# Center of maximal compact

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?

• My recollection is that all maximal compact subgroups of $SL(2,R)$ are isomorphic to the circle group $T$, and so the answer would appear to be no. – Yemon Choi Jan 14 '12 at 9:21
• It's perhaps interesting to note that $\dim Z(K)/(K\cap Z(G)) \leq 1$, so you're not so far off. – Allen Knutson Jan 16 '12 at 22:38

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.