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 slN(Cq) with Cq=Cq[t0 +/- 1,t1 +/- 1] an irrational quantum 2-torus, the simple affine vertex algebra Lsl infinity(l,0) with l a positive integer, and Levi subgroups GLI of GLl(C). First, we give a canonical isomorphism between the category of integrable restricted slN(Cq)-modules of level l and that of equivariant quasi Lsl infinity(l,0)-modules. Second, we classify irreducible N-graded equivariant quasi Lsl infinity(l,0)-modules. Third, we establish a duality between irreducible N-graded equivariant quasi Lsl infinity(l,0)-modules and irreducible regular GLI-modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible N-graded equivariant quasi Lsl infinity(l,0)-module. Fifth, we completely determine the following branchings: (i) The branching from Lsl infinity(l,0)circle times Lsl infinity(l ',0) to Lsl infinity(l+l ',0) for quasi modules. (ii) The branching from slN(Cq) to its Levi subalgebras. (iii) The branching from slN(Cq) to its subalgebras slN(Cq[t0 +/- M0,t1 +/- M1]).