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

TL;DR

This paper computes the 2-parameter Green functions for 8-dimensional spin groups with disconnected center, providing new insights into their character formulas in finite reductive groups.

Contribution

It determines all 2-parameter Green functions for 8-dimensional spin groups with disconnected center, a case previously not well-understood.

Findings

Explicit computation of Green functions for specific spin groups

Identification of properties of Green functions in disconnected center cases

Advancement in understanding Lusztig induction in these groups

Abstract

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.