An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups

Abstract

This note extends the classical theorem of Minkowski on lattice points and convex bodies in $R^n$ to 2-step simply connected nilpotent Lie groups with a $Q$-structure. This includes all groups of Heisenberg type. More generally (and more naturally), it works for any simply connected nilpotent Lie group with a $Q$-structure whose Lie algebra admits a grading of length 2. Here a new invariant associated with the grading occurs which we call the degree. It explains why some directions are more equal than others.