In this talk, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a Beurling-Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the underlying Lie group. For this purpose, we need to introduce a particular dense subalgebra of the Beurling-Fourier algebra, which we call an analytic subalgebra. We first introduce an analytic subalgebra allowing a ``local" solution for general connected Lie groups. We will demonstrate that a ``global" solution is also possible for simply-connected nilpotent Lie groups.