Imprimitive irreducible modules for finite quasisimple groups

TL;DR

This paper classifies imprimitive irreducible modules of finite quasisimple groups, showing most are induced and providing explicit classifications for certain groups, advancing understanding of their representation theory.

Contribution

It proves that imprimitive irreducible modules over fields of characteristic zero are Harish-Chandra induced and classifies these modules for specific groups, extending prior knowledge.

Findings

Most irreducible modules are imprimitive asymptotically.

Classification of imprimitive modules for classical and sporadic groups.

Explicit listing of imprimitive characters for low-rank groups.

Abstract

Motivated by the maximal subgroup problem of the finite classical groups we begin the classification of imprimitive irreducible modules of finite quasisimple groups. We obtain our strongest results for modules over fields of characteristic 0, although much of our analysis carries over into positive characteristic. If G is a finite quasisimple group of Lie type, and K an algebraically closed field of cross characteristic, we prove that an imprimitive irreducible KG-module is Harish-Chandra induced. We then specialize to the case when K has characteristic 0 and apply Harish-Chandra philosophy to classify irreducible Harish-Chandra induced modules in terms of Harish-Chandra series, as well as in terms of Lusztig series. One of the surprising outcome of our investigations is the fact that, asymptotically, most of the irreducible KG-modules are imprimitive in this situation. For exceptional groups of Lie type of rank at most 2, and for sporadic groups, we list all ordinary irreducible imprimitive characters.