4 fixed TeX in second-to-last paragraph

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.

3 deleted 1 characters in body

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 c \in [0,1/a(\lambda)] or c=1/m for a positive integer m with m Taking the n \rightarrow \infty limit of all this should be possible; I am running out of time again. 2 added 2863 characters in body; added 15 characters in body If you take Strictly speaking, these papers classify the unitary irreducibles in category O. I don't know whether there is a limit as n goes to infinity 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 S_n gl_n correspond to unitary modules for the Cherednik algebrayou get . At least for the symmetric group, the question of when there is a picture faithful unitary module in O is not very similar interesting: there is always one (either L_c(triv) or L_c(sign) will work). But if one is to that of 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 TL algebrasits support is a subvariety of V. The faithful unitaries should be the unitaries with full support (I have no idea whynot 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

Taking the $n \rightarrow \infty$ limit of all this should be possible; I am running out of time again.

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