A functional model for the tensor product of level 1 highest and level -1 lowest modules for the quantum affine algebra Uq(sT2)

Let V (Λi) (resp., V (−Λj)) be a fundamental integrable highest (resp., lowest) weight module of Uq(ŝl2). The tensor product V (Λi)⊗ V (−Λj) is filtered by submodules Fn = Uq(ŝl2)(vi ⊗ vn−i), n ≥ 0, n ≡ i − j mod 2, where vi ∈ V (Λi) is the highest vector and vn−i ∈ V (−Λj) is an extremal vector. We show that Fn/Fn+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 (Λi) ⊗ V (−Λj) by the filtration (Fn)n≥0. The subspace of V (Λi)⊗V (−Λj) of sl2-weight m is mapped to a certain space of sequences (Pn,l)n≥0,n≡i−j mod 2,n−2l=m, whose members Pn,l = Pn,l(X1, . . . , Xl|z1, . . . , zn) are symmetric polynomials in Xa and symmetric Laurent polynomials in zk, 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.