Fock Space of Level infinity and Characters of Vertex Operators

We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level $\mathfrak{F}^{\infty}$. Then, we prove a duality between $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$ of Howe type, which provides a decomposition of $\mathfrak{F}^{\infty}$ into irreducible representations with joint highest weight vector for $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}$. The decomposition of the Fock space $\mathfrak{F}^{\infty}$ into highest weight representations provides a method to calculate and interpret the extended trace.