D’Alembert’s and Wilson’s equations on Lie groups

Abstract

Let G be a connected nilpotent Lie Group. We show that the solutions of the short and the long version of d’Alembert’s equation on G have the same form as on abelian groups. Furthermore we show that any solution of Wilson’s equation $$ f(xy) + f(xy^{-1}) = 2f(x)g(y) $$ in the case g ≠ 1 has the form $$ f = A(m + \check{m})/2 + B(m-\check{m})/2 $$ and $$ g = (m + \check{m})/2 $$ where $$ m : G \rightarrow {\mathbb C}^* $$ is a homomorphism and A and B are complex constants. Finally we solve Jensen’s equation $$ f(xy) + f(xy^{-1}) = 2f(x) $$ on a semidirect product of two abelian groups.