This note extends the classical theorem of Minkowski on lattice points and convex bodies in ℝn 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.
Cite this article
Martin Moskowitz, An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups. Port. Math. 67 (2010), no. 4, pp. 541–546DOI 10.4171/PM/1877