The Grothendieck ring of complex varieties $K(Var\_\mathbb C)$ is the free abelian group generated by isomorphism classes $[X]$ of $\mathbb C$-varieties, modulo the scissor relation $[X]=[Z]+[X\setminus Z]$ for every closed subvariety $Z\subset X$. The product is given by $[X]\cdot [Y]=[X\times\_\mathbb CY]$. There is a similar construction for stacks, leading to a ring $$K(St\_\mathbb C).$$ One has to restrict to stacks with affine stabilizers. This construction is treated for example in section 3 of [this paper by Bridgeland](https://arxiv.org/pdf/1002.4372v1.pdf). One can show that $K(St\_\mathbb C)$ can be obtained from $K(Var\_\mathbb C)$ by inverting the classes of all special groups, or *equivalently* all classes $[GL\_d]$ for $d\geq 1$, or *equivalently*, the classes $\{\mathbb L,\mathbb L^i-1:i\geq 1\}$, where $\mathbb L=[\mathbb A^1]$.


I am missing some basic point because it seems to me that all classes $[G]$ (for $G$ special) are invertible in 
$K(St\_\mathbb C)$ so that if I have a quotient stack $X/G$ its class will be $$[X/G]=[X]/[G]\in K(St\_\mathbb C)$$ regardless of the action of $G$ on $X$, and this is confusing. Here are a couple of examples I would like to understand:


1. What is the class of $GL\_2/GL\_2$, where $GL\_2$ acts on itself by conjugation? I would like this to be different from $[pt]=1$.
2. What is the class of weighted projective space $\mathbb P(a\_0,\dots,a\_{n-1})=(\mathbb C^{n}\setminus 0)/\mathbb C^\*$? I would like this to be different from the class $[\mathbb P^{n-1}]=1+\mathbb L+\cdots+\mathbb L^{n-1}$.


Thank you!



What is the motivic class of a quotient stack?
Yes: if $G$ is a special group, then $[X/G] = [X]/[G]$ in the Grothendieck group of stacks. This is the analogue of the fact that if $X \to E$ is a $G$-torsor over an algebraic variety (and $G$ is still special) then $[X] = [E]\cdot [G]$ in the Grothendieck group of varieties. In the category of stacks, the map $X \to X/G$ is always a $G$-torsor, regardless of the action of $G$ on $X$. Does this help?


Qiaochu's toy example is very relevant but also somewhat orthogonal to this situation: finite groups are not special, and $[X/G] = [X]/[G]$ is almost never true in the Grothendieck group of stacks (or varieties, for that matter) if $G$ is finite. For example, $\mathbb G\_m$ is a finite étale double cover of itself, but its class in the Grothendieck group is nonzero. A more interesting example is that the class of $BG$ is in many cases (but not always!) just the class of a point: see "A geometric invariant of a finite group" by Torsten Ekedahl.


