Weights of restricted modules of some Cartan type Lie algebras

Question

Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$. Denote by $L_{[p]} $ the minimal $p$-envelope of $L$ and let ${\mathfrak t}$ be a 2-dimensional torus of $L_{[p]}$. For a nontrivial irreducible restricted $L_{[p]}$-module $V$ we denote by $\Gamma(V, {\mathfrak t})$ the set of weights of $V$ with respect to ${\mathfrak t}$.

In Section 10.7 of the book [H. Strade: Simple Lie algebras over fields of positive characteristic. II. Classifying the absolute toral rank two case] it is showed that $\Gamma(V, {\mathfrak t}) \cup$ {0} is a 2-dimensional vector space over the prime field $\mathbb{F}_p$.

In some cases the 0 weight is actually missing in $V$. For instance, this happens when $L$ is the Hamiltonian algebra $H(2;\underline{1};\Phi(\tau))^{(1)}$ and $V$ is the adjoint module. On the other hand, in this case does there exist a nontrivial restricted irreducible module $V$ for $L_{[p]}$ such that the 0 weight is in $\Gamma(V, {\mathfrak t})$?

Answer 1

Such modules do exist for $p>3$. The Lie algebra $L_p={\rm Der}\ H(2;\underline{1};\Phi(\tau))^{(1)}$ has a filtration of depth $1$ such that ${\rm gr}(L)$ is a graded Lie algebra of type $\rm H$. This can be seen by embedding it into $W(2;\underline{1})$ as explained in Strade's paper in Can. J. Math. , vol. 43, 1991, 580-616. As $TR(L)=2$ the Lie algebra $L_p$ contains a $2$-dimensional torus, $T$ say. Any $2$-dimensional torus of $L_p$ must intersect trivially with $(L_p)_{(0)}$ because any nonzero torus of $(L_p)_{(0)}$ is $1$-dimensional and has no zero weight on $L_p/(L_p)_{(0)}$ (here $(L_p)_{(0)}$ is the zero component of the filtration). Passing to corresponding graded algebras and modules and using some results on representations of $H(2;\underline{1})^{(2)}$ mentioned in Jim's answer one can see that all but two of the irreducible restricted representations of $L_p$ are induced from irreducible restricted representations of $(L_p)_{(0)}$. The equality $L_p=T\oplus (L_p)_{(0)}$ then shows that all possible $T$-weights (including zero) occur in such modules with the same multiplicity.

As a historical remark I should add that irreducible representations of finite dimensional graded Cartan type Lie algebras of type ${\rm W, S,H}$ were first studied in the late 70s by Ya. Krylyuk, a former PhD student of Kostrikin. Unfortunately, the results of his PhD thesis were buried in the cemetery of "secondary scientific and technical information" called VINITI and remained completely unknown to the rest of the world (the was no ${\rm arXiv}$ at the time). Wikipedia now has a comprehensive article about VINITI.

Answer 2

Though I suspect the answer to the question may be yes, it's hard to document this one way or the other based on the limited literature dealing with modular representations of Lie algebras of Cartan type. It's been difficult to systematize this kind of study, other than by trying to imitate some features found in the representations of Lie algebras of simple algebraic groups. (Much of the work by Strade and others has gone instead into the classification problem.)

I can at least point to some of the relevant work already done. There are papers (probably hard to access) by Guang Yu SHEN, for example: Graded modules of graded Lie algebras of Cartan type. III. Irreducible modules. Chinese Ann. Math. Ser. B 9 (1988), no. 4, 404–417. Then there is a series of papers by Randall R. Holmes (and collaborators), such as: Simple restricted modules for the restricted Hamiltonian algebra. J. Algebra 199 (1998), no. 1, 229–261. Roughly speaking, the emphasis has been on constructing or classifying simple modules starting with those induced from suitable proper subalgebras. But there is not as much explicit weight structure on view here as in the classical case. While I don't know all the details of these papers, I'd caution against expecting to find easy answers.

Beyond simple modules there are of course many others, including projective modules. In this direction Holmes, Nakano, and recently Jantzen have made significant progress. But the methods and results still have a lot of gaps. Contacting Holmes directly (at Auburn University) might be helpful for your specific question.

P.S. To me the (restricted) Cartan types have always been a bit disorienting, but each does involve one of the familiar classical types as the 0-part of a natural grading. For instance, the Hamiltonian Lie algebra involves a symplectic Lie algebra, which provides a suitable torus (along with weights) to get things started. But there are different approaches in the literature.