Alternative methods to construct G‑equivariant Rockland splittings

Determine whether there exists a method—potentially via the Heisenberg calculus for multifiltered manifolds as developed in Yuncken [51]—to construct G‑equivariant Rockland splittings in the sense of Definition 6.2 for graded Rockland sequences arising from BGG complexes, specifically for the SL(3, F)‑equivariant BGG sequence on the flat parabolic manifold SL(3, F)/B(F), where standard Heisenberg calculus does not yield such splittings.

Background

The paper establishes K‑homology classes for graded Rockland sequences (including BGG complexes) when suitable G‑equivariant Rockland splittings exist. The main analytic tool employed is the Heisenberg calculus on Carnot manifolds.

For the flat parabolic manifold SL(3, F)/B(F), the authors note that while the BGG sequence is SL(3, F)‑equivariant, there is no obvious way—using the standard Heisenberg calculus—to produce SL(3, F)‑equivariant Heisenberg splittings; only splittings for subgroups of rank one or less are accessible. This motivates the search for alternative methods to construct G‑equivariant Rockland splittings.

The authors suggest that a Heisenberg calculus adapted to multifiltered manifolds (as in [51]) might address the obstruction seen in higher‑rank settings, but they do not know a method that achieves this. Later, a no‑go theorem shows the standard Heisenberg approach fails in higher rank, reinforcing the need for new techniques.