Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.)
I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is homogeneous Suslin. Let $T$ be some tree so that $A = p[T]$. Suppose $\mathbb{P}$ is a small forcing in $V_{\omega + \omega}$. (I am most interested by forcing whose conditions are reals like Sacks forcing, Cohen forcing, or Random forcing.) Let $G \subseteq \mathbb{P}$ be $\mathbb{P}$-generic for $V$. Let $A_G = p[T]^{V[G]}$. The question is
Is there an elementary embedding $j : L(A,\mathbb{R}) \rightarrow L(A_G,\mathbb{R}^{V[G]})$?
Ultimately, I would like, under $\mathsf{AD}_\mathbb{R}$, to have some generic absoluteness about statements involving $A$ (as defined as $p[T]$).
This statement also resemble results that are proved in choice setting assuming large cardinals.
The only ideas I have are: In Solovay's paper about independence of DC, he showed that under $\mathsf{AD}_\mathbb{R}$, for all $A \subseteq \mathbb{R}$, $(A,R)^\sharp$ exists. In $\mathsf{AD}_\mathbb{R}$, $(A,\mathbb{R})^\sharp$ is homogeneously Suslin. So $(A,\mathbb{R})^\sharp = p[U]$ for some homogeneously Suslin tree $U$. I am hoping perhaps one can show that $p[U]^{V[G]} = (A_G,\mathbb{R}^{V[G]})^\sharp$. Therefore, $(A,\mathbb{R})^\sharp \subseteq (A_G, \mathbb{R}^{V[G]})^\sharp$. I believe this will imply that there is an elementary embedding from $L(A,\mathbb{R})$ into $L(A_G,\mathbb{R}^{V[G]})$.
I would like to know if the embedding above can exists under $\mathsf{AD}_\mathbb{R}$. Can my vague argument possibly work? Are there other ways to obtain this embedding or the desired generic absoluteness. Thanks for any information and references.
