On polynomial functions on non-conmmutative groups

TL;DR

This paper explores different classes of polynomial-like functions on topological groups, analyzing when they coincide and extending the study to representations beyond the regular one, with implications for non-commutative group analysis.

Contribution

It introduces a comparison of two polynomial function classes on topological groups and extends the analysis to general group representations, including Montel type conditions.

Findings

Classes coincide for many groups but not all

Montel type versions are considered for generating subsets

Approach uses group representation theory

Abstract

Let $G$ be a topological group. We investigate relations between two classes of "polynomial like" continuous functions on $G$ defined, respectively, by the conditions (1) $\Delta_h^{n+1}f=0$ for every $h \in G$, and (2) $\Delta_{h_{n+1}} \Delta_{h_{n}}\cdots \Delta_{h_{1}}f=0$, for every $h_1,\cdots, h_{n+1} \in G$. It is shown that for many (but not all) groups these classes coincide. We consider also Montel type versions of the above conditions - when (1) and (2) hold only for steps $h$ in a generating subset of $G$. Our approach is based on the study of the counterparts of the discussed classes for general representations of groups (instead of the regular representation).