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A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$.

For a definition and discussion of positive definite function see here.

I've often wished I had a collection of diverse examples of positive definite functions on groups, for the purpose of testing various conjectures. I hope the diverse experience of the participants of this forum can help me collect a list of such examples.

To clarify what I'd like to see:

What is an example of a positive definite function on a group $G$ that is not easily seen to be a coefficient of a unitary representation of $G$? What are some positive definite functions that arise in contexts sufficiently removed from studying the coefficients of unitary representations?

Also, the weirder the group $G$ the better.

Edit: There is now a version of this question on MO.

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    As per request, I've made the question CW.2011-08-13
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    I guess my favourite example is $\frac{1}{1+x^2}$ on $\mathbb{R}$. But it's not quite clear what exactly you're looking for: explicit positive elements of the Fourier algebra or would a function of the form $f \ast \tilde{f}$ for $f \in L^2(G)$ already be satisfactory for you? I find the exposition on positive definite functions in appendix C of [Bekka-de la Harpe-Valette](http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf) quite nice, but sect 13.4 and the following chapters of Dixmier's book on $C^\ast$-algebras is still the best source in my opinion (few explicit examples, though).2011-08-14
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    let me clarify, Theo.2011-08-14
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    Thanks for the tag edits, someone!2011-08-14
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    @Theo: Thanks for the comments, too!2011-08-14
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    Jon: I thought it would be convenient to have the version of the question here and on MO cross-linked. I hope you don't mind the two minor edits.2011-09-07
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    Not at all t.b. Thanks for doing this!2011-09-26

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