Character of irreducible representations restricted to finite order elements - an asymptotic formula

Let G be a connected reductive group over the complex numbers and let $T\subset G$ be a maximal torus. For any $t\in T$ of finite order and any irreducible representation $V\left(\lambda \right)$ of G of highest weight λ, we determine the character $\mathrm{ch}(t,V(\lambda \left)\right)$ by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety ${(G/P)}^t\subset G/P$ (for any parabolic subgroup P). This together with Wirtinger's theorem gives an asymptotic formula for $\mathrm{ch}(t,V(n\lambda \left)\right)$ when n goes to infinity.