Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra? For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions:
• The (universal envelopping algebra of the) Virasoro algebra is the $\star$-algebra $Vir_c$ generated by elements $L_n$, ($n \in \mathbb{Z}$), subject to the relations
$$
[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0},
$$
and with $\star$-structure $L_n^* = L_{-n}$.
• The Temperley-Lieb algebra is the $\star$-algebra $TL_{\delta}$ with generators $U_i$ ($i \in \mathbb{Z}$) and relations :  


*

*$U_i^2 = \delta U_i$   and $*$-structure $U_i^* =  U_i$.

*$U_iU_{i+1}U_i=U_i$  and $U_iU_{i-1}U_i=U_i$   

*$U_i U_j=U_j U_i$  for  $|i-j|\ge 2$    


Both $Vir_c$ and $TL_{\delta}$ depend on a parameter.
These are the numbers $c$ and $\delta \in \mathbb{R}$.  

Let's call a representation $\rho$ of a $\star$-algebra on a Hilbert space unitary if $\rho(x^*)=\rho(x)^*$.
We are interested in the unitary representations of $Vir_c$ and $TL_{\delta}$. In the case of the Virasoro algebra, we further restrict to positive energy representations, i.e., unitary representations in which the spectrum of $L_0$ is positive.
Depending on the value of the parameters $c$ or $\delta$, three things can happen:  
1. Discrete series (only) of quotient of Verma modules are unitary and positive energy.
2. Continuum of Verma modules are unitary, positive energy representations.
3. The Verma modules are not unitary.    
Now here's the striking thing:
$\begin{array}{c|c|c|c|c|c|c}
    & \text{Discrete series} & \text{Continuum} & \text{Others}  \newline 
                             \hline
Vir_c & c \in \{ 1-\frac{6}{m(m+1)}   \vert   m = 2,3,4 \ldots \}  &c \in [1,\infty) & \text{non-unitary}  \newline
                             \hline
TL_\delta & \delta\in \{ 2\cos\big(\frac\pi m\big)\quad  \vert  \quad m = 2,3,4 \ldots \}  &\delta \in [2,\infty) & \text{non-unitary}   
\end{array}$    
The parameters $c$ and $\delta$ belong to a countable set (discrete series) exhibiting an accumulation point, followed by a continuum. 

  
*
  
*Is it pure coincidence that those two algebras exhibit such similar behaviour?    
  
*Is there some natural map from $Vir_c$ to $TL_{\delta}$, or vice-versa?  
  
*Is there any way of linking the values $c\in 1-\frac{6}{m(m+1)}$ and $\delta\in 2\cos(\frac\pi m)$?  
  
*Are there other algebras exhibiting a similar phenomenon?
  

 A: I think it is not a coincidence, although the only relationship I can think of is a bit distant.  Roughly, it goes: From positive energy representations of affine Kac-Moody algebras one gets certain values of $c$ in the discrete series.  The TL algebras appear as centralizer algebras for quantum $\mathfrak{sl}_2$ (i.e. $End(V^{\otimes n})$ for $V$ the "vector" representation).  At roots of unity one gets the TL discrete series (here $\delta$ is the $q$-dimension of $V$.  On the other hand, the level-preserving tensor product on reps. of the affine Kac-Moody algebra of type $A$ gives a fusion category equivalent to the category one obtains from the quantum group situation (due to Finkelberg, although Lepowsky tells me there is a small gap that can be fixed using VOAs).
So I guess I am saying that there is a sort of Schur-Weyl duality relationship.  This is not restricted to the type $A$ situation, for example, BMW algebras exhibit similar behavior which corresponds to the type $BCD$ quantum groups (or affine Kac-Moody algebras).  
Probably the appropriate language to use is that of tensor categories associated with quantum groups.  At roots of unity one gets unitary reps (see Wenzl or Xu's work on this) giving a discrete series which (at least combinatorially) corresponds to level-preserving fusion products for Kac-Moody algebras, which then are responsible for the Virasoro algebra situation.  For non-roots of unity one still gets a continuous series of unitary reps.
A: The Cherednik algebra has a similar classification into discrete and unitary series: see arXiv:1106.5094 and arXiv:0901.4595.  Strictly speaking, these papers classify the unitary irreducibles in category O.  I don't know whether there is a larger category in which contravariant forms will exist, but anyway for the symmetric group category O will be closely tied to affine Lie algebras (thus to Virasoro) by the Arakawa-Suzuki functor, and to Hecke (thus TL algebras) by the Knizhnik-Zamolodchikov functor (which actually identifies O with the category of q-Schur modules for most values of the parameter).  Maybe the Cherednik algebra can serve as a bridge between them: Etingof conjectures (true by case by case check for the symmetric group) that KZ of a unitary module is unitary, and it is true (again case by case) that via Arakawa-Suzuki the unitary modules (i.e. integrable modules) for affine $gl_n$ correspond to unitary modules for the Cherednik algebra. 
At least for the symmetric group, the question of when there is a faithful unitary module in O is not very interesting: there is always one (either $L_c(triv)$ or $L_c(sign)$ will work).  But if one is to make the connection to TL and the Virasoro algebra work probably one needs more detail.
Every Cherednik algebra module is in particular a module over a ring C[V] of polynomial functions on a vector space V, and its support is a subvariety of V.  The faithful unitaries should be the unitaries with full support (I have not checked this, though one direction is obvious).
In the (much simpler) case of the Cherednik algebra of the symmetric group $S_n$, the algebra depends on one parameter c, which we may assume positive.  The irreducibles in O are indexed by irreducible $S_n$-modules, and therefore by partitions of n.  Writing $a(\lambda)$ for the largest hook length of the partition $\lambda$ and $b(\lambda)$ for a certain smaller hook length (see the paper of Etingof/Stoica for the precise def'ns), the corresponding irreducible $L_c(\lambda)$ is unitary iff $\lambda=(1^n)$ (corresponding to the sign representation), or $c \leq a(\lambda)$ or $c=1/m$ for a positive integer $m$ with $m \leq b(\lambda)$.  The continuous part of the unitary set is precisely the closure of the set where the corresponding standard module is irreducible and unitary (this much is not surprising: the condition for the contravariant form to be positive definite on the standard module is open, and it's obviously pos. def. at $0$).
The module $L_c(\lambda)$ has full support iff: $c$ is not rational or $c=k/m$ and the partition is $m$-regular: the differences $\lambda_i-\lambda_{i+1}$ are strictly less than $m$.  Thus $L_c(\lambda)$ is unitary of full support iff (1) $\lambda=(1^n)$, (2) $\lambda=(n)$ and $0 \leq c < 1/n$, (3) $\lambda \neq (n),(1^n)$ is a rectangle and $c \in [0,1/a(\lambda)]$ or $c=1/m$ for a positive integer $m$ with $m<b(\lambda)$, (4) $\lambda$ is not a rectangle and $c \in [0,1/a(\lambda)]$ or $c=1/m$ for a positive integer $m$ with  $m \leq b(\lambda)$.
Taking the $n \rightarrow \infty$ limit of all this should be possible; I am running out of time again.  
