Euler characteristics of linear symplectic quotients and O (2) -spaces

We give explicit computations of the -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 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 -Euler characteristic depends only on the group and not on the representation.