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Let $G(q)$ be a finite group of Lie type over a field with $q$ elements,where $q$ is a prime power. The Green functions of $G(q)$, as defined byDeligne and Lusztig, are known in \textit{almost} all cases by work ofBeynon--Spaltenstein, Lusztig und Shoji. Open cases exist for groups ofexceptional type ${^2\!E}_6$, $E_7$, $E_8$ in small characteristics. We proposea general method for dealing with these cases, which procedes by a reduction tothe case where $q$ is a prime and then uses computer algebra techniques. Inthis way, all open cases in type ${^2\!E}_6$, $E_7$ are solved, as well as atleast one particular open case in type $E_8$.

Computing Green functions in small characteristic