Newton-Okounkov bodies for Carnot groups

To a graded additive semigroup of lattice points one associates a convex body, called a Newton-Okounkov body (NO body for short :), which encodes information about the asymptotic behavior of the semigroup. This relatively simple construction applied to algebraic geometry situations has proven to be very fruitful, e.g. it gives a simple proof of the celebrated Hodge inequality. In this talk, we introduce a noncommutative analogue of Newton–Okounkov bodies for graded semigroups in lattices of Carnot groups. Using Carnot dilations, we associate to such a semigroup a compact body and identify it with a sub-Finsler unit ball. We prove asymptotic counting and equidistribution results for the graded pieces. The talk should be accessible to graduate students in algebra, geometry or analysis. This is a joint work in progress with Eugene Eyeson.