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 $V(\lambda )$ of $G$ of highest weight $\lambda$, we determine the character $ch(t,V(\lambda ))$ 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 $(G\mathrm{/}P{)}^t\subset G\mathrm{/}P$ (for any parabolic subgroup $P$). This together with Wirtinger’s theorem gives an asymptotic formula for $ch(t,V(n\lambda ))$ when $n$ goes to infinity.