Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ by the action of $G = \text{Aut}(C'/C)$. Now, let $\rho:G\to GL(W)$ be a finite dimensional complex representation of $G$. Below, we identify locally free coherent sheaves with holomorphic vector bundles.
Let $\underline W$ be the trivial vector bundle on $C'$ with fibre $W$. Define the vector bundle $W^\rho$ on $C$ to be the subsheaf of $\pi_*\underline W$ whose sections over $U\subset C$ are the $G$-equivariant holomorphic functions $U' = \pi^{-1}(U)\to W$. We want to compute $c_1(W^\rho)$ as follows. We have a natural map $\varphi:\pi^*W^\rho\to\underline W$ on $C'$ (coming from the adjunction counit $\pi^*\pi_*\underline W\to\underline W$) which is an injective map of coherent sheaves. Now, by looking at the zeros of the determinant of $\varphi$ (which occur exactly at the critical points of $\pi$), we can figure out the value of $c_1(W^\rho) = \frac1{|G|}c_1(\pi^*W^\rho) = -\frac1{|G|}\cdot\dim H^0(C',\text{coker }\varphi)$.
Carrying out this computation explicitly, we seem to get the following answer. Given any branch point $p\in C$ of $\pi$, pick a preimage $p'\in C$. Let $G_{p'}\subset G$ be the (necessarily cyclic) stabilizer group of $p'$, of order $n_p$. For $0\le i<n_p$, define the numbers $w_{p,i} := \dim\text{Hom}^{G_{p'}}((T_{p'}C)^{\otimes i},W)$ and set $w_p:=\sum_{0\le i<n_p}\frac{i}{n_p}w_{p,i}\in\mathbb Q$. We then get $c_1(W^\rho) = -\sum_p w_p$, where the sum is over the branch points of $\pi$. Is the result of this computation correct and can I verify it by comparing it to some well-known/basic theorem? (I tried to calculate explicitly in local holomorphic coordinates where the map is given by $z\mapsto z^{n_p}$ and got the above answer.)
I find it quite interesting that the sum of the rational numbers $w_p$ is an integer. But clearly, these numbers can be defined without referring to the vector bundle $W^\rho$ or its Chern class. Is there some direct way in which we could prove this integrality statement?
This question is motivated by trying to understand the index computation (Theorem 4.1) in Chris Wendl's paper on super-rigidity and equivariant transversality (https://arxiv.org/abs/1609.09867).