*EDIT: There are serious problems with the definition below; see the comment thread below for those problems and some thoughts on addressing them. I'm leaving the question up for now since I think the **idea** is still interesting, but it is definitely broken at the moment.*




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Suppose $\mathbb{P}$ is a forcing notion and $\nu$ is a $\mathbb{P}$-name. Say that a $\mathbb{P}$-name $\mu$ is *$\nu$-invariant* if, whenever $G, H$ are $\mathbb{P}$-generic and $\nu[G]=\nu[H]$, then $\mu[G]=\mu[H]$. OK fine, that's nonsense, but it's easy to fix: formally, $\mu$ is $\nu$-invariant if whenever $\mathbb{Q}$ is a forcing notion and $\gamma\_0,\gamma\_1$ are $\mathbb{Q}$-names which are forced to be $\mathbb{P}$-generic and have $\nu[\gamma\_0]=\nu[\gamma\_1]$, then it is forced that $\mu[\gamma\_0]=\mu[\gamma\_1]$. 


Say further that a name $\mu$ is *hereditarily $\nu$-invariant* if it and each name in its transitive closure is $\nu$-invariant, and let $HI\_\nu$ denote the class of hereditarily $\nu$-invariant $\mathbb{P}$-names.


If $G$ is $\mathbb{P}$-generic, then it's not hard to show that $\{\mu[G]: \mu\in HI\_\nu\}$ is a model of ZF; call it "$V[G]\_\nu$". My question is this: 



> 
> Does the model $V[G]\_\nu$ have a snappy description as HOD of something, or as a symmetric submodel? By Grigorieff we know that it is such a model; but I don't immediately see how to *describe* it nicely as such.
> 
> 
> 


(I'm particularly interested in the case when $\nu$ is a name for a set of reals, if that matters.)


I'm interested in the $HI\_\nu$-construction because it gibes nicely with ideas from computability theory - specifically on a recasting of Steel forcing I've used, but also other things - and it makes a couple arguments I'm working on nicer. Of course, it has no broader mathematical value since there are provably no models of this form which aren't of better-known forms; but it's still something I find neat.



Invariant names and submodels of forcing extensions
The original definition has a problem with the hereditary requirement, unless your name is something particularly nice, e.g. a name for a subset of the ground model. But in the latter case, you just get a model of $\sf ZFC$, since you essentially get $V[A]$ for some $A\subseteq V$.


In the comments you suggest instead considering something which is invariant under interpreting some subset of $\nu$ correctly, but this turns out to be quite similar to just the usual symmetric extensions: you have a name, and you consider all the automorphisms which preserve it (not necessarily actual equality, generic equality is all the same here), and then you look at things which get preserved when some subset of $\nu$ is being interpreted "correctly" rather than the whole set.


This just translates to an automorphism group of $\Bbb P$, and some normal filter of subgroups. At least in the "relatively intuitive case".


