Changing the weight space for an eigenvariety

Question

Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing overconvergent eigenforms for the group $G$ with a corresponding weight space $\mathcal{W}^G$ (this could be for modular forms, Hilbert modular forms or whatever). The eigenvariety data includes a weight map from the eigenvariety $\mathcal{E}^G$ to $\mathcal{W}^G$.

Say we have a larger weight space $\mathcal{W} \supset \mathcal{W}^G$ with some homomorphism going from $\mathcal{W}^G$ to $\mathcal{W}$. Are there conditions on this homomorphism (and maybe further conditions) that would allow one to obtain an eigenvariety over $\mathcal{W}$?

Answer 1

I think this question is based on a misconception. “Being an eigenvariety” isn’t a rigorously defined property of a space (or map of spaces) which you could prove to hold or to not hold. It’s more like an umbrella term for a collection of constructions which all do roughly the same kind of thing, and which spaces are covered by that term is a matter of convention. (If I say that some rigid variety is proper, and my colleague says it isn’t, then one of us is wrong; but if I say it is an eigenvariety and he says it isn’t, then that is just a difference of interpretation, about which we can agree to disagree.)

It’s certainly an interesting and rewarding question to ask what other spaces eigenvarieties have natural maps to (e.g. various kinds of local or global Galois deformation spaces). But debating whether or not the space with the new map “is an eigenvariety or not” doesn’t really make sense.