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](https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf) 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 $P\_\theta(k) \backslash P\_\theta(k) wP\_\Omega(k)$?



The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$
I originally gave an [answer](https://mathoverflow.net/a/315163) that relied on a claim in [Casselman - Introduction to the theory of admissible representations of $p$-adic reductive groups](https://personal.math.ubc.ca/%7Ecass/research/pdf/p-adic-book.pdf) that, as you pointed out by linking to [Errata for Casselman's unpublished notes](https://mathoverflow.net/questions/306675/errata-for-casselmans-unpublished-notes), is wrong; so I deleted the answer. Here's another try.


I hope that I may use $\Theta$ instead of $\theta$ for a subset of the set of simple roots.


The natural map $(w^{-1}P\_\Theta w \cap P\_\Omega)\backslash P\_\Omega \to P\_\Theta\backslash P\_\Theta w P\_\Omega$ is an isomorphism of varieties, so $P\_\Theta\backslash P\_\Theta w P\_\Omega$ is a quasi-projective variety. That seems like it must not be what you were asking for; if not, please let me know, so that I can try to fix this part of the answer.


Since $w^{-1}P\_\Theta w \cap P\_\Omega$ is smooth, we have that $P\_\Omega(k^\text{sep}) \to ((w^{-1}P\_\Theta w \cap P\_\Omega)\backslash P\_\Omega)(k^\text{sep})$, and hence $P\_\Omega(k^\text{sep}) \to (P\_\Theta\backslash P\_\Theta w P\_\Omega)(k^\text{sep})$, is surjective.


Since $(w^{-1}P\_\Theta w \cap P\_\Omega)U\_\Omega$ is a pseudo-parabolic subgroup of $P\_\Omega$, in the sense of [Conrad, Gabber, and Prasad - Pseudo-reductive groups, 2nd edition](https://doi.org/10.1017/CBO9781316092439), Definition 2.2.1, we have that $P\_\Omega(k) \to ((w^{-1}P\_\Theta w \cap P\_\Omega)U\_\Omega\backslash P\_\Omega)(k)$ is surjective by Lemma C.2.1 there. Let $g$ be a point of $(P\_\Theta w P\_\Omega)(k^\text{sep})$ whose image in $(P\_\Theta\backslash P\_\Theta w P\_\Omega)(k^\text{sep})$ is ($k$-)rational. Then there is some $h \in P\_\Omega(k)$ such that $g$ belongs to $(P\_\Theta w U\_\Omega h)(k^\text{sep}) = (P\_\Theta w U\_\Omega)(k^\text{sep})h$. (If you're uncomfortable with pseudo-parabolic subgroups, then it's just a bit of fiddling to handle the genuine parabolic subgroup $w^{-1}P\_\Theta w \cap M\_\Omega$ of $M\_\Omega$ instead.)


It remains to show that $U\_\Omega(k) \to (P\_\Theta\backslash P\_\Theta w U\_\Omega)(k)$ is surjective. This is where we come to the [claim](https://personal.math.ubc.ca/%7Ecass/research/pdf/p-adic-book.pdf#page=11) of Casselman that isn't true for the full quotient $P\_\Theta\backslash P\_\Theta w P\_\Omega$—but we've got down to, as it were, ‘the unipotent part’, where it's fine! We have that $P\_\Theta\backslash P\_\Theta w U\_\Omega$ is isomorphic, as a variety, to $(w^{-1}P\_\Theta w \cap U\_\Omega)\backslash U\_\Omega$. Since both $w^{-1}P\_\Theta w \cap U\_\Omega$ and $U\_\Omega$ are smooth, connected, unipotent subgroups of $G$ that are normalized by $M\_\emptyset = C\_G(A\_\emptyset)$, and since the weights of $A\_\emptyset$ on $w^{-1}P\_\Theta w \cap U\_\Omega$, and on $U\_\Omega$, form closed subsets of the relative root system of $G$ with respect to $A\_\emptyset$, each is directly spanned by relative root groups ([Borel - Linear algebraic groups](https://doi.org/10.1007/978-1-4612-0941-6), Proposition 21.9(ii)). In particular, since both sets of roots are divisible (in the sense that, if $a$ is a relative root such that $2a$ is a weight of $A\_\emptyset$ on either group, then $a$ is also a weight of $A\_\emptyset$ on that group), we have that there is a section, *as varieties*, of $U\_\Omega \to (w^{-1}P\_\Theta w \cap U\_\Omega)\backslash U\_\Omega$, hence of $U\_\Omega \to P\_\Theta\backslash P\_\Theta w U\_\Omega$.


All together, we have shown that $P\_\Omega(k) \to (P\_\Theta\backslash P\_\Theta w P\_\Omega)(k)$ is surjective, so that we may identity $(P\_\Theta\backslash P\_\Theta w P\_\Omega)(k)$ with $P\_\Theta(k)\backslash P\_\Theta(k)w P\_\Omega(k)$, as you desired.


