Let $X$ be a tame DM stack over $\mathbb{C}.$ Let $IX$ denote the inertia stack of $X.$ Let $K(IX)$ denote the Grothendieck group of vector bundles on $IX.$ By the discussion on page 20 of Toen's paper "Riemann-Roch Theorems for Deligne-Mumford Stacks", there is a map
$$\rho: K(IX) \to K(IX)_\mathbb{C}.$$
Let $V$ be a vector bundle on $IX$. Then $\rho$ maps $V$ to $\sum_{\zeta \in \mu_\infty} \zeta V^{(\zeta)},$ where $\mu_\infty$ is the union of all $\mu_n$ for integers $n \geq 1,$ i.e., $\mu_\infty$ consists of all roots of unity, and $V^{(\zeta)}$ is the eigenbundle of $V$ corresponding to an eigenvalue $\zeta.$
Is $\sum_{\zeta \in \mu_\infty} \zeta V^{(\zeta)}$ a finite sum?
Edit: The map $\rho$ is well-defined if the sum above is finite. So the question is whether $\rho$ is always well-defined.