This question is mainly a reference request.

I was wondering if there exists such a thing as the *symmetric product* of a quotient stack $[U/G]$, and in particular I would be interested in the motivic class of this object. For example, if $U=\textrm{pt}$ and $G=\mathbb C^\times$, and if one can make sense of $\textrm{Sym}^n(\textrm{B}\mathbb C^\times)$, then one can also write the motivic generating series $$A(t)=\sum_{n\geq 0}\bigr[\textrm{Sym}^n(\textrm{B}\mathbb C^\times)\bigr]\cdot t^n\in K(\textrm{St}_{\mathbb C})[[t]].$$
Has this been studied somewhere?

Thanks!