  The 1$^-$ excitations of the three--body halo nucleus $^{11}$Li are
investigated. We use adiabatic hyperspherical expansion and solve the Faddeev
equations in coordinate space. The method of complex scaling is used to compute
the resonance states. The Pauli forbidden states occupied by core neutrons are
excluded by constructing corresponding complex scaled phase equivalent two-body
potentials. We use a recently derived neutron--core interaction consistent with
known structure and reaction properties of $^{10}$Li and $^{11}$Li. The
computed dipole excited states with $J^\pi=1/2^+$, $J^\pi=3/2^+$, and
$J^\pi=5/2^+$ have energies ranging from 0.6 MeV to 1.0 MeV and widths between
0.15 MeV and 0.65 MeV. We investigate the dependence of the complex energies of
these states on the $^{10}$Li spectrum. The finite spin 3/2 of the core and the
resulting core-neutron spin-spin interaction are important. The connection with
Coulomb dissociation experiments is discussed and a need for better
measurements is pointed out.
