I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:

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][1].

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. I'd like a collection of quirky beasts...





  [1]: http://eom.springer.de/P/p073890.htm