Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula

TL;DR

This paper derives an asymptotic formula for the character of irreducible representations of complex reductive groups evaluated at finite order elements, using the Lefschetz Trace Formula and fixed point analysis.

Contribution

It explicitly determines fixed point components and their normal bundles to compute characters and provides an asymptotic formula for scaled highest weights.

Findings

Explicit character formulas for finite order elements

Asymptotic behavior of characters as weights grow large

Application of Lefschetz Trace Formula to representation theory

Abstract

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 $(G/P)^t\subset G/P$ (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.