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 $\mathfrak{g}$ of left-invariant vector fields on a Lie group $\mathbb G$andweassumethat$S$Liegenerates$\mathfrak g$\mathrm{.}Wesaythatafunction$f:\mathbb G\to \mathbb R$(ormoregenerallyadistributionon$\mathbb G$)is$S$-polynomialifforall$X\in S$thereexists$k\in \mathbb N$ such that the iterated derivative $X^{k}f$ 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 $X\in S$, they form a finite-dimensional vector space. Second, if $\mathbb{G}$ 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 $\mathfrak{g}$ are equivalent notions.