On bases of certain Grothendieck groups

TL;DR

This paper extends the construction of a second basis for the Grothendieck group from split reductive groups to nonsplit special orthogonal groups, broadening the applicability of the previous framework.

Contribution

It introduces a new basis for the Grothendieck group of nonsplit special orthogonal groups, generalizing prior results for split reductive groups.

Findings

Established a second basis for nonsplit special orthogonal groups

Extended previous work from split to nonsplit cases

Provided algebraic tools for further representation theory studies

Abstract

In a previous paper we have defined a second basis of the Grothendieck group of a split reductive group over a finite field. In this paper we extend this to the case of nonsplit special orthogonal groups.