Equivariant bundles and isotropy representations

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split Γ-spaces. We show that equivariant principal G-bundles over split Γ-CW complexes X can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex A = Γ\X is a graph, with all edge stabilizers toral subgroups of Γ, we obtain a purely combinatorial classification of bundles with structural group G a compact connected Lie group. If G is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof-May-Segal [18] and Goresky-Kottwitz-MacPherson [10]