The center–cohomology isomorphism conjecture for the small quantum group

Let ${}^0{{\mathcal G}{\mathfrak{r}} }^\zeta_{\gamma_\ell}$ be the affine Springer fiber appearing in Theorem A, let $\mathrm{u}_\zeta$ be the small quantum group, and let $\check\Lambda$, $\check T$, $W_{\ell,{\operatorname{ex}\nolimits}}$, and $\check G$ denote the associated coweight lattice, torus, extended affine Weyl group, and Langlands-dual group. The cohomology algebra carries the embedding $\mathbf{a}$ into the center of the indicated module category.

Center–cohomology isomorphism conjecture. The algebra embedding

$$\mathbf{a}:H^\bullet({}^0{{\mathcal G}{\mathfrak{r}} }^\zeta_{\gamma_\ell})\hookrightarrow Z(\mathrm{u}_\zeta\operatorname{-mod}\nolimits^{\check\Lambda})$$

is an isomorphism. Moreover, it restricts to isomorphisms

$$H^\bullet({}^0{{\mathcal G}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{\ell{\check\Lambda}}\simeq Z(\mathrm{u}_\zeta)^{\check T},$$ $$H^\bullet({}^0{{\mathcal G}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\operatorname{ex}\nolimits}}}\simeq Z(\mathrm{u}_\zeta)^{\check G}.$$

This conjecture asserts that the cohomology of the affine Springer fiber realizes the relevant centers of the small quantum group and its module category. Its first isomorphism and the $\check T$-invariant specialization are stated earlier as conjectured extensions of the known embedding; the $W_{\ell,{\operatorname{ex}\nolimits}}$-invariant statement extends a conjecture of Bezrukavnikov–Qi–Shan–Vasserot, known in the principal-block case in type $A$.