The authors consider the rational quantum Knizhnik–Zamolodchikov (qKZ) equation associated with the Lie algebra \(\text{ sl}(2)\). The qKZ equation with values in a tensor product of Verma modules \(V(\lambda _1)\otimes \cdots \otimes V(\lambda _n)\) of the algebra \(\text{ sl}(2)\) was solved before. (Here \(\lambda _i\in {\mathbb C}\) are highest weights of the Verma modules.) These solutions \(\Psi (z,\lambda)\), where \(z=(z_1.\cdots ,z_n)\), \(\lambda =(\lambda _1,\cdots ,\lambda_n)\), are meromorphic functions written in terms of hypergeometric integrals. The authors endow the module \(V(\lambda_i)\) with an evaluation Yangian module structure with complex evaluation parameter \(z_i\) and denote the corresponding Yangian module by \(V(z_i,\lambda_i)\). Let \(V(z,\lambda)=V(z_1,\lambda _1)\otimes \cdots \otimes V(z_n,\lambda _n)\). The space of hypergeometric solutions can be naturally identified with the corresponding tensor product of Verma modules \(V_q(\lambda _1)\otimes \cdots \otimes V_q(\lambda _n)\) of the quantum group \(U_q\text{ sl}(2)\), where \(q=\exp (\pi i/p)\) and \(p\) is the step of the qKZ equation. The authors endow the module \(V_q(\lambda _i)\) with an evaluation structure of affine quantum group \(U_q{\widehat{\text{ gl}(2)}}\) module with complex evaluation parameter \(z_i\) and denote it by \(V_q(z_i,\lambda _i)\). Let \(V_q(z,\lambda)= V_q(z_1,\lambda _1) \otimes \cdots \otimes V_q(z_n,\lambda _n)\). Then the hypergeometric solutions define a hypergeometric map \(\text{ qKZ}(z,\lambda ;p): V_q(z,\lambda)\to V(z,\lambda)\). The parameters \(z,\lambda ,p\) are called generic if the \(U_q{\widehat{\text{ gl}(2)}}\) module \(V_q(z,\lambda)\) is irreducible. For generic values of parameters the hypergeometric map is an isomorphism of vactor spaces. The authors derive properties of the hypergeometric map for these values of parameters and describe all singularities of hypergeometric solutions of the qKZ equations.
For the entire collection see [Zbl 0932.00043].