  We give a group-theoretic interpretation of the AdS/CFT correspondence as
relation of representation equivalence between representations of the conformal
group describing the bulk AdS fields $\phi$ and the coupled boundary fields
$\phi_0$ and ${\cal O}$. We use two kinds of equivalences. The first kind is
equivalence between bulk fields and boundary fields and is established here.
The second kind is the equivalence between coupled boundary fields. Operators
realizing the first kind of equivalence for special cases were given by Witten
and others - here they are constructed in a more general setting from the
requirement that they are intertwining operators. The intertwining operators
realizing the second kind of equivalence are provided by the standard conformal
two-point functions. Using both equivalences we find that the bulk field has in
fact two boundary fields, namely, the coupled boundary fields. Thus, from the
viewpoint of the bulk-boundary correspondence the coupled fields are on an
equal footing. Our setting is more general since our bulk fields are described
by representations of the Euclidean conformal group $G=SO(d+1,1)$, induced from
representations $\tau$ of the maximal compact subgroup $SO(d+1)$ of $G$. From
these large reducible representations we can single out representations which
are equivalent to conformal boundary representations labelled by the conformal
weight and by arbitrary representations $\mu$ of the Euclidean Lorentz group
$M=SO(d)$, such that $\mu$ is contained in the restriction of $\tau$ to $M$.
Thus, our boundary-to-bulk operators can be compared with those in the
literature only when for a fixed $\mu$ we consider a 'minimal' representation
$\tau=\tau(\mu)$ containing $\mu$.
