Harmonic functions on nilpotent groups (Q5943608)

Let \(G\) be a locally compact group with left Haar measure \(\lambda _{G}\) and \(M_{G}\) the convolution algebra of bounded measure. \(M(G)\) contains \(L^{1}(G)\) as an ideal so \(M(G)\) acts on the right of \(L^{1}(G)\). Consequently we have the adjoint action of \(M(G)\) on \(L^{\infty}(G)\) given by \((\mu \circ F,\xi) =(F,\xi ^{*}\mu)\) where \(\mu \in M(G)\), \(\xi \in \) \(L^{1}(G)\), \(F\in L^{\infty }(G)\) and \((F,\xi) =\int F(g)\xi (g) d\lambda _{G}(g)\). If \(G\) is a probability measure on \(G\), then \(F\in L^{\infty }(G)\) is said to be \(\sigma \)-harmonic if \(\sigma \circ F=F\). It is easy to see that constant functions are \(\sigma\)-harmonic and, more generally, if \(\sigma\) is supported on a closed subgroup \(H\) of \(G\) and \(F\) is constant on the left cosets of \(H\), then \(F\) is \(\sigma\)-harmonic. In the present paper, the author studies conditions that all \(\sigma\)-harmonic functions are constant. A necessary condition for this is that the support of \(\sigma\) generates \(G\) as a closed group, that is that \(\sigma\) is not supported on any proper closed subgroup. The question arises of whether this condition is sufficient. It is known that in some particular cases (Choquet and Deny - for abelian groups, Dynkin and Malyutov - when the semigroup generated by the support of \(\sigma\) is \(G \)) the answer is positive. \textit{C.-H. Chu} and \textit{T. Hilberdink} [Integral Equations Oper. Theory 26, 1-13 (1996; Zbl 0878.43001)] proved that the answer is also positive for right uniformly continuous functions on nilpotent groups, and \textit{H. Furstenberg} [Proc. Sympos. Pure Math., Vol. 26, 193-229, A.M.S., Providence, R.I. (1973; Zbl 0289.22011)] in the case where \(G\) is nilpotent and \(\sigma\) satisfies the condition: for each \(g\in G\) there is a positive integer \(n\) such that \(L^{1}(g\sigma ^{n})\cap L^{1}(\sigma ^{n})\neq \emptyset\). In the present paper, the author shows that for nilpotent groups of class 2 the condition that \(G\) is the closed subgroup generated by the support of \(\sigma\) implies that all \(\sigma\)-harmonic functions are constant, but unfortunately the author's proof does not carry over to nilpotent groups of higher class than 2 without supplementary hypotheses. However the author establishes a result (Theorem 4.5) which includes his above mentioned result and Furstenberg's result, using non-probabilistic methods.