  We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family
of simple positive energy representations for the simple affine vertex algebra
V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations
are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1
and include in particular all admissible simple highest weight modules and all
simple modules induced from sl_2. Any such simple module in the minimal
nilpotent orbit has bounded weight multiplicities.
