  We study the loop spaces of the symmetric powers of an orbifold and use our
results to define equivariant power operations in Tate K-theory. We prove that
these power operations are elliptic and that the Witten genus is an H_oo map.
As a corollary, we recover a formula by Dijkgraaf, Moore, Verlinde and Verlinde
for the orbifold Witten genus of these symmetric powers. We outline some of the
relationship between our power operations and notions from (generalized)
Moonshine.
