An extended trace formula for 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 FL. Then, we prove a duality between glL and aL gl(infinity) of Howe type, which provides a decomposition of FL into irreducible representations with joint highest weight vector for gl(infinity) and a(infinity). The decomposition of the Fock space FL into highest weight representations provides a method to calculate and interpret the extended trace. (c) 2022 Elsevier Inc. All rights reserved.