Chen–Zhu conjecture on irreducible components of affine Deligne–Lusztig varieties

Let $G$ be a reductive group over the relevant local field, let $b$ and $\mu$ be as in the paper, and let $X_\mu(b)$ be the associated affine Deligne–Lusztig variety. Write $\mathbb J_b$ for the $\sigma$-centralizer of $b$, let $\underline{\lambda}_G(b)$ be the best integral approximation of the Newton point of $b$, and let $\mathrm{MV}_\mu(\underline{\lambda}_G(b))$ denote the union of Mirkovi\c-Vilonen cycles indexed by coweights mapping to $\underline{\lambda}_G(b)$. Let $V_\mu(\underline{\lambda}_G(b))$ be the sum of the corresponding weight spaces in the irreducible representation $V_\mu$ of the dual group.

Chen–Zhu conjecture. There exists a natural bijection

$$\mathbb J_b \backslash \operatorname{Irr} X_\mu(b) \cong \operatorname{MV}_\mu(\underline{\lambda}_G(b)).$$

In particular,

$$|\mathbb J_b \backslash \operatorname{Irr} X_\mu(b)| = |\operatorname{MV}_\mu(\underline{\lambda}_G(b))| = \dim V_\mu(\underline{\lambda}_G(b)).$$

This conjecture predicts a parametrization of the $\mathbb J_b$-orbits of irreducible components of affine Deligne–Lusztig varieties by Mirkovi\c-Vilonen cycles, with the cardinality governed by the corresponding weight multiplicity in the geometric Satake representation.