# 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 $\mathcal{E}_i$.