Let $G$ be (the points of) a connected, reductive group over a local field $k$ with maximal split torus $S$, minimal parabolic $P_0$, and Weyl group $W = N_G(S)/Z_G(S)$. Let $\Delta$ be the corresponding set of simple roots, and for $\Theta, \Omega \subset \Delta$, let $P_{\Theta}, P_{\Omega}$ be the corresponding parabolic subgroups. Let $C(w) = P_{\Theta}wP_{\Omega}$, so that $G$ is the disjoint union

$$\bigcup\limits_{w \in W_{\Theta} \backslash W/W_{\Omega}} C(w)$$

Define a partial ordering on $W_{\Theta} \backslash W/W_{\Omega}$ by setting $w < w'$ if $C(w) \subseteq \overline{C(w')}$. Let

$$G_w = \bigcup\limits_{w < w'} C(w')$$

Is this an open set in $G$? This is claimed in Casselman's notes on representation theory, but I don't see why this should be the case.


Okay now that I have thought about this some more, it actually falls out pretty formally. The cells $C(w_0)$ are the orbits of the group $P_{\Theta} \times P_{\Omega}$ acting on $G$, so each closure $\overline{C(w')}$ is a disjoint union of cells $C(w_0)$. By definition of the order relation, $\overline{C(w')}$ is the disjoint union of the $C(w_0)$ with $w_0 < w'$. Now

$$G - G_w = \bigcup\limits_{\substack{w' \in W_{\Theta} \backslash W/W_{\Omega}\\w \not< w'}} C(w')$$


$$\overline{G - G_w} =\bigcup\limits_{\substack{w' \in W_{\Theta} \backslash W/W_{\Omega}\\w \not< w'}} \overline{C(w')} = \bigcup\limits_{w \not< w'} \bigcup\limits_{w_0 < w'} C(w_0)$$

If we take one of the cells $C(w_0)$, for some $w' \not< w$ and $w_0 < w'$, then of course $w$ is not less than $w_0$, so $C(w_0)$ is contained in $G - G_w$. Thus $G - G_w$ is equal to its closure.

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