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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?

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Take the Laplacian of a Cayley graph for the group. – Qiaochu Yuan May 31 '12 at 19:14
When you say compact groups you mean compact connected Lie groups, right? Finite groups are compact, after all. – Qiaochu Yuan May 31 '12 at 19:14
@Qiaochu Indeed, thanks. – Grant Rotskoff Jun 1 '12 at 2:12
up vote 9 down vote accepted

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).

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Thanks, very helpful perspective. – Grant Rotskoff Jun 1 '12 at 2:15

It's not clear to me that the Laplacian of the Cayley graph of the group is the right object. You might want to start by looking at Audrey Terras's book Fourier Analysis on Finite Groups and Applications (if you haven't already). She does not define the Laplacian of a finite group (but does talk about Cayley graphs and their Laplacians.)

To elaborate, on symmetric spaces in general, there is typically a whole algebra of differential operators which commute with the group action, not just polynomials in the Laplacian, which are relevant for the harmonic analysis (see e.g. Harmonic Analysis on Symmetric Spaces and Applications, I, II.)

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Thank you for the recommendations. Terras' book looks promising. – Grant Rotskoff Jun 1 '12 at 2:16
Is there any relation between Laplacian of a Cayley graph and center of group algebra? – Alexander Chervov Jun 2 '12 at 18:38

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