Central character of an associative-algebra module ($\theta :Z(A)\to \mathbb{C}$)

A module over a complex associative algebra $A$ has central character $\theta$ if every element $z$ of the centre acts as the scalar $\theta (z)$, where $\theta$ is a unital algebra homomorphism. A Verma module has a central character: the centre preserves its one-dimensional highest-weight space, and centrality extends that scalar action from the cyclic generator to every vector. A nonzero homomorphism between two modules with scalar central actions forces their central characters to coincide. A generalized central character allows each $z-\theta (z)$ to act locally nilpotently instead of as zero.