A functional model for the tensor product of level 1 highest and level -1 lowest modules for the quantum affine algebra U_q(sl_{2}^)

Let $V({\mathrm{\Lambda }}_i)$ (resp., $V(-{\mathrm{\Lambda }}_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q({\widehat{sl}}_2)$. The tensor product $V({\mathrm{\Lambda }}_i)\otimes V(-{\mathrm{\Lambda }}_j)$ is filtered by submodules $F_n=U_q({\widehat{sl}}_2)(v_i\otimes {\bar{v}}_{n-i})$, $n\ge 0, n\equiv i-j\bmod 2$,where$v_i\in V(\Lambda_i)$isthehighestvectorand$\bar{v}{n-i}\in V(-\Lambda_j)$isanextremalvector\mathrm{.}Weshowthat$F_n/F{n+2}$ is isomorphic to the level 0 extremal weight module $V(n({\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_0))$. Using this we give a functional realization of the completion of $V(\Lambda_i)\otimes V(-\Lambda_j)$bythefiltration$(F_n)_{n\geq0}$\mathrm{.}Thesubspaceof$V(\Lambda_i)\otimes V(-\Lambda_j)$of$sl_2$-weight$m$ is mapped to a certain space of sequences $(P_{n,l}{)}_{n\ge 0,n\equiv i-j2,n-2l=m}$, whose members $P_{n,l}=P_{n,l}(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_l\mathrm{\mid }{z}_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ are symmetric polynomials in $X_a$ and symmetric Laurent polynomials in $z_k$, with additional constraints. When the parameter $q$ is specialized to $\sqrt{-1}$, this construction settles a conjecture which arose in the study of form factors in integrable field theory.