Talk:Satake isomorphism

"grassmannian"

What does this mean?

It's easy to see that $Gr=G(K)/G(O)$ is grassmannian.

It seems as if "grassmannian" is being used as an adjective, but I don't know what that adjective means. If this sentence is trying to say $Gr=G(K)/G(O)$ is a Grassmannian, I still don't understand it: in what sense is it a Grassmannian? I think of a Grassmannian a reductive algebraic group modulo a maximal parabolic. Is that the case here? John Baez (talk) 08:51, 14 September 2022 (UTC)

I had the same question. After some further reading, here's one way to see $Gr$ as something like a Grassmannian for $G=GL_{n}$. In that case, the points of $Gr$ can be identified with $O$-lattices of maximal rank in $K^{n}$, which is something like the $O$-points of a 'Grassmannian of subspaces' for the quasicoherent module $K^{n}$ of $O$. I *think* it's even explicitly an inductive limit of honest Grassmannians (of quotients) of $O_{i}^{*}$, where $O_{i}$ runs through the finitely generated $O$-submodules of $K^{n}$ and $O_{i}^{*}$ is the $O$-linear dual.

Something similar should also work for other groups, where one adds some algebraic conditions on the lattices (i.e. unimodular for SL).

I feel that the article here is hardly comprehensible to anyone not already familiar with the subject, but I'm not familiar enough to write a better version. Sloth sisyphos (talk) 12:44, 10 May 2023 (UTC)