  The string theory in the Penrose limit of AdS_2 x S^2 is investigated. The
specific Penrose limit is the background known as the Nappi-Witten spacetime,
which is a plane-wave background with an axion field. The string theory on it
is given as the Wess-Zumino-Novikov-Witten (WZNW) model on non-semi-simple
group H_4. It is found that, in the past literature, an important type of
irreducible representations of the corresponding algebra, h_4, were missed. We
present this "new" representations, which have the type of continuous series
representations. All the three types of representations of the previous
literature can be obtained from the "new" representations by setting the
momenta in the theory to special values. Then we realized the affine currents
of the WZNW model in terms of four bosonic free fields and constructed the
spectrum of the theory by acting the negative frequency modes of free fields on
the ground level states in the h_4 continuous series representation. The
spectrum is shown to be free of ghosts, after the Virasoro constraints are
satisfied. In particular we argued that there is no need for constraining one
of the longitudinal momenta to have unitarity. The tachyon vertex operator,
that correspond to a particular state in the ground level of the string
spectrum, is constructed. The operator products of the vertex operator with the
currents and the energy-momentum tensor are shown to have the correct forms,
with the correct conformal weight of the vertex operator.
