The Gelfand spectrum of a Fourier algebra is known to be able to detect underlying group structure. We will introduce a weighted version of Fourier algebras and continue the same line of research, namely focusing on their Gelfand spectra. We suspect that the resulting Gelfand spectrum can detect the complexification of Lie groups. We will demonstrate that the suspicion is actually true in some cases focusing on the real line (as a motivating example) and the Heisenberg group. We will end the seminar with some open problems.