Title:

Extensions of simple modules for the universal Chevalley groups and its parabolic subgroups

The modular representation theory of Chevalley groups is still in a tentative stage (see the introduction of [8]). As far as this topic is concerned, we know: the indexing set for simple modules, the linkage principle, the strong linkage principle, the blocks, etc. In this thesis two problems have been solved. Both of them deal with the extension of a simple module by another simple one. The first problem deals with the extension group between simple Gmodules when G is the universal Chevalley group, and describes this group for type A2• The second one investigates the blocks of the parabolic subgroups of the universal Chevalley groups which is highly related to the extension problem. The wide open problem of describing the modular representations of Chevalley groups, and the solution of the above mentioned problems recall to mind the Hindu fable of the blind men and the elephant as written by J.G. Saxe, however see the paragraph below. It was six men of Indostan to learning much inclined., Who went to see the Elephant (though all of them were blind) That each by observation might satisfy his mind. The First approached the Elephant, and happening to fall Against his broad and sturdy side, at once began to bawl: "God bless me! but the Elephant is very like a wall!" The Second, feeling of the tusk, cried: "Ho! what have we here So very round and smooth and sharp? To me ' tis very clear This wonder of an Elephant is very like a spear!" The Third approached the animal, ana happening to take The squirming trunk within his hands, thus boldly up and spake: "I see," quoth he, "the Elephant is very like a snake!" The Fourth reached out an eager hand, and felt about the knee. "What most this mighty beast is like is mighty plain", quoth he; "'Tis very clear the Elephant is very like a tree!" (The Fifth, who chawed. to touch the ear, said: "E'er, the blindest man Can tell what this resembles rest; deny the fact who can, This marvel of an Elephant is very like a fan!" The Sixth no sooner had begun about the beast to grope, Then, seizing on the swinging tail that fell within his scope, "I see," quoth he, "tiie Elephant is very like a rope!" And so these men of Indostan disputed loud and long, Each in his own opinion exceeding stiff and strong, Though each was partly in the right, and all were in the wrong! One way to study the representation theory of a group is to get hold of the simple modules. The modular representations of the Chevalley groups (and its parabolic subgroups) are not necessarily completely reducible, so the extension problem appears naturally. The natural question is, if V is a module (with two composition factors say), when is it completely reducible? Conversely, given two simple modules , L9 what modules V may be constructed with , l > 2 as its composition factors, and when do these extensions split? Another important aspect of the extension problem is Anderson's conjucture (conjucture 7.2 of [4]), which may be very strongly connected with Lusztig's conjuncture on the character of the simple modules (problem IV of [33]). This thesis consists of five chapters. Since we cannot put a sharp line between the blocks and the extensions, the first chapter is meant to be a preliminary for both our problems, and also it presents the necessary background. The second chapter deals with the extension group in general (when G is the universal Chevalley group), and puts some relations between the extension functor and Jantzen's translation functor. In the third and fourth chapters we Investigate this functor when G is of type A2 > In the third one we determine the functor Ext1U1 between simple U1modules, where is the restricted enveloping algebra of the Lie algebra of G. The extensions Ext 1G( L (μL(λ)) i.e. between simples, have been determined in the fourth chapter. Finally, in the fifth chapter we determine the blocks of the parabolic subgroups of the universal Chevalley groups. Throughout this thesis, the notations dim and x are abbreviations for dimK and respectively xK. The symbols N, Z, Θ, IR and T will denote the natural, integral, rational, real, and the complex numbers respectively. Modules for the affine algebraic groups will always mean the rational ones defined in Section 1.1. A submodule or a direct summand may mean isomorphic to a submodule or to a direct summand. Finally, the end of the proofs (if any), definitions, examples, etc., will be marked thus.
