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

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