Function theory related to the group PSL2(R)

Abstract

We study analytic properties of the action of PSL2(R) on spaces of functions on the hyperbolic plane. The central role is played by principal series representations. We describe and study a number of different models of the principal series, some old and some new. Although these models are isomorphic, they arise as the spaces of global sections of completely different equivariant sheaves and thus bring out different underlying properties of the principal series. The two standard models of the principal series are the space of eigenfunctions of the hyperbolic Laplace operator in the hyperbolic plane (upper half-plane or disk) and the space of hyperfunctions on the boundary of the hyperbolic plane. They are related by a well-known integral transformation called the Poisson transformation. We give an explicit integral formula for its inverse. The Poisson transformation and several other properties of the principal series become extremely simple in a new model that is defined as the space of solutions of a certain two-by-two system of first-order differential equations. We call this the canonical model because it gives canonical representatives for the hyperfunctions defining one of the standard models. Another model, which has proved useful for establishing the relation between Maass forms and cohomology, is in spaces of germs of eigenfunctions of the Laplace operator near the boundary of the hyperbolic plane. We describe the properties of this model, relate it by explicit integral transformations to the spaces of analytic vectors in the standard models of the principal series, and use it to give an explicit description of the space of C∞-vectors.