Geometric Casselman–Shalika conjecture

Let $G$ be the reductive group, $T$ a maximal torus, and $\rho$ the half-sum of positive roots used to define the affine Grassmannian strata. For a dominant coweight $\lambda\in X_*(T)$ and coweights $\nu,\mu\in X_*(T)$ such that $\nu+\mu$ is dominant, let $\MV_{\lambda,\nu}$, $\cA_\lambda$, $h_\mu^{\lambda,\nu}$, and $\cL_\psi$ denote the mixed-characteristic geometric Casselman–Shalika objects appearing in the paper, and let $V^\alpha$ be the algebraic representation of $\widehat{G}$ of highest weight $\alpha$. Geometric Casselman–Shalika conjecture. There is a canonical isomorphism

$$H^i_c(\MV_{\lambda,\nu}, \cA_\lambda \otimes (h_\mu^{\lambda,\nu})^*\cL_\psi) \xrightarrow{\sim} \begin{cases} \operatorname{Hom}_{\widehat{G}}(V^\lambda \otimes V^\mu, V^{\mu+\nu})(-\langle \rho,\nu\rangle) & i = \langle 2\rho,\nu\rangle \text{ and } \mu \in X_*(T)_+ \\ 0 & \text{otherwise}. \end{cases}$$

This conjecture is presented as a generalization of the geometric Casselman–Shalika formula to mixed characteristic; it identifies compactly supported cohomology of the relevant Mirković–Vilonen-type space with a representation-theoretic multiplicity, including the indicated degree and Tate twist. Its resolution is not supplied in the source.