The distribution of symmetry of a naturally reductive nilpotent Lie group

TL;DR

This paper investigates the symmetry distribution in naturally reductive nilpotent Lie groups, showing it aligns with invariant distributions from fixed vectors, extending known results from compact spaces and exploring quotient structures.

Contribution

It extends the understanding of symmetry distributions from compact to nilpotent Lie groups and analyzes the quotient by the symmetry foliation.

Findings

Symmetry distribution coincides with invariant distribution from fixed vectors.

Extension of known results from compact to nilpotent Lie groups.

Analysis of quotient spaces by the symmetry foliation.

Abstract

We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry.