a property of the characters for center of universal enveloping algebra

Question

Let $\mathfrak g$ be a complex simple Lie algebra. We fix Cartan subalgebra $\mathfrak h$ and a system of positive roots $\Psi$ for the root system of the pair $(\mathfrak g, \mathfrak h).$ For each $ \lambda \in \mathfrak h^\star$, via Harish-Chandra map, we obtain a character $\chi_\lambda$ of the center of the universal enveloping algebra for $\mathfrak g.$ Is there $D$ in the center of $U(\mathfrak g)$ so that the map $$ \lambda \mapsto \chi_\lambda (D)$$ is injective when restricted to the subset of dominant integral elements?. For $sl_2$ the Casimir element answers the question.