Irreducible Representations of the Generalized Jacobson-Witt Algebras

Abstract

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= 𝔄(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 χ when ht (χ) < min {pni-pni-1| 1 ≤ i ≤ m} for n=(n1,…,nm), 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 χ-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.