Some simple modules for the restricted Cartan-type Lie algebras
This paper investigates modules for simple restricted Cartan-type Lie algebras over an algebraically closed field of characteristic \(p \geq 5\). Such an algebra \(\mathfrak g\) has a natural restricted grading \(\mathfrak g = \sum_i \mathfrak g_i\). If a \(\mathfrak g\)-module \(M\) has character \(\chi\), then the height of \(\chi\) is defined as \(\min \{ i \geq t \mid \chi (\mathfrak g^i) = 0\}\) where \(t = \min \{ i \mid \mathfrak g_i \not= 0\}\) and \(\mathfrak g^i = \sum_{j\geq i} \mathfrak g_j\). The authors' main result is that every simple module having character height at most one for a restricted Cartan-type Lie algebra \(\mathfrak g\) can be realized as a quotient of a module obtained by starting with a simple module \(S\) for the homogeneous component of degree zero in the natural grading of \(\mathfrak g\), extending the action trivially to positive components and inducing up to \(\mathfrak g\). Furthermore, the induced module is already simple if \(S\) is not restricted, or if it is restricted and its maximal vector does not have a weight belonging to a certain very small set of exceptional weights. The proofs involve considerable computation using the structures of the four types of Cartan-type algebras.
scientific article; zbMATH DE number 1787488
scientific article