Let G be a connected reductive group and K a maximal compact subgroup of G, is it true that the center of K is the intersection of the center of G with K?
To elaborate on Yemon's comment: a noncompact, irreducible Hermitian symmetric space is of the form $G/K$, where $G$ is a noncompact simple Lie group with trivial center, and $K$ is a maximal compact subgroup with nondiscrete center; see e.g. Thm 6.1 in Chap. VIII of S. Helgason, ``Differential geometry, Lie groups and symmetric spaces'', Academic Press, 1978. So there are lots of examples where the two centers are un-related.