Let $\mathfrak{g}$ = $\mathfrak{gl}\_{\infty}$. 


To each positive integer $k$ one can associate the level $k$ Fock space $\mathcal{F}\_{k}$.


For a dominant weight $\lambda$ of level $k$, one can define an action of $\mathfrak{g}$ on $\mathcal{F}\_{k}$ 


so that it has a highest weight vector of weight $\lambda$ which generates the corresponding irreducible $\mathfrak{g}$-module. Let us denote $\mathcal{F}\_{k}$ with this action as 


$\mathcal{F}\_{k}(\lambda)$. 


My question is about realizing these spaces and actions. In the lectures of Kac and Raina "Highest-weight representations of infinite dimensional Lie algbebras", there is a construction of $\mathcal{F}\_{1}(\lambda)$ (here $\lambda$ is a fundamental weight) as a subspace of the semi-infinite wege space. In this realization the action of $\mathfrak{g}$ is just the natural action on wedge products. Is there an analogous realization of 


$\mathcal{F}\_{k}(\lambda)$  

for higher $k$? What about for 


$\mathfrak{g}$ = $\hat{\mathfrak{sl}}\_{p}$? 



Realizing higher level Fock spaces
For $\mathfrak{gl}\_{\infty}$ the answer is easy, you realize the Fock space as a direct limit of polynomial representations of finite $\mathfrak{gl}\_n$ modules. You can read about the construction [here](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=842325&vfpref=html&r=3&mx-pid=2473335). I worked on the $\hat{sl}\_p$ case a few years ago, but got stuck. If you could do it, it would be very nice.


