# The $\mu$-ordinary stratum and the Newton stratum

Here the following is stated:

Consider a pair $$(G,\mu)$$ with $$G/\mathbf{Q}_p$$ reductive and $$\mu \in X_{\ast}(G_{\overline{\mathbf{Q}_p}})$$ a minuscule cocharacter (defined over the reflex field $$E$$ of its conjugacy class), so we have an associated flag variety $$\mathcal{F}\ell_{G,\mu}$$ defined over $$E$$ as usual. Caraiani-Scholze defined a Newton stratification of this space, with the strata $$\mathcal{F}\ell^{b}_{G,\mu}$$ indexed by the Kottwitz set $$B(G,\mu)$$. Sean’s question is whether or not the $$\mu$$-ordinary stratum always coincides with the set $$\mathcal{F}\ell_{G,\mu}(E)$$. I have a proof when $$G$$ is quasisplit (and maybe it works more generally, I haven’t checked), but it uses fancy pants things like shtukas, diamonds, etc. Surely there is a direct argument.

Is there a reference (either a published paper or a preprint) where the statement above is proved using shtukas and diamonds? It does not have to be by this particular person, any reference would do. It has to use these notions. Alternatively, if somebody is feeling generous about their time, you can just post a complete proof as an answer.

I am an inexperienced mathematician who is desperate to get into the mathematics of “fancy pants things” so for any statement not involving “fancy pants things” that has a proof using “fancy pants things” I try to understand that proof. I have tried to ask the author by commenting on the blog post, did not work.