On global SL(2,R) symmetries of differential operators

This paper studies the Lie symmetries of the equation∂x2+ax-1∂x+b∂tf(x,t)=0.Generically the symmetry group is sl(2,R). In particular, we show the local action of the symmetry group extends to a global representation of SL(2,R) on an appropriate subspace of smooth solutions. In fact, every principal series is realized in this way. Moreover, this subspace is naturally described in terms of sections of an appropriate line bundle on which the given differential operator is intimately related to the Casimir element.