Microlocal lifts of eigenfunctions on hyperbolic surfaces and trilinear invariant functionals

Abstract

S. Zelditch introduced an equivariant version of a pseudo-differential calculus on a hyperbolic Riemann surface. We recast his construction in terms of trilinear invariant functionals on irreducible unitary representations of PGL(2,R). This allows us to use certain properties of these functionals in the study of the action of pseudo-differential operators on eigenfunctions of the Laplacian on hyperbolic Riemann surfaces.