It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A physics reference of this fact in the path integral formulation, see this paper, Sec. 2.2 of Naculich-Schnitzer. Related math references are in this paper of Hasegawa and this paper of Nakanishi and Tsuchiya. Physically, the $U(N)_k$ modes are constructed as k independent copies of the $U(N)_1$ modes which can be found inside $U(N \cdot k)_1$. The coset corresponds to freezing the other currents out by a Lagrange multiplier, which corresponds to throwing out the other currents.
Also, $U(M)_1$ is well-known to be isomorphic to the theory of $M$ independent Dirac Fermion CFTs.
As such, is there a way to understand the (integrable) modules of $U(N)_k$ or their characters in terms of $N \cdot k$ free fermions?
If a simple example like $U(2)_2$ could be explained (or pointed to a reference) that would be great as well.
