Principle subspace for bosonic vertex operator ϕ√2m(z) and Jack polynomials

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Let ϕ√2m(z)=∑ₙ∈ aₙ z⁻ⁿ⁻ᵐ, m∈ be bosonic vertex operator, L some irreducible representation of the vertex algebra ₍ₘ₎, associated with one-dimensional lattice, generated by vector l, l,l =2m. Fix some extremal vector v∈ L. We study the principle subspace [aᵢ]ᵢ∈· v and its finitization [aᵢ]ᵢ>ₙ· v. We construct their bases and find characters. In the case of finitization basis is given in terms of Jack polynomials.