On Auslander-Reiten components and simple modules for finite groups of Lie type

Abstract

Let kG be the group algebra of a finite group G over an algebraically closed field k of characteristic p, where p is a prime. Let B be a p-block of kG with defect group AE(B) =G D. Then B is called to be of wild representation type, if D is neither cyclic, nor dihedral, semi-dihedral or a generalized quaternion group. Let 2 be a connected component of the stable Auslander-Reiten quiver 0s(B) of a wild p-block B of kG. In [6] K. Erdmann has shown that the tree class of 2 is A1. It is therefore natural to ask where a simple kG-module S of a wild p-block B can be found in its Auslander-Reiten component 2(S). The first author proved in [11] that a simple kG-module S of a wild p-block B must lie at the end of 2(S) if G is p-solvable. Moreover, he showed that, if there is some simple module which does not lie at the end, then there exist several simple modules lying at the end and having uniserial projective covers. We hope that this result may give a device for determining where simple modules lie in 0s(B). In this paper, we prove a similar assertion for wild p-blocks of a finite group G of Lie type, when p is the defining characteristic of G. Our main result is as follows. Theorem. Let G be a finite group of Lie type defined over a field K of characteristic p. Let B be a block of G with full defect of wild representation type. Then any simple kG-module S of B lies at the end of its Auslander-Reiten component 2(S).