I have a basic question concerning Atiyah and Schmid's paper "A Geometric Construction of the Discrete Series for Semisimple Lie Groups", Inventiones mathematicae 42 (1977): 1-62".
I will use the same notation as in the paper: $G$ is a semisimple Lie group with finite centre, $K$ its maximal compact subgroup, and $\Gamma$ an appropriately chosen discrete subgroup of $G$. It is stated (page 12) that the index of a Dirac operator on $\Gamma\backslash G/K$ twisted by a representation $V$ of $K$ is given by $\int_{\Gamma\backslash G/K} f(\Theta,\Phi)$, where $f$ is a polynomial, $\Phi$ the curvature of the bundle $\cal{V}$, formed using $V$, and $\Theta$ the cuvature of $\Gamma\backslash G/K$. This means that $f(\Theta,\Phi)=C(V)dx$ is then a constant multiple of the volume form.
Question: it is then stated (page 14) that $C(V)$ depends in a polynomial fashion on the highest weight $\mu$ of the representation $V$ of $K$, which "follows easily from the Weyl character formula and the relations between characters and characteristic classes."
Would someone be able to supply a reference where I can find the relations between characters and characteristic classes being referred to? Cheers.
