  We extend classical results of Kostant and al. on multiplets of
representations of finite-dimensional Lie algebras and on the cubic Dirac
operator to the setting of affine Lie algebras and twisted affine cubic Dirac
operator. We prove in this setting an analogue of Vogan's conjecture on
infinitesimal characters of Harish-Chandra modules in terms of Dirac
cohomology. For our calculations we use the machinery of Lie conformal and
vertex algebras.
