Virasoro symmetries of the extended Toda hierarchy

We prove that the extended Toda hierarchy of [1] admits a nonabelian Lie algebra of infinitesimal symmetries isomorphic to half of the Virasoro algebra. The generators L-m, mgreater than or equal to-1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP1 Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.