Yes, the central character of the even $K$-types is trivial, and of the odd ones is the sign character.
Also, this can easily be seen from the classification of irreducible representation on Hilbert spaces.
More generally, the category of rep on Hilbert spaces of a locally compact group $G$ with trivial central character coincides with that of the rep theory on Hilbert spaces of a locally compact group $G/Z$, here $Z$ being the center of $G$.
Quote from Wikipedia: In 1973, Lepowsky showed that any irreducible (g,K)-module X is isomorphic to the Harish-Chandra module of an irreducible representation of G on a Hilbert space.