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 g of left-invariant vector fields on a Lie group G and we assume that S Lie generates g . We say that a function f:G→R (or more generally a distribution on G ) is S -polynomial if for all X∈S there exists k∈N such that the iterated derivative Xkf 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∈S , they form a finite-dimensional vector space. Second, if 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 g are equivalent notions.