Multiple q-zeta values and traces

Abstract

Let $(a)_\infty = (a; q)_\infty = \prod_{n=0}^\infty (1-aq^n)$. An elegant result of Bloch and Okounkov [BO] states that if $x = e^z$, then $$ \frac{(xq)_\infty (x^{-1}q)_\infty}{(q)_\infty^2}, $$ which appears in various traces in representation theory and algebraic geometry, is a formal power series in $z^2$ whose coefficient for $z^{2k}$ is a quasi-modular form of weight $2k$. Quasi-modular forms are special types of multiple $q$-zeta values. In this paper, we generalize this result of Bloch and Okounkov and prove that certain other traces are related to multiple $q$-zeta values. A simple case of our main results asserts that if $x = e^z$ and $y = e^w$, then $$ \frac{(xq)_\infty (yq)_\infty}{(q)_\infty (xyq)_\infty}, $$ which appears in [CW, Theorem 5] as a trace (the deformed Bloch-Okounkov $1$-point function), is a formal power series in $z$ and $w$ whose coefficient for $z^mw^n$ is a multiple $q$-zeta value (in the sense of [BK3, Oko]) of weight $(m+n)$.