Positive definite function zoo 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 "diagonal" 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. I'd like a collection of quirky beasts...
 A: For me, often positive-definite functions arise as kernel functions in machine learning
A small list can be found at this link
Also, I would also add one of the classic books on this subject:
Harmonic analysis on semigroups by Christian Berg, Jens Peter Reus Christensen, Paul Ressel.
A: I guess the example below provides one answer to your first question.
A famous positive-definite function** is the one in the Bessis-Moussa-Villani conjecture:
Let $A$ and $B$ be $n \times n$ Hermitian matrices. Then the function
$$\phi(t) = \mbox{trace}(e^{A+i t B}),$$
is a positive-definite function. 
** Conjectured to be positive-definite, though apparently it has been proved very recently; however, until that has been verified independently, I will adhere to the safety of the word "conjecture"
A: Perhaps you are already aware of this, but I thought I'd mention it for other interested google-enabled readers. 


*

*Infinitely divisible distributions are one place where positive-definite functions come up (Lévy processes, Lévy-Khintchine formula, etc., are also relevant keywords)

*Infinite divisibility in Free Probability is another related place.
