Orateur
Description
Let $G$ be a semisimple Lie group with finite centre, $K$ a maximal compact subgroup and $P$ a parabolic subgroup of $G$. We present a new construction of Poisson transforms between vector bundle valued differential forms on the homogeneous parabolic geometry $G/P$ and its corresponding Riemannian symmetric space $G/K$ which is tailored to the exterior calculus and can be fully described by invariant elements in finite dimensional representations of reductive Lie groups. Furthermore, we show how these transforms are compatible with several invariant differential operators, which induce a strong connection between Bernstein-Gelfand-Gelfand complexes on $G/P$ and twisted deRham complexes on $G/K$. Finally, we consider the special case of the real hyperbolic space and its conformal boundary and discuss Poisson transforms of differential forms with values in the bundle associated to the standard representation $\mathbb{R}^{n+1,1}$ of $G = SO(n+1,1)_0$.
