Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let G be a connected reductive complex algebraic group with split real form ( G , σ ) . Consider a strict wonderful G ‐variety X equipped with its σ‐equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G ‐vector bundle over X . In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E given in terms of the spherical roots of X , and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points.