We study families of infinite-dimensional algebras that are similar to semisimple Lie algebras as well as symplectic reflection algebras. Infinitesimal Hecke algebras are deformations of semidirect product Lie algebras, and we study two families over $\mathfrak{sl}(2)$ and $\mathfrak{gl}(2)$. Both of them have a triangular decomposition and a nontrivial center, which allows us to define and study the BGG Category $\mathcal{O}$ over them - including a (central character) block decomposition, and an analog of Duflo’s Theorem about primitive ideals. We then discuss certain related setups.