The Schur indices of the cuspidal unipotent characters of the finite chevalley groups $E_{7}(q)$

Abstract

We show that the two cuspidal unipotent characters of a finite Chevalley group $E_7(q)$ have Schur index $2$, provided that $q$ is an even power of a (sufficiently large) prime number $p$ such that $p\equiv 1 \bmod 4$. The proof uses a refinement of Kawanaka's generalized Gelfand--Graev representations and some explicit computations with the \textit{CHEVIE} computer algebra system.