Do we know under which conditions the Kunen tree (Recall the Kunen tree provides an analysis of the equivalence classes of functions $f: \omega_1 \to \omega_1$ with respect to the normal measure $W^1_1$ on $\omega_1$ under $AD$) is homogeneous, if it is? Do we know if and under which hypothesis the Martin tree (it is a generalization of the Kunen tree to all the projective hierarchy) is homogeneous? Any reference treating their homogeneity will be appreciated. Thx.
Let me answer my question which I stumbled on while looking at my profile. The answer is yes. Actually assuming $AD$, if $\kappa$ is less than the supremum of the Suslin cardinals then every tree $T$ on $\omega \times \kappa$ is weakly homogeneous (Martin and Woodin).
Also assuming that we have a Woodin $\delta$, a tree $T$ on $\omega \times \alpha$, $\alpha$ some ordinal, then there is a $\kappa < \delta$ such that in the generic extension $V[G]$ where $G$ is generic for $Col(\omega, \kappa)$, $T$ is $< \delta$ weakly homogeneous.
One can modify the assumption and just assume that there is a pair $T$ and $U$ of $\delta^+$ absolutely complementing trees.