Dimensions of the Simple Restricted Modules for the Restricted Contact Lie Algebra

According to [2], the finite-dimensional restricted simple Lie algebras over an algebraically closed field of characteristic p > 7 are either classical or of Cartan type. Lusztig has a conjecture (see [8, p. 294]) which provides a formula for the dimensions of the simple modules for the classical algebras. (In this paper, modules are assumed to be left, finitedimensional and restricted.) Anderson, Jantzen, and Soergel [1] have recently shown that for p À 0 the Lusztig conjecture follows from an analogous conjecture (also due to Lusztig) for certain quantum groups. They have also pointed out that this latter conjecture is implied by recent work of Kazhdan and Lusztig and a paper by Casian, although Casian’s paper has not yet convinced all of its readers. The simple modules for the Cartan-type algebras fall into two classes. There is a large class (see [11] and [5]) consisting of modules that are induced from simple modules for the homogeneous component of degree zero extended trivially to positive components (constructed as “mixed products” in [11]). The degree-zero homogeneous components of Cartan-type algebras are either classical or nearly so. Therefore, the Lusztig conjecture can be used to get a formula for the dimensions of the modules in this class. The small class consists of modules corresponding to “exceptional” weights. In [11] the dimensions of these modules were determined for three of the four Cartan-type algebras, namely, the Witt, special and hamiltonian algebras. (There the exceptional weights turned out to be fundamental dominant weights.) In this paper, we obtain the dimensions (assuming p is not too small) in the remaining case of the contact algebra. In the process, we determine the contragredients of the simple modules.