Infinite-level Fock spaces, crystal bases, and tensor product of extremal weight modules of type $A_{+\infty }$

Abstract

We study the category $\mathcal{C}$ generated by extremal weight modules over $U_q({\mathfrak{gl}}_{>0})$. We show that $\mathcal{C}$ is a tensor category, and give an explicit description of the socle filtration of tensor product of any two extremal weight modules. This follows from the study of Fock space ${\mathcal{F}}^{\infty }\otimes \mathcal{M}$ of infinite level, which has commuting actions of a parabolic $q$-boson algebra and $U_p({\mathfrak{gl}}_{>0})$ with $p=-q^{-1}$. It contains a (semisimple) limit of the fermionic Fock space ${\mathcal{F}}^n$ of level $n$, which has a $q$-analogue of Howe duality often called level-rank duality. To describe the socle filtration of ${\mathcal{F}}^{\infty }\otimes \mathcal{M}$, we introduce the notion of a saturated crystal valuation, whose existence was observed for example in the embedding of an extremal weight module into a tensor product of fundamental weight modules of affine type due to Kashiwara and Beck-Nakajima.

76 pages, the proof of Theorem 8.9 is revised, statements and proofs of Theorem 8.31 are revised, comments added on Lemma 2.1, typos corrected, minor corrections