Fix an integer . Let denote the moduli space of triples where is a vector bundle of rank on the Fargues-Fontaine curve which is trivial at all geometric points, and is an injection which is an isomorphism outside the closed Cartier divisor at infinity.
Brain teaser a. Prove that is a locally spatial diamond over with a Weil descent datum to .
Now, let be the division algebra over of invariant , and let be an irreducible representation of whose local (inverse) Jacquet-Langlands correspondent is supercuspidal. Note that acts on by its natural identification with .
Brain teaser b. Prove that the geometric etale cohomology of satisfies the following:
if is orthogonal, and if is not orthogonal.
Here denotes the Langlands parameter of .
It is probably not fair to call these brain teasers. Anyway, here is one big hint: the infinite-level Lubin-Tate space for is naturally a -torsor over , by trivializing the bundles .