Imprimitive Irreducible Modules for Finite Quasisimple Groups

Imprimitive Irreducible Modules for Finite Quasisimple Groups Synopsis

Motivated by the maximal subgroup problem of the finite classical groups the authors begin the classification of imprimitive irreducible modules of finite quasisimple groups over algebraically closed fields $K$. A module of a group $G$ over $K$ is imprimitive, if it is induced from a module of a proper subgroup of $G$.

The authors obtain their strongest results when ${\rm char}(K) = 0$, although much of their analysis carries over into positive characteristic. If $G$ is a finite quasisimple group of Lie type, they prove that an imprimitive irreducible $KG$-module is Harish-Chandra induced. This being true for $\mbox{\rm char}(K)$ different from the defining characteristic of $G$, the authors specialize to the case ${\rm char}(K) = 0$ 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. The authors determine the asymptotic proportion of the irreducible imprimitive $KG$-modules, when $G$ runs through a series groups of fixed (twisted) Lie type. One of the surprising outcomes of their investigations is the fact that these proportions tend to $1$, if the Lie rank of the groups tends to infinity.

For exceptional groups $G$ of Lie type of small rank, and for sporadic groups $G$, the authors determine all irreducible imprimitive $KG$-modules for arbitrary characteristic of $K$.