Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

We establish an explicit correspondence of certain Arthur packets between real unitary groups and $p$-adic symplectic or orthogonal groups. This allows one to compute Arthur packets of real unitary groups by translating results from the $p$-adic side. A main ingredient in our proof is an explicit relation between Zuckerman's translation functor on the real side and the Jacquet functor on the $p$-adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and $p$-adic groups. Our approach also allows one to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and $p$-adic sides.