Filtrations of the modules for Chevalley groups arising from admissible lattices
Let \(G\) be the simply connected Chevalley group over a field \(k\) of characteristic \(p > 0\) associated with a semisimple complex Lie algebra \(\mathfrak g\). Let \(M\) be an admissible lattice in a finite-dimensional irreducible \(\mathfrak g\)-module \(V(\lambda)\) and \(\overline M\) a rational \(G\)-module obtained from \(M\) through reduction modulo \(p\). The author proves that \(\overline M\) is indecomposable if all weight spaces of \(V(\lambda)\) are 1-dimensional. He constructs a filtration of \(\overline M\) generalizing the Jantzen filtration for a Weyl module and the Andersen filtration for a cohomology module in the generic situation. It turns out that the filtration layers are self-dual under the transpose dual. In particular they are semisimple if the corresponding Weyl module is multiplicity free. Then the groups of type \(A_ 1\) are considered, in which case a description of the module graph of \(\overline M\) is given. It is shown that the nondirected graph associated to the module graph of \(\overline M\) is independent of \(M\). All the possible graphs for \(\overline M\) can be characterized from the module graph \(H^ 0_ k(\lambda)\) in a purely combinatorial manner. All \(\overline M\) with maximal Loewy length, which is the Loewy length of \(H^ 0_ k(\lambda)\) are classified.
finite-dimensional irreducible \(\mathfrak g\)-module
scientific article; zbMATH DE number 4214255
scientific article