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Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of highest-weight submodules, which are generically some reducible quotients of Verma modules. I am wondering how one could study the properties of those submodules. In particular, if $\mathfrak{g}=\oplus_{n\in\mathbb{Z}}\mathfrak{g}_n$ is the usual grading by imaginary root, and $\mathfrak{g}_\pm=\oplus_{\pm n>0}\mathfrak{g}_n$, I'm wondering how to understand when those submodules and $V$ itself are $\mathfrak{g}_-$-free.

Now, let me be very specific. Consider $\mathfrak{g}$ to be the affine $A_r$ at a certain level $k$. It admits a free-field realization using the so-called symplectic boson introduced in [1] (it works for arbirary algebra, but let us restrict to $A_r$). The Fock space is generated by a set of oscillators $q^a_s$, where $s\in \mathbb{Z}+\frac{1}{2}$ and $a$ runs over the basis of some real symplectic representation of $A_r$. Let $\Omega^{ab}$ be the corresponding symplectic form. The oscillators satisfy: $$ [q^a_r,q^b_s]=\Omega^{ab}\delta_{r,-s}. $$ The module $V$ is generated by acting with these oscillators on the vacuum $|0\rangle$ satisfying $q^a_s|0\rangle=0, s>0$. The generators of the affine algebra are given by $J_n^A = \sum_{s\in\mathbb{Z}+1/2} R^A_{ab}:q^a_{-s}q^b_{s+n}:$, $A=1\dots r^2+2r$, where $R^A_{ab}$ is a symplectic representation of $A_r$, and $::$ is the usual "creation-annihilation" normal ordering meaning that in $:q^a_{-s}q^b_s:$, the q with positive lower index appears written on the right. This sum always reduces to a finite sum when acting on vectors in $V$.

After glimpsing though some literature on Wakimoto modules, I've had an impression that such symplectic representations have mostly not been studied (perhaps because they are not unitary?). I'll be happy to be corrected if this is not the case.

The question that interests me the most is how $V$ decomposes into highest-weight modules and what are their properties. It is clear that $V$ has a lot of highest-weight vectors, like $|0\rangle$ or $v_a q^a_{-1/2}|0\rangle$ (where $v_a$ are chosen appropriately), which generate such submodules. Such highest-weight vectors also generate finite-dimensional representations of $\mathfrak{g}_0$, i.e. $A_r$ itself (for example, $q^a_{-1/2}|0\rangle$ transform in the representation given by $R^A_{ab}$). Because of that, the highest-weight module of $\mathfrak{g}$ built on such a vector is not a Verma module -- all singular vectors at the level $n=0$ (where $n$ refers to grading by imaginary root) vanish. But it might happen that no extra relations appear at other levels, and though the module is not $\mathfrak{n}_-$-free (as the Verma module would be), it could still be $\mathfrak{g}_-$-free. To understand when this happens is one of the main questions I have.

One case in which I expect this to happen but cannot find the proof is if $\mathfrak{g}_0=su(2)$ and the representation $R^A_{ab}$ is a sum of 8 copies of the fundamental representation (I don't know if 8 is important here or if it is just a sufficiently large number). A similar case when the statement should also hold is $su(N)$ with $R^A_{ab}$ being the sum of $2N$ copies of fundamental and $2N$ copies of anti-fundamental representation. In these cases I only understand how to study several submodules (using the Kaz-Kazhdan determinant) but not the most general ones.

One case in which I expect this not to happen (i.e. $V$ should not be $\mathfrak{g}_-$-free) is the case when $R^A_{ab}$ is a sum of two copies of the adjoint representation of $A_r$.

Is anything known about this?

[1] Goddard, P.; Olive, D.; Waterson, G. Superalgebras, symplectic bosons and the Sugawara construction. Comm. Math. Phys. 112 (1987), no. 4, 591--611. http://projecteuclid.org/euclid.cmp/1104160054.

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