Difference equations associated with groups, representative functions.
Let \(G\) be a group and \(H\) be its subgroup. The function \(f: G\to\mathbb{C}\) is called right \(H\)-representative (respectively left) if the complex vector space generated by the right translates \(f_h\) (respectively left), \(h\in H\) is finite-dimensional. If \(H = G\), \(f\) is called representative. The author, inspired by the results of \textit{J.-P. Bézivin} and \textit{F. Gramain} [Ann. Inst. Fourier 43, No. 3, 791--814 (1993; Zbl 0796.39006)], establishes the link with the system of difference equations with constant coefficients. Let \(G\) be a connected complex Lie group and \(H\) be its subgroup. If every complex valued entire function on \(G\) and right or (left) \(H\)-representative is representative, then every entire \(H\)-invariant function is constant. The converse is also true if \(G\) is abelian or nilpotent.
scientific article; zbMATH DE number 2123571