The twining character formula for reductive groups

Jackson Hopper

Published 2022-04-07Version 1

Let $\widehat{G}$ be a split, connected, reductive group with an outer automorphism $\sigma$ preserving some pinning. Jantzen's twining character formula relates the trace of a natural action of $\sigma$ on a highest-weight representation $V_{\mu}$ of $\widehat{G}$ to the character of a corresponding highest-weight representation $(V_{\sigma})_{\mu}$ of a related connected, reductive group, $\widehat{G^{\sigma, \circ}}$. The formula appears to be well-known when $\widehat{G}$ is an adjoint group, but the general case, where $\widehat{G}$ is only assumed to be split, connected, and reductive, seems to be missing from the literature. This paper proves that the formula holds for all split, connected, reductive groups, following Hong's geometric proof for the case $\widehat{G}$ is adjoint. This paper thus fills a gap in the literature, and proves some results of independent interest along the way.