# 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

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

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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.

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I guess I should say that it is straighforward to extend the $\mathfrak{gl}_\infty$-module I mentioned above to $a_\infty$ in a way analogous to the Kac-Raina level 1 extension. So, you get an action of $\hat{\mathfrak{sl}}_p$ on the Fock space. The problem is that it is not generally irreducible as a $\hat{\mathfrak{sl}}_p$-module. –  David Hill Oct 4 '10 at 19:38
Thank you very much for this link. I think this is very close to what I was looking for, although I haven't had a chance to look at it closely yet. A question: from what I understand the level k Fock space has a basis indexed by k multi-partitions. Is this "visible" in your construction? –  Oded Yacobi Oct 4 '10 at 20:37
I'll have to think about it more, but my first instinct is that the answer is no. The $k$-multipartition description comes from tensoring together $k$ level 1 representations, each parametarized by partitions. In my paper, I skipped the big space and defined the action in one shot. Then again, maybe the translation is not too bad? I'll think some more. –  David Hill Oct 4 '10 at 21:14

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 $q=0$"

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"

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.

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Thanks! Does the higher level Fock space carry a rep of $U_{q}(\hat{\mathfrak{sl}}_{n})$ only for generic $q$, or also for $q=1$ or $q$ root of unity? –  Oded Yacobi Oct 5 '10 at 0:07
The papers I mentioned deal with generic $q$. I think one should be able to make sense of the construction at $q=1$ though. This is just because the formulas for the actions of $E_i$ and $F_i$ on the standard basis of Fock space make sense at $q=1$. See Theorem 2.1 in Uglov's paper. These seem to make sense at other roots of unity as well...although certainly the structure of the representation would be much much complicated in those cases. –  Peter Tingley Oct 5 '10 at 1:58

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.

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