By G. D. James
Publisher: Cambridge University Press
Print Publication Year: 1984
Online Publication Date:May 2010
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511661921.001
This essay concerns the unipotent representations of the finite general linear groups GLn (q). An irreducible unipotent representation is, by definition, a composition factor of the permutation representation of GLn (q) on a Borel subgroup, and the ordinary irreducible unipotent representations may be indexed by partitions λ of n, as may the ordinary irreducible representations of the symmetric group. The remarkable feature is that the representation theory of over an arbitrary field appears to be the case “q = 1” of the subject we study here.
The most important results are undoubtedly the Submodule Theorem (Chapter 11) and the Kernel Intersection Theorem (Chapter 15), but there seems to have been no previous work on the representation modules for the unipotent representations of GLn (q), so we claim originality for all the results apart from those whose source is quoted or which are obviously known (Chapters 3 – 8).
Chapters 1 and 2 set the scene, by outlining the connection between and representations of GLn (q) over fields of characteristic dividing q, and by giving examples of the situation to be considered later. The preliminary results which we need are derived in Chapters 3 – 8. Thereafter, we assume that the characteristic of our ground field K does not divide q, but otherwise K is arbitrary.