Solving the index problem for (curved) Bernstein–Gelfand–Gelfand sequences

Abstract We study the index theory of curved Bernstein–Gelfand–Gelfand (BGG) sequences in parabolic geometry and their role in K -homology and noncommutative geometry. The BGG-sequences fit into K -homology, and we solve their index problem. We provide a condition for when the BGG-complex on the flat parabolic geometry G / P of a semisimple Lie group G fits into G -equivariant K -homology by means of Heisenberg calculus. For higher rank Lie groups, we prove a no-go theorem showing that the approach fails.