Gerhard Hiss | Akateeminen Kirjakauppa

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 $K$.