Double Dirichlet series and quantum unique ergodicity of weight 1/2 Eisenstein series

The problem of quantum unique ergodicity (QUE) of weight 1/2 Eisenstein series for {Gamma}_0(4) leads to the study of certain double Dirichlet series involving GL2 automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.