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

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.