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Tensor products and injectives for groups of Lie type
Abstract
Let G be a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic p, T a maximal split torus of G defined over the field of p elements, and $X\sb{n}(T)$ the set of n-restricted weights. Let G(n) denote the finite group consisting of the set of fixed points of the nth Frobenius morphism of G. In the first chapter the summands containing the highest weight space of tensor products of G-modules with n-restricted highest weight are studied. It is shown that for certain tensor products these summands are isomorphic to the indecomposable injectives in the category of $G\sb{n}T$-modules, where $G\sb{n}$ is the kernel of the nth Frobenius morphism on G. In the second chapter we study the reduction modulo p of the virtual ordinary characters introduced by Deligne and Lusztig for the finite group G(n). This is of interest because it leads to the decomposition of the irreducible ordinary characters after reduction modulo p. According to Brauer's reciprocity law these decomposition numbers determine the Cartan invariants of G(n). Jantzen has shown that the generic decomposition pattern of the Deligne-Lusztig characters corresponds to the decomposition of the induced $G\sb{n}T$-modules with p-regular highest weight. We prove here a similar result for composition factors with p-singular highest weight of Deligne-Lusztig characters that are located "deep" inside an alcove wall. Our method of proof yields Jantzen's result in most cases. Proper Deligne-Lusztig characters that are located "deep" inside an alcove wall are studied in Chapter 3 for the rank two groups. Typically such a character splits into a regular ordinary character and a semisimple ordinary character. We look at the distribution of the extremal composition factors introduced by Humphreys after reduction modulo p. In the last section we express the Brauer character of the injective hull of the trivial module in terms of ordinary characters for the group of type $G\sb2$ over the field of p elements.
Subject Area
Mathematics
Recommended Citation
Pillen, Cornelius, "Tensor products and injectives for groups of Lie type" (1992). Doctoral Dissertations Available from Proquest. AAI9219480.
https://scholarworks.umass.edu/dissertations/AAI9219480