Graded hypoellipticity of BGG sequences

TL;DR

This paper extends hypoellipticity theory to filtered manifolds, develops a pseudodifferential calculus, and applies it to analyze the graded hypoellipticity of BGG sequences in parabolic geometries.

Contribution

It generalizes the Rockland criterion and BGG machinery to broader filtered manifolds, establishing graded hypoellipticity of BGG sequences.

Findings

Extended Rockland criterion to filtered manifolds

Constructed a pseudodifferential calculus on filtered manifolds

Proved BGG sequences are graded Rockland sequences

Abstract

This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein--Gelfand--Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.