group over a field $F$. Let $A_0$ be a maximal $F$-split torus, and $P$ a maximal $F$-parabolic subgroup containing $A_0$. Let $W = N_G(A_0)/Z_G(A_0)$ be the Weyl group of $A_0$, and let $w \in W$. I think that $G$ is the disjoint union of the double cosets $$G = \bigcup\limits_{w \in W} PwN$$

where $N$ is the unipotent radical of $P$. Is there any way to compute the dimension of a particular double coset $PwN$? Or perhaps generalize the following result:

Assume $F$ is algebraically closed, let $T$ be a maximal torus of $G$ with Weyl group $W = N_G(T)/T$, and let $B$ be a Borel subgroup containing $T$ with unipotent radical $U$. If $w = nT \in W$, let $S_w = \{\alpha_1, ... , \alpha_m\}$ be the set of negative roots with respect to $B$ which are made positive by $w$, where $w$ acts on $X(T)$ by $w.\chi(t) = \chi(n^{-1}tn)$. Let

$$U_w' = \{ u_1 \cdots u_m : u_i \in U_{w.\alpha_i} \}$$

where $U_{\alpha_i}$ is the root subgroup of $w.\alpha_i$. Then $U_w'$ is a closed, connected subgroup of $U$ of dimension $m$, and the product map

$$B \times U_w' \rightarrow BwU, (b,u) \mapsto bwu$$

is an isomorphism of varieties. In particular, the dimension of $BwU$ is $\textrm{Dim } B + \ell(w^{-1})$, where $\ell$ is the length function.

minimalparabolic $F$-subgroup, right? In other words, $P$ is a Borel $F$-subgroup (but $W$ is the relative Weyl group, not the absolute one)? Do you want to just cover $G(F)$ by $P(F)wN(F)$'s, or really something geometric? The Borel-Tits relative structure theory on $F$-points provides a "Bruhat decomposition" on $F$-points generalizing what you describe for algebraically closed $F$, allowing any field and no quasi-split hypothesis (using a minimal parabolic $F$-subgroup). That $W$ is a constant $F$-group equal to $N_G(A_0)(F)/Z_G(A_0)(F)$ is a real theorem. $\endgroup$ – nfdc23 Dec 7 '16 at 3:58