The goal of this paper is to study the representations of reduced universal enveloping algebras of Lie algebras arising from a reductive algebraic group over an algebraically closed field of prime characteristic.
The author starts by considering the general setup of a finite-dimensional restricted Lie algebra \({\mathfrak g}\) over an arbitrary ground field of prime characteristic which is graded by a free abelian group \(Y\) of finite rank. Under the assumption that the \(p\)-character \(\chi\) vanishes on the homogeneous components of \({\mathfrak g}\) with non-zero degree, the \(\chi\)-reduced universal enveloping algebra of \({\mathfrak g}\) is also a finite-dimensional \(Y\)-graded algebra. Several basic properties of the graded modules over such algebras are proved in the preliminary section. This also gives a more down-to-earth version of the author’s earlier approach to this topic.
For the rest of the paper \({\mathfrak g}\) is the Lie algebra of a connected reductive algebraic group over an algebraically closed field of prime characteristic. Moreover, it is assumed that the \(p\)-character \(\chi\) has so-called standard Levi form, so that the preliminary section can be applied (e.g. to show results on the duals of baby Verma modules and on the structure of projective indecomposables). The author constructs a certain filtration for every baby Verma module and proves a sum formula for these filtrations. The sum formula is then used to derive a strong linkage principle for baby Verma modules and to show that certain simple modules occur with multiplicity one as composition factors in certain baby Verma modules. The latter yields some properties of translation functors. It should be noted that many features of these representations are similar to those of the Bernstein-Gel’fand-Gel’fand category of a finite-dimensional complex semisimple Lie algebra or to the representations of reductive algebraic groups in prime characteristic.
The results obtained so far can be applied to compute the characters and dimensions of simple modules in several cases. In the last section this is demonstrated in one case for \(B_2\) which avoids the brute force computations in an earlier preprint of the author. All cases (including all cases of rank two) checked by the author confirm G. Lusztig’s hope in [Represent. Theory 1, 207-279 (1997; Zbl 0895.20031)] that the multiplicities of composition factors of baby Verma modules are given by values at 1 of certain polymomials.