  We study the properties of conformal operators in the SL(2) sector of planar
N=4 SYM and its supersymmetric SL(2|2) extension. The correlation functions of
these operators and their form factors with respect to asymptotic on-shell
states are determined in the appropriate limit by two different polynomials
which can be identified as eigenstates of the dilatation operator in the
coordinate and momentum representations, respectively. We argue that, in virtue
of integrability of the dilatation operator, the two polynomials satisfy a
duality relation -- they are proportional to each other upon an appropriate
identification of momenta and coordinates. Combined with the conventional N=4
superconformal symmetry, this leads to the dual superconformal symmetry of the
dilatation operator. We demonstrate that this symmetry is powerful enough to
fix the eigenspectrum of the dilatation operator to the lowest order in the
coupling. We use the relation between the one-loop dilatation operator and
Heisenberg spin chain to show that, to lowest order in the coupling, the dual
symmetry is generated by the Baxter Q-operator in the limit of large spectral
parameter.
