A functional model for the tensor product of level 1 highest and level-1 lowest modules for the quantum affine algebra U_q(sl₂)

Let V(Λ_i) (resp., V(-Λ_j)) be a fundamental integrable highest (resp., lowest) weight module of U_q(sl₂). The tensor product V(Λ_i)⊗V(-Λ_j) is filtered by submodules F_n=U_q(sl₂)(v_i ⊗v_n-i), n≥0, n≡i-j mod 2, where v_i∈V(Λ_i) is the highest vector and v_n-i∈V(-Λ_j) is an extremal vector. We show that F_n/F_n+2 is isomorphic to the level 0 extremal weight module V(n(Λ₁-Λ₀)). Using this we give a functional realization of the completion of V(Λ_i)⊗V(-Λ_j) by the filtration (F_n)_n≥0. The subspace of V(Λ_i)⊗V(-Λ_j) of sl₂-weight m is mapped to a certain space of sequences (P_n,l)_{n≥0,n≡i-jmod2,n-2l=m}, whose members P_n,l=P_n,l (X₁,...,X_l z₁,...,z_n) are symmetric polynomials in X_a and symmetric Laurent polynomials in z_k, with additional constraints. When the parameter q is specialized to -1, this construction settles a conjecture which arose in the study of form factors in integrable field theory.