Resonance Relations, Holomorphic Trace Functions and Hypergeometric Solutions to qKZB and Macdonald-Ruijsenaars Equations

Abstract

The resonance relations are identities between coordinates of functions ψ(λ) with values in tensor products of representations of the quantum group 𝒰q(sl2)⁠. We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker–Akhiezer type, satisfying the resonance relations. We give an alternative representation–theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group 𝒰q(sl2)⁠, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weylanti–invariant hypergeometric solutions of the qKZB equations with values in a tensor product of finite–dimensional 𝒰q(sl2) -modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of 𝒰q(sl2)-invariants, and when q is a root of unity is computed by the Verlinde algebra.