Abstract. To distinguish the contributions to the generalized Hurwitz number  of the source Riemann surface with different genus, by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants, we define the genus expanded  cut-and-join operators, which can form  a differential operator algebra. Moreover, the differential operator algebra  is isomorphic to the central subalgebra of the symmetric group algebra for the finite case,  and is isomorphic is isomorphic to the central subalgebra  of infinite
symmetric group algebra  for the shifted case.  As an application, we get some  differential equations for the generating functions of the Hurwitz numbers  for the source Riemann surface with different genus, thus we can express the generating functions in  terms of the genus expanded  cut-and-join operators.