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.

  • $\begingroup$ Surely $P$ is meant to be a minimal parabolic $F$-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
  • $\begingroup$ To be more specific about the distinction of decomposition on geometric vs. $F$-rational points, suppose $F$ admits a quadratic Galois extension $F'/F$ and let $G$ be the quasi-split special unitary group ${\rm{SU}}_3(F'/F)$ that has $F$-rank equal to 1. Then the relative Weyl group has order 2, but if $B$ is a Borel $F$-subgroup then $G$ cannot be covered by two $B$-double-cosets in view of the Bruhat decomposition over $\overline{F}$ and the size of the absolute Weyl group exceeding 2. More motivation and context (and precision) seems necessary in this question. $\endgroup$ – nfdc23 Dec 7 '16 at 4:33
  • $\begingroup$ In the situation I am looking at, $P$ is a maximal parabolic subgroup, and there is an argument I am trying to understand that involves looking at the a double coset $PwN$ to determine dimension of a certain subgroup of the unipotent radical of $N$. Perhaps it's best I just ask a new question. $\endgroup$ – D_S Dec 7 '16 at 5:00
  • $\begingroup$ OK. Since you mention viewing the situation as that of generalizing the case of a Borel subgroup over an algebraically closed field, it is indeed unclear what is being sought. For the quasi-split ${\rm{SU}}_3(F'/F)$ with $F$-rank equal to 1 (where $F'/F$ is a quadratic Galois extension), the Borel $F$-subgroups are also the maximal (proper) parabolic $F$-subgroups, so it is a counterexample to your question as posed. $\endgroup$ – nfdc23 Dec 7 '16 at 5:37

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