The group theoretic interpretation of harmonic analysis was born out of the observation that the discrete Fourier transform on a signal of length $n$ was precisely the Fourier transform of the finite group $\mathbb{Z}/n\mathbb{Z}.$ I was thinking about the extent to which this analogy holds, but I was having trouble with the Laplacian, which, of course, occupies a central role in harmonic analysis. In particular, there doesn't seem to be a generalization of the operator to finite groups (though certainly there is for compact connected Lie groups). 


Does anyone know of such an object?



Is there any way to generalize the Laplacian to finite groups?
One natural generalization is the center of the group algebra (i.e., algebra of class functions - so in the abelian case you consider, just the group algebra itself). In the continuous case there are many different notions of group algebra, depending on the class of functions you consider, but if you allow a broad enough interpretation of the group algebra this will include the case of the Laplacian (and its higher analogues) in harmonic analysis. These appear from the action of the center of the universal enveloping algebra (which itself can be interpreted via invariant differential operators on the group or distributions supported at the identity). I think it's not egregious to claim that any version of Fourier transform for a group (or symmetric space) in particular simultaneously diagonalizes all of these commuting operators (in the finite abelian case that's literally all it does). However I don't think there's a canonical single operator generalizing the Laplacian (outside of the setting say of simple Lie groups where we take the quadratic Casimir, but that doesn't give something interesting in the finite context).


