Fargues' conjecture for irreducible representations of the Weil group of GL_n

Let $E$ be the local field underlying the Weil group $W_E$, let $G=\mathrm{GL}_n$, and let $B(G)$ denote its Kottwitz set. Write $\mathrm{Bun}_n$ for the moduli stack of $G$-bundles and $D_{\mathrm{lis}}(\mathrm{Bun}_n,\overline{\mathbb{Q}}_\ell)$ for the corresponding lisse derived category. For $b\in B(G)$, let $j_b$ denote the inclusion of the corresponding stratum, let $G_b(E)$ be the associated inner form, and let $\mathcal{F}_{\pi}$ be the sheaf attached to a smooth representation $\pi$ of $G_b(E)$. Let $j$ denote the inclusion of the open stratum. Fargues' conjecture for $\mathrm{GL}_n$. For each irreducible continuous $\overline{\mathbb{Q}}_\ell$-representation $\mathbb{L}$ of $W_E$ of degree $n$, there exists an object $\mathrm{Aut}_{\mathbb{L}}\in D_{\mathrm{lis}}(\mathrm{Bun}_n,\overline{\mathbb{Q}}_\ell)$ such that: (1) $\mathrm{Aut}_{\mathbb{L}}$ is a Hecke eigensheaf with eigenvalue $\mathbb{L}$; (2) $\mathrm{Aut}_{\mathbb{L}}$ is cuspidal, in particular $\mathrm{Aut}_{\mathbb{L}}\cong j_!(j^*\mathrm{Aut}_{\mathbb{L}})$; and (3), for $b\in B(G)$ basic,

$$j_b^*\mathrm{Aut}_{\mathbb{L}}\cong\mathcal{F}_{\mathrm{LL}_b(\mathbb{L})},$$

where $\mathrm{LL}_b(\mathbb{L})$ is the generalized Jacquet–Langlands correspondent of $\mathbb{L}$, a smooth irreducible $\overline{\mathbb{Q}}_\ell$-representation of $G_b(E)$. This is the special case of Fargues' conjecture relevant to the paper; its status is not specified here, and the precise Hecke-eigensheaf condition is given elsewhere in the source.