Polynomial and horizontally polynomial functions on Lie groups

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset S of the algebra [Formula: see text] of left-invariant vector fields on a Lie group [Formula: see text] and we assume that S Lie generates [Formula: see text] . We say that a function [Formula: see text] (or more generally a distribution on [Formula: see text] ) is S-polynomial if for all [Formula: see text] there exists [Formula: see text] such that the iterated derivative [Formula: see text] is zero in the sense of distributions. First, we show that all S-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent k in the previous definition is independent on [Formula: see text] , they form a finite-dimensional vector space. Second, if [Formula: see text] is connected and nilpotent, we show that S-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of [Formula: see text] are equivalent notions.