  We study the adjoint and coadjoint representations of a class of Lie group
including the Euclidean group. Despite the fact that these representations are
not in general isomorphic, we show that there is a geometrically defined
bijection between the sets of adjoint and coadjoint orbits of such groups. In
addition, we show that the corresponding orbits, although different, are
homotopy equivalent. We also provide a geometric description of the adjoint and
coadjoint orbits of the Euclidean and orthogonal groups as a special class of
flag manifold which we call a Hermitian flag manifold. These manifolds consist
of flags endowed with complex structures equipped to the quotient spaces that
define the flag.
