Brain teaser: mysterious moduli and local Langlands

Fix an integer n>1. Let X denote the moduli space of triples (\mathcal{E}_1, \mathcal{E}_2,f) where \mathcal{E}_i is a vector bundle of rank n on the Fargues-Fontaine curve which is trivial at all geometric points, and f: \mathcal{E}_1 \oplus \mathcal{E}_2 \to \mathcal{O}(1/2n) is an injection which is an isomorphism outside the closed Cartier divisor at infinity.

Brain teaser a. Prove that X is a locally spatial diamond over \breve{\mathbf{Q}}_p with a Weil descent datum to \mathbf{Q}_p.

Now, let D be the division algebra over \mathbf{Q}_p of invariant 1/2n, and let \tau be an irreducible representation of D^\times whose local (inverse) Jacquet-Langlands correspondent is supercuspidal. Note that D^\times acts on X by its natural identification with \mathrm{Aut}(\mathcal{O}(1/2n)).

Brain teaser b. Prove that the geometric etale cohomology of X satisfies the following:

R\Gamma_c(X_{\mathbf{C}_p},\overline{\mathbf{Q}_\ell})\otimes_{D^\times} \tau \cong \varphi_{\tau}[1-2n](\tfrac{1-2n}{2}) if \tau is orthogonal, and R\Gamma_c(X_{\mathbf{C}_p},\overline{\mathbf{Q}_\ell})\otimes_{D^\times} \tau \cong 0 if \tau is not orthogonal.

Here \varphi_\tau denotes the Langlands parameter of \tau.

It is probably not fair to call these brain teasers. Anyway, here is one big hint: the infinite-level Lubin-Tate space for \mathrm{GL}_{2n} is naturally a \mathrm{GL}_n(\mathbf{Q}_p)^2-torsor over X, by trivializing the bundles \mathcal{E}_i.


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