Principal subspace for the bosonic vertex operator φ{symbol}_{sqrt(2 m)} (z) and Jack polynomials

Let φ{symbol}_{sqrt(2 m)} (z) = ∑_{n ∈ Z} a_n z^{- n - m}, 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 Z l, 〈 l, l 〉 = 2 m. Fix some extremal vector v ∈ L. We study the principal subspace C [a_i]_{i ∈ Z} ṡ v and its finitization C [a_i]_{i > N} ṡ v. We construct their bases and find characters. In the case of finitization, the basis is given in terms of Jack polynomials.