  For any closed complex manifold $X$, we calculate the Poincar\'{e} and Hodge
polynomials of the delocalized equivariant cohomology $H^*(X^n, S_n)$ with a
grading specified by physicists. As a consequence, we recover a special case of
a formula for the elliptic genera of symmetric products in
Dijkgraaf-Moore-Verlinde-Verlinde \cite{Dij-Moo-Ver-Ver}. For a projective
surface X, our results matches with the corresponding formulas for the Hilbert
scheme of X^[n]. We also give geometric construction of an action of a
Heisenberg superalgebra on $\sum_{n \geq 0} H^{*,*}(X^n, S_n)$, imitating the
constructions for equivariant K-theory by Segal \cite{Seg} and Wang \cite{Wan}.
There is a corresponding version for $H^{-*, *}$.
