Relative BGG sequences; I. Algebra

TL;DR

This paper introduces a relative version of Kostant's harmonic theory for Lie algebra homology, enabling new geometric and representation-theoretic applications involving nested parabolic subalgebras.

Contribution

It develops a relative harmonic theory and proves a corresponding Kostant's theorem, extending Lie algebra homology results to nested parabolic subalgebras.

Findings

Realization of representations with lowest weight via relative homology

All weights in an affine Weyl orbit can be realized as relative homology groups

Applications to invariant differential operators in differential geometry

Abstract

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with lowest weight in one (regular or singular) affine Weyl orbit. In the regular case, we show how all the weights in the orbit can be realized as relative homology groups (with different coefficients). These results are motivated by applications to differential geometry and the construction of invariant differential operators.