Let $G$ be a group. I call a function $\ell\colon G\to [0,\infty)$ a conditionally negative definite (cnd) length function if
- $\ell(g)=0$ iff $g=e$,
- $\ell(g)=\ell(g^{-1})$,
- $\sum_{g,h\in G}\overline{\alpha(g)}\alpha(h)\ell(g^{-1}h)\leq 0$ whenever $\alpha\colon G\to\mathbb C$ has finite support and $\sum_{g\in G}\alpha(g)=0$.
Here is a reformulation that might be closer to what people in geometric group theory study: If $\ell\colon G\to [0,\infty)$ is a cnd length function, then there exists a (real) Hilbert space $H$, an orthogonal representation $\pi$ of $G$ on $H$ and a map $b\colon G\to H$ satisfying the $1$-cocycle identity $$ b(gh)=b(g)+\pi(g)b(h) $$ such that $\ell(g)=\lVert b(g)\rVert^2$. The triple $(H,\pi,b)$ is essentially uniquely determined by $\ell$ if we assume $H$ to be minimal, i.e., $H$ is the closed linear hull of $\{\pi(g)b(h)\mid g,h\in G\}$.
Conversely, if $(H,\pi)$ is an orthogonal representation of $G$, $b\colon G\to H$ satisfies the $1$-cocycle identity from above, $b(g)=0$ iff $g=e$ and $\lVert b(g)\rVert=\lVert b(g^{-1})\rVert$, then $\ell(g)=\lVert b(g)\rVert^2$ defines a cnd length function on $G$. So there seems to be some connection between cnd length functions and group cohomology, but I am missing the background in this field.
By Schönberg's theorem, these functions are related to isometric embeddings into Hilbert spaces, and there is also a connection to NC Lévy processes on the group von Neumann algebra $L(G)$.
Now let $G$ be a free group of rank at least $2$. If we take the usual symmetric generating set, then the graph distance to the neutral element in the Cayley graph is a cnd length function by a result of Haagerup.
Question: Is there a full characterization of cnd length functions on non-abelian free groups? If not, can someone at least give some more examples?