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. 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:

- 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 $[pt]=1$.
- 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!