The cone conjecture for automorphic forms on Hodge-type Shimura varieties

Let $S_K$ be the special fiber of a Hodge-type Shimura variety, with associated morphism

$$\zeta\colon S_K\to \mathop{\text{$G$-\tt Zip}}\nolimits^\mu.$$

For a weight $\lambda\in X^*(T)$, let

$$C_K(\overline{\mathbb{F}}_p)=\{\lambda\in X^*(T)\mid H^0(S_K,\mathcal V_I(\lambda))\neq 0\}$$

and

$$C_{\operatorname{zip}}=\{\lambda\in X^*(T)\mid H^0(\mathop{\text{$G$-\tt Zip}}\nolimits^\mu,\mathcal V_I(\lambda))\neq 0\}.$$

For a cone $C\subset X^*(T)$, write

$$\langle C\rangle=\{\lambda\in X^*(T)\mid \exists N\geq 1,\ N\lambda\in C\}.$$

The cone conjecture. For any Hodge-type Shimura variety, one has

$$\langle C_K(\overline{\mathbb{F}}_p)\rangle=\langle C_{\operatorname{zip}}\rangle.$$

The inclusion $C_{\operatorname{zip}}\subset C_K(\overline{\mathbb{F}}_p)$ follows by pullback along the surjective morphism $\zeta$. The conjecture asserts that this inclusion becomes an equality after saturation, relating automorphic forms on Shimura varieties to the group-theoretical theory of $G$-zips.