  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.
