Let $k$ be a $p$-adic field, $G$ a connected reductive group over $k$ with minimal parabolic $P_0$ containing a maximal split torus $A_0$. Let $W = N_G(A_0)(k)/Z_G(A_0)(k)$ be the Weyl group, and $S \subset W$ the simple reflections from $P_0$.

For $\theta, \Omega \subset S$, we have the standard parabolic subgroups $P_\theta, P_\Omega$.

Each double coset $P_{\theta}w P_{\Omega}$, for $w \in W$ with a $k$-rational representative, is a locally closed subvariety of $G$, with $P_\theta wP_\Omega(k) = P_\theta(k)wP_\Omega(k)$.

How do we know that the quotient $P_\theta \backslash P_\theta wP_\Omega$ is an algebraic variety over $k$? This variety and its dimension are considered in Casselman's notes on representation theory, Chapter 6. I do not know the general theory of quotients of algebraic group actions which would make sense out of something like this.

Once this variety is made sense of, can we say that its $k$-rational points coincides with $P_\theta(k) \backslash P_\theta(k) wP_\Omega(k)$?