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

TL;DR

This paper investigates the index theory of curved Bernstein-Gelfand-Gelfand sequences in parabolic geometry, establishing conditions for their role in $K$-homology and demonstrating limitations for higher rank Lie groups.

Contribution

It provides a criterion for when BGG-sequences fit into $G$-equivariant $K$-homology and proves a no-go theorem for higher rank cases.

Findings

Established a condition for BGG-sequences in flat parabolic geometry to fit into $K$-homology.

Applied Heisenberg calculus to analyze the index problem.

Proved a no-go theorem showing the approach fails for higher rank Lie groups.

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.