The present paper contains two interrelated developments. First are proposed new generalized Verma modules. They are called k Verma modules, k ∈ IN , and coincide with the usual Verma modules for k = 1. As a vector space a k Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(G), where G is the subalgebra of lowering generators in the sta...