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)