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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.
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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