  In this paper, we investigate the spectrum of a class of weighted Laplacians
on Cayley graphs and determine under what conditions the corresponding
eigenspaces are generically irreducible. Specifically, we analyze the spectrum
on left-invariant Cayley graphs endowed with an invariant metric, and we give
some criteria for generically irreducible eigenspaces. Additionally, we
introduce an operator that is comparable to the Laplacian and show that the
same criterion holds.
