Simple Restricted Modules for the Restricted Hamiltonian Algebra

The simple restricted modules for the restricted simple Cartan-type Lie algebras fall into two classes. There is a large class consisting of modules induced from simple modules for the subalgebra of nonnegative components. Then there is the small class of exceptional simple modules. Explicit constructions of these exceptional modules and formulas for their dimensions were obtained by Shen [9] for the Witt and special algebras and by Holmes [3] for the contact algebra (for sufficiently large characteristic). In this paper, we take up the case of the hamiltonian algebra. Already in [9] Shen has made considerable progress in this case, especially in considering the map δ of Section 3 below. However, we feel that some proofs are incomplete and some statements are incorrect. In particular, we believe that the formula given for the dimensions of the exceptional modules is inaccurate. We provide a new formula under the assumption of sufficiently large characteristic (see 5.10). For more detailed comments comparing Shen’s results with ours, see the remarks following 3.2, 4.8, 5.8, and 5.10.