Affine Deligne–Lusztig induction conjecture for general-position characters of anisotropic tori

Let $G=\operatorname{GL}_n$ and let $b=1$, so that $J_1(F)=G(F)$. For a suitable pair $(f,\underline{w}_f)$, write

$$R_{f,\underline{w}_f,1}\colon \operatorname{Rep}_{\overline{{\mathbb Q}}_\ell}(\tilde{I}_{f,\underline{w}_f}/I^f)\longrightarrow \operatorname{Rep}_{\overline{{\mathbb Q}}_\ell}(G(F))$$

for the affine Deligne–Lusztig induction map, and suppose that $\tilde{I}_{f,\underline{w}_f}$ surjects onto the relevant torus. Affine Deligne–Lusztig induction conjecture. The collection of these maps has the following properties: (A) if $\mathfrak{T}$ is anisotropic modulo the center of $G$, and $\chi$ is a character of $\mathfrak{T}(F)$ in sufficiently general position, then there are $f,\underline{w}_f$ such that $\tilde{I}_{f,\underline{w}_f}\twoheadrightarrow\mathfrak{T}(F)$ and $R_{f,\underline{w}_f,1}(\chi)$ is an irreducible supercuspidal representation of $G(F)$; (B) the map $\chi\mapsto R_{f,\underline{w}_f,1}(\chi)$ in (A) is injective up to Galois conjugation; and (C) this map coincides with automorphic induction realized via cuspidal types. This proposes a geometric realization of ramified automorphic induction and predicts that the resulting affine Deligne–Lusztig cohomology recovers precisely the relevant irreducible supercuspidal representations, with uniqueness up to Galois conjugation. The statement is formulated for $\operatorname{GL}_n$ and $b=1$; the source notes that analogous assertions for more general reductive groups may involve endoscopic phenomena.