01068nas a2200109 4500008004300000245005500043210005500098520073100153100002000884700001800904856003600922 2004 en_Ud 00aVirasoro Symmetries of the Extended Toda Hierarchy0 aVirasoro Symmetries of the Extended Toda Hierarchy3 aWe prove that the extended Toda hierarchy of \\\\cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\\\\geq -1$ 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 $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/2544