The character of an ultrafilter $U$, denoted $\chi(U)$, is the minimal size of an $A \subseteq U$ such that $(\forall x \in U ) (\exists y \in A) y \subseteq x$. This cardinal characteristic has been studied for ultrafilters on $\omega$. For all nonprincipal $U$, $\omega_1 \leq \chi(U)$, and it is known that, consistently, there exists nonprincipal $U$ with $\chi(U) < 2^\omega$. My question is, for normal ultrafilters $U$ on a measurable cardinal $\kappa$, is it possible that $\chi(U) < 2^\kappa$?

Here are some relevant papers about ultrafilters on $\omega$, by Shelah, Brendle, and Hart:

MR1686797, MR0987317, MR2365799, MR2847327

(available through http://www.ams.org/mathscinet/)