Polynomial and horizontally polynomial functions on Lie groups

TL;DR

This paper extends the concept of polynomial functions on Lie groups, showing that under certain conditions, these functions are analytic, finite-dimensional, and equivalent to Leibman polynomials in connected nilpotent groups.

Contribution

It introduces the notion of $S$-polynomial functions on Lie groups, proves their analyticity and finite-dimensionality, and establishes equivalences among different polynomial concepts in nilpotent groups.

Findings

$S$-polynomial functions are analytic and form finite-dimensional spaces.

In connected nilpotent groups, $S$-polynomial functions coincide with Leibman polynomials.

Various polynomial notions are equivalent in connected nilpotent Lie groups.

Abstract

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 $\mathbb G$ and we assume that $S$ Lie generates $\mathfrak g$. We say that a function $f:\mathbb G\to \mathbb R$ (or more generally a distribution on $\mathbb G$) is $S$-polynomial if for all $X\in S$ there exists $k\in \mathbb N$ 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.