A characterization of extenders of HOD

Assume $AD+V=L(\mathbb{R})$. Let $\kappa=\utilde{\delta}^2_1$, the supremum of all $\utilde{\Delta}^2_1$ prewellorderings. We prove that extenders on the sequence of \H that have critical point $\kappa$ are generated by countably complete measures. This provides a partial reversal of Woodin's result that the $<\Theta$-strongness of $\kappa$ in \H is witnessed by $\kappa$-complete ultrafilters on $\k$. The aforementioned characterization of extenders works in a more general setting for all cutpoint measurable cardinals of \H in all models of determinacy where the fine structural analysis of \H has been carried out. For example, it holds in the minimal model of the Largest Suslin Axiom. It also gives a simple proof of a theorem of Steel that the successor members of the Solovay sequence are cutpoints in \H (in models where \H analysis is carried out).