Multipliers, $W$-algebras and the growth of generalized polynomial identities

Let $A$ be a $W$-algebra over a field $F$ of characteristic zero, where $W$ is any $F$-algebra. We first develop a comprehensive theory of generalized identities independent of the algebraic structure of $W$, using the multiplier algebra of $A.$ Then, we investigate the generalized variety generated by the $k\times k$ matrix algebra with a suitable action, proving that it exhibits almost polynomial growth of the generalized codimensions. Furthermore, we characterize the generalized varieties of almost polynomial growth generated by finite dimensional $W$-algebras. Finally, we provide a counterexample to the Specht property of generalized $T_W$-ideals in characteristic zero.