description of functions of conditionally negative type on a group

Question

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?

Answer 1

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.

Answer 2

Sorry for the overdue answer! I think it should be added that, on some classes of groups, there are integral representations of conditionally negative definite functions, known as Levy-Khinchin formulae. E.g, for locally compact abelian groups, see:

MR0420769 (54 #8781) Forst, Gunnar The Lévy-Hinčin representation of negative definite functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), no. 4, 313–318.