Extended affine Lie algebras, affine vertex algebras, and general linear groups

In this paper, we explore natural connections among the representations of the extended affine Lie algebra s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) with C q = C q [ t 0 ± 1 , t 1 ± 1 ] \mathbb {C}_q=\mathbb {C}_q[t_0^{\pm 1},t_1^{\pm 1}] an irrational quantum 2 2 -torus, the simple affine vertex algebra L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) with ℓ \ell a positive integer, and Levi subgroups G L I \mathrm {GL}_{\mathbf {I}} of G L ℓ ( C ) \mathrm {GL}_\ell (\mathbb {C}) . First, we give a canonical isomorphism between the category of integrable restricted s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) -modules of level ℓ \ell and that of equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -modules. Second, we classify irreducible N \mathbb N -graded equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -modules. Third, we establish a duality between irreducible N \mathbb N -graded equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -modules and irreducible regular G L I \mathrm {GL}_{\mathbf {I}} -modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible N \mathbb N -graded equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -module. Fifth, we completely determine the following branchings: (i) The branching from L s l ∞ ^ ( ℓ , 0 ) ⊗ L s l ∞ ^ ( ℓ ′ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0)\otimes L_{\widehat {\mathfrak {sl}_\infty }}(\ell ’,0) to L s l ∞ ^ ( ℓ + ℓ ′ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell +\ell ’,0) for quasi modules. (ii) The branching from s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) to its Levi subalgebras. (iii) The branching from s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) to its subalgebras s l N ^ ( C q [ t 0 ± M 0 , t 1 ± M 1 ] ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q[t_0^{\pm M_0},t_1^{\pm M_1}]) .