Cardinal arithmetic in inner models

Question

I know that regular cardinals get computed weirdly in inner models of determinacy (this comes up in Jackson's analysis of the projective ordinals $\delta^1_n$); this is a question about a specific case of this.

Suppose $V$ contains a proper class of Woodins. Can we still have $\omega_2^{L(\mathbb{R})}=\omega_2$? If so, how high up can we have $\omega_n^{L(\mathbb{R})}=\omega_n$?

Answer 1

The second question admits an easy negative answer: in models of $\sf AD$ every $\omega_n$ is singular for finite $n>2$.

Kleinberg claimed in the following paper that this is due to Martin, but without reference. He also proved it as Corollary 2.2.

E. M. Kleinberg, ${\rm AD}\vdash $ “the $\aleph _{n}$ are Jonsson cardinals and $\aleph _{\omega }$ is a Rowbottom cardinal”, Ann. Math. Logic 12 (1977), no. 3, 229--248. MR 469769, Zbl 0378.02032.