The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$

Question

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 $P_\theta(k) \backslash P_\theta(k) wP_\Omega(k)$?

Answer 1

I originally gave an answer that relied on a claim in Casselman - Introduction to the theory of admissible representations of $p$-adic reductive groups that, as you pointed out by linking to Errata for Casselman's unpublished notes, is wrong; so I deleted the answer.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, 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 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, 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.