  We construct a canonical Thom isomorphism in Grojnowski's equivariant
elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with
vanishing Borel-equivariant second Chern class, which is natural under
pull-back of vector bundles and exponential under Whitney sum. It extends in
the complex-analytic case the non-equivariant sigma orientation of Hopkins,
Strickland, and the author. The construction relates the sigma orientation to
the representation theory of loop groups and Looijenga's weighted projective
space, and sheds light even on the non-equivariant case. Rigidity theorems of
Witten-Bott-Taubes including generalizations by Kefeng Liu follow.
