Principal subspace for the bosonic vertex operator and Jack polynomials

Let , m∈N, be a bosonic vertex operator and L be some irreducible representation of the vertex algebra A(m) associated with the one-dimensional lattice Zl, 〈l,l〉=2m. Fix some extremal vector v∈L. We study the principal subspace C[ai]i∈Z⋅v and its finitization C[ai]i>N⋅v. We construct their bases and find characters. In the case of finitization, the basis is given in terms of Jack polynomials.