Transverse parabolic structures and transverse BGG sequences

Manifolds endowed with a parabolic geometry in the sense of Cartan come with natural sequences of differential operators and their analysis provide the so called (curved) BGG sequence of {\v C}ap, Slov{\'a}k and Sou{\v c}ek. The sequences involved do not form an elliptic complex in the sense of Atiyah but enjoy similar properties. The proper framework to study these operators is the filtered calculus associated to the natural filtration of the tangent bundle induced by the parabolic geometry. Such analysis was carried over by Dave and Haller in a very general setting. In this article we use their methods associated with the transversal index theory for filtered manifolds developped by the author in a previous paper to derive curved BGG sequences for foliated manifolds with transverse parabolic geometry.