Irreducible Representations of the Generalized Jacobson-Witt Algebras

Let L be the generalized Jacobson-Witt algebra W(m; n) over an algebraically closed field F of characteristic p > 3, which consists of special derivations on the divided power algebra R = u(m; n). Then L is a so-called generalized restricted Lie algebra. In such a setting, we can reformulate the description of simple modules of L with the generalized p-character chi when ht(chi) < min{p(ni) - p(ni-1) vertical bar 1 <= i <= m} for n - (n(1), ... , n(m)), which was obtained by Skryabin. This is done by introducing a modified induced module structure and thereby endowing it with a so-called (R, L)-module structure in the generalized x-reduced module category, which enables us to apply Skryabin's argument to our case. Simple exceptional-weight modules are precisely constructed via a complex of modified induced modules, and their dimensions are also obtained. The results for type W are extended to the ones for types S and H.