Projective functors for algebraic groups.
Let \(G\) be a semi-simple split simply connected algebraic group over an algebraically closed field of characteristic \(p>0\). \textit{W. Soergel} [J. Pure Appl. Algebra 152, No. 1-3, 311-335 (2000; Zbl 1101.14302)] introduced a certain subcategory of \(G\)-modules which in part gives information about the multiplicities of certain simple \(G\)-modules in induced (or standard) modules. Soergel's category is relative to the Steinberg module. In this paper, the author generalizes Soergel's category and defines a subcategory for each weight. The definition is based on certain linkage conditions among the composition factors of modules. These categories are defined not only for \(G\) but also for each subgroup \(G_rT\) where \(G_r\) denotes the \(r\)-th Frobenius kernel and \(T\) is a maximal torus. The first main result of the paper is that these categories have projective functors. This makes use of previous work of the author [Proc. Lond. Math. Soc., III. Ser. 90, No. 3, 711-737 (2005; Zbl 1063.17005)] on projective functors. Next, relationships are found amongst these categories. In particular, if \(p\) is greater than twice the Coxeter number of the root system, then the categories for \(G_rT\) (corresponding to a given weight) are all equivalent, independent of \(r\). Further, the category for \(G\) can be identified with a certain subcategory for \(G_1T\). This is then used to give a description of the category for \(G\) in terms of the Bernstein-Gelfand-Gelfand category \(\mathcal O\). Finally, these categories are used to obtain information on tilting modules for \(G\). More precisely, information is given on the character of an indecomposable tilting module after deformation to the corresponding quantum group (quantized enveloping algebra) in terms of characters of indecomposable tilting modules over the quantum group.
scientific article; zbMATH DE number 5142202
scientific article