A cardinal $\kappa$ is huge iff there is $\lambda>\kappa$ and a $\kappa$-complete normal ultrafilter on $P_{\leq \kappa}(\lambda)$, or, equivalently, on the set of families of subsets of $\lambda$ of order-type $\kappa$. [1] I want to know whether one can further assume that $U$ is concentrated on something smaller? For example, can one further assume that $U$ is concentrated on $<\kappa$-directed families of sets of size $\kappa$? Only on those directed families that contain $P_{\kappa}(\lambda)$? Covering families as above ?

In notation: Let $U$ be such an ultrafilter. Let $D_\kappa=\{X\in P_{\leq\kappa}(\lambda): \forall S\subseteq X(|S|<\kappa\implies \cup S \in X\} $ be the subset of all $\kappa$-directed subsets of $P_{\leq \kappa}(\lambda)$.
Is $D_\kappa\in U $ ? Let $D'_\kappa=\{X\in D_\kappa: P_{<\kappa}(\lambda)\subseteq X\}$ be the subset of all
$\kappa$-directed subsets of $P_{\leq \kappa}(\lambda)$ containing all sets of size less than $\kappa$.
Let $C'_\kappa=\{X\in D_\kappa: \forall S\in P_{\leq\kappa}(\lambda)\exists x\in X(S\subseteq x)\}$.

Is $D'_\kappa\in U$ ? Is $C'_\kappa\in U$? Can one find such an $U$ that the answers are positive?

I am also looking for any references describing in detail ultrafilter characterizations of large cardinals.

[1] The Ultrafilter Characterization of Huge Cardinals Robert J. Mignone Proceedings of the American Mathematical Society Vol. 90, No. 4 (Apr., 1984), pp. 585-590 http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733411-6/S0002-9939-1984-0733411-6.pdf