Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let $\mathbf{G}$ be a connected reductive complex algebraic group with split real form $(G,\sigma )$. Consider a strict wonderful $\mathbf{G}$-variety $\mathbf{X}$ equipped with its $\sigma$-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 $E$, 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.