Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:


1) $\psi(x,x)=0$


2) $\psi(y,x)=\psi(x,y)$


3) for any elements $x\_1,...x\_n$ and all real numbers $c\_1,...,c\_n$, with $c\_1+...+c\_n=0$, the following inequality holds:
$$
\sum\_{i=1}^{n}\sum\_{j=1}^{n}
c\_ic\_j\psi(x\_i,x\_j)\leq 0.
$$


Let $G$ be a discrete group.
Recall that a function $G\rightarrow \mathbb{R}$ is conditionally of
negative type if the kernel $\psi$, defined by $\psi(g,h)= \psi(h^{−1}g)$ is conditionaly of negative type.



> 
> Does there exist class of discrete groups which admit an explicit description of functions which are conditionaly of negative type?
> 



description of functions of conditionally negative type on a group 
If you add the condition that Jesse mentioned in the comment above, it is a theorem that such functions are always realized from an affine isometric actions of the group $G$ on a Hilbert space. More precisely, suppose $G$ acts continuously by affine isometries on a Hilbert space $H$. Now, define


$$ \psi (g)= \| g \cdot x-x \|^2 $$


for an arbitrary point $x$ in the affine space $H$. First, it is easy to see that such functions are always negative definite. More difficult is to show that any negative definite function can be obtained in this way.


You can read more about them in Kazhdan's Property (T) by Bekka, de la Harpe and Valette.


