Weight-Shifting Operators of Hypergeometric Type for Maass Forms

Abstract: This paper explicitly constructs a family of weight-shifting operators mapping from the space of weight-k smooth automorphic functions, A_k(Gamma), to the space of weight-t automorphic functions, A_t(Gamma). These operators are defined as integral transforms whose kernel, K, is the product of the covariant factor P given in Definition 1.2 and a function, F, which is invariant under the diagonal action of SL(2, R). The key assumption in the operator's construction is the spectral condition posited in Hypothesis 1.1, namely that the kernel K is an eigenfunction of the weight-t hyperbolic Laplacian Delta_t with respect to the first variable. This partial differential equation reduces to an ordinary differential equation for the invariant function F, which is shown to be equivalent to the Hypergeometric Differential Equation (HDE). The main result of this paper, presented in Theorem 2.1, is the explicit determination of the HDE parameters (a,b,c) as functions of the initial automorphic data (k, t, q) and the operator's spectral parameter lambda_K. This work presents an analytic methodology for the explicit construction of an intertwining operator that maps a Laplacian eigenspace to its corresponding eigenspace, demonstrating how the kernel is determined from the automorphic data via the Hypergeometric Differential Equation.