Euler characteristics of linear symplectic quotients and $\operatorname{O}(2)$-spaces

We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the circle and real and unitary representations of the real $2\times 2$ orthogonal group. In the case of translation groupoids associated to linear symplectic quotients of representations of a arbitrary compact Lie group $G$, we show that unlike the other cases, the $\Gamma$-Euler characteristic depends only on the group and not on the representation.