The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am wondering if there are any useful analogues of this theory (say in characteristic 0) for maps into Artin stacks with infinite stabilizers.

Obviously some strong restrictions must apply. For instance, maps from curves into the classifying stack $B\mathbb{C}^*$ of $\mathbb{C}^*$ correspond to line bundles on the source curve, and so the stack of these maps cannot be separated without some strong condition.

Has anyone thought about what could be done to build a useful theory of stable maps into some types of Artin stack with infinite stabilizers? Are there some conditions on the target stack so that the resulting stack of stable maps, whatever those are, would form a nice stack itself: say, proper? Deligne-Mumford? with projective coarse moduli space? Or are these things completely unreasonable to hope for?

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    $\begingroup$ For stacks of the form $BG$ with $G$ semisimple (or reductive), you should look at the work of Martens-Thaddeus. In my opinion, this work is the moral equivalent of 2-pointed twisted stable maps from genus 0 curves to $BG$. $\endgroup$ – Jason Starr Dec 19 '13 at 20:07
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    $\begingroup$ Yes. Or rather, the fiber of the evaluation map to $BG \times BG$. Of course it's unreasonable to hope for a Deligne-Mumford stack in general (as the stack itself times $\overline{M}_g$ appears as the moduli space of constant stable maps), but one can ask this of fibers of the evaluation map. $\endgroup$ – Michael Thaddeus Dec 20 '13 at 23:38

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