  In this paper we first consider the Hamiltonian action of a compact connected
Lie group on an $H$-twisted generalized complex manifold $M$. Given such an
action, we define generalized equivariant cohomology and generalized
equivariant Dolbeault cohomology. If the generalized complex manifold $M$
satisfies the $\bar{\partial}\partial$-lemma, we prove that they are both
canonically isomorphic to $(S\g^*)^G\otimes H_H(M)$, where $(S\g^*)^G$ is the
space of invariant polynomials over the Lie algebra $\g$ of $G$, and $H_H(M)$
is the $H$-twisted cohomology of $M$. Furthermore, we establish an equivariant
version of the $\bar{\partial}\partial$-lemma, namely
$\bar{\partial}_G\partial$-lemma, which is a direct generalization of the
$d_G\delta$-lemma for Hamiltonian symplectic manifolds with the Hard Lefschetz
property.
  Second we consider the torus action on a compact generalized K\"ahler
manifold which preserves the generalized K\"ahler structure and which is
equivariantly formal. We prove a generalization of a result of Carrell and
Lieberman in generalized K\"ahler geometry. We then use it to compute the
generalized Hodge numbers for non-trivial examples of generalized K\"ahler
structures on $\C\P^n$ and $\CP^n$ blown up at a fixed point.
