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

Moskowitz, Martin

doi:10.4171/PM/1877

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Portugaliae Mathematica

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Volume 67, Issue 4, 2010, pp. 541–546

DOI: 10.4171/PM/1877

Published online: 2010-11-24

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

Martin Moskowitz^[1]

(1) The CUNY Graduate Center, New York, USA

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.

Keywords: Simply connected nilpotent group, Lie algebra, log-lattice, grading, degree, shrinking automorphism, homogeneous subadditive norm, volume, convex body

Moskowitz Martin: An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups. Port. Math. 67 (2010), 541-546. doi: 10.4171/PM/1877