The 2-parameter Green functions for 8-dimensional spin groups

The 2-parameter Green functions occur as a crucial ingredient in the character formula for Lusztig induction in finite reductive groups. Still, very little is known about these functions, in particular in the case of groups arsing from algebraic groups with disconnected centre. We collect some basic properties and then apply these, together with some explicit computations, to determine all 2-parameter Green functions for 8-dimensional spin groups in odd characteristic, whose centre is disconnected of order~4.