Scattering theory for twisted automorphic functions

Abstract:

The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with an irreducible unitary representation $\rho$ and satisfying $u(\gamma z)=\rho (\gamma )u(z)$. The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, $R_-$ and $R_+$, for the solution operator. The scattering operator, which maps $R_-f$ into $R_+f$, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of $\rho$ is one, the elements of the scattering operator cannot vanish. However when $\dim (\rho )>1$ this is no longer the case.