The relative Langlands program, as developed by Sakellaridis and Venkatesh, conjectures relationships between spherical varieties and automorphic L-functions. Locally, for X an affine spherical variety, the IC complex of the formal arc space of X is conjecturally related to special values of local L-functions. We make this relation precise by showing that a nearby cycles of the IC complex determines a crystal, in the sense of Kashiwara, and this crystal is conjecturally (and often provably) the crystal basis of a finite-dimensional representation of the dual group. This is joint work with Yiannis Sakellaridis.