An extended trace formula for vertex operators

Abstract

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 F∞. Then, we prove a duality between gl∞ and a∞=glˆ∞ of Howe type, which provides a decomposition of F∞ into irreducible representations with joint highest weight vector for gl∞ and a∞. The decomposition of the Fock space F∞ into highest weight representations provides a method to calculate and interpret the extended trace.