Structural conjectures for endomorphism rings of degenerate representations

Let $e$ be a finite sum of primitive idempotents in the center $Z_n$ of $\operatorname{Rep}_{\mathcal{O}}(G_n)$, let $\mathcal{P}_{\lambda}$ be the associated projective object, let $eE_{\lambda}$ be its endomorphism ring after applying $e$, and let $eZ_{\lambda}$ be the quotient of $eZ_n$ acting faithfully on $eE_{\lambda}$. Let $\eta$ range over the minimal primes of $eZ_{\lambda}$. Structural conjecture. The following assertions hold: (1) $eZ_{\lambda}$ is reduced and flat over $\mathcal{O}$; (2) for every minimal prime $\eta$, the map $eZ_{\lambda}\to eE_{\lambda}$ becomes an isomorphism after localization at $\eta$; (3) the natural map

$$eE_{\lambda}\longrightarrow\prod_{\eta}(eE_{\lambda})_{\eta}$$

is injective, so in particular $eE_{\lambda}$ is reduced, commutative, and flat over $\mathcal{O}$; and (4) $eE_{\lambda}\otimes_{\mathcal{O}}\overline{\mathcal{K}}$ is a smooth $\overline{\mathcal{K}}$-algebra. These properties refine the paper's characteristic-zero description of the endomorphism rings; their validity in the stated integral setting is conjectural.