Realizing higher level Fock spaces 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 
$\mathfrak{g}$ = $\hat{\mathfrak{sl}}_{p}$?   
 A: Higher level Fock spaces have been studied in the context of the quantum affine algebra $U_q(\widehat{sl}_n)$. There is a "higher level Fock space" representation for this algebra whose underlying space looks like semi-infinite wedge space. I believe the original reference is Jimbo, Miwa, Misra and Okado "Combinatorics of representations of $U_q(\widehat{sl}_n)$ at $q=0$"
http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=AUCN&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Jimbo&s5=Miwa&s6=Misra&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq,
although there the wedge space structure is not clear. That is explained in Uglov's paper "Canonical bases of higher level $q$-deformed Fock space and Kahzdan Lusztig polynomials"
http://arxiv.org/abs/math/9905196.
Higher level Fock space is more complicated then the level 1 case. For instance many different irreducible representation occur as direct summands of Fock space. In order to get a realization of a single irreducible highest weight representation, you need to pick off the irreducible subrepresentation generated by a certain overall highest weight vector. On the level of representations, this is difficult. However, in the "crystal limit" (i.e. at $q=0$), this can be done quite easily. The basis of the resulting representation is naturally indexed by $\ell$ tuples of partitions (where $\ell$ is the level), satisfying a couple conditions. This fact has been useful in studying crystal bases of these higher level representations. 
A: 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. 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.
A: Representation theory of direct limit Lie algebras (like $\mathfrak{gl}_\infty$) has been studied extensively by Dimitrov, Penkov, and Styrkas. You can find their papers on the arXiv.
