Due to the relation with the invariant subspace problem, the structure of compressed shift operator is one of the important problems in operator theory. This report focuses on the reducibility of the compressed shift operator on the Hardy space over the bidisk via the geometric approach. We give a complete characterization for the compressed shift operator to be a Cowen-Douglas operator. Then we extend the notion of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and the connection between the reducibility of compressed shift operator and generalized Cowen-Douglas operator are established.