Cartan Invariants for the Restricted Toral Rank Two Contact Lie Algebra

Restricted modules for the restricted toral rank two contact Lie algebra are considered. Contragredients of the simple modules, Cartan invariants, and dimensions of the simple modules and their projective covers are determined. Let L be a finite dimensional restricted Lie algebra. All L-modules in this paper are assumed to be left, restricted and finite dimensional over the defining field. Each simple L-module has a projective cover; the multiplicities of the composition factors of the various projective covers are called Cartan invariants. Here, we use the method of [3] to compute the Cartan invariants for the restricted toral rank two contact Lie algebra K(3, 1). To carry out the computation, one needs to know the simple modules and their multiplicities as composition factors of certain induced modules. In [4]–which considered restricted contact Lie algebras of arbitrary toral rank–it was shown that these multiplicities are generically one, that is, the induced modules are, with a few exceptions, simple (see 1.1 below). Although it is not known at the time of this writing, it is expected that (for arbitrary toral rank) the few exceptional induced modules will not be simple. At least this is the case for the algebra K(3, 1) as will be shown in this paper (see 6.1). In addition to the Cartan invariants for K(3, 1) we will compute the dimensions of the simple modules which will give in turn the dimensions of their projective covers. Also, we determine the contragredient of each simple module. I thank the referee for the improvement in 2.3(2) and its proof as well as for other useful comments. 1. Statement of Main Results Let F be an algebraically closed field of characteristic p > 2 and let n = 2r + 1 with r ∈ N. For 1 ≤ k ≤ n let εk be the n-tuple with jth component δjk (Kronecker delta). Set A = {a = ∑n k=1 akεk | 0 ≤ ak < p} ⊂ Z. For a, b ∈ A, define ( a b ) := ∏ k ( ak bk ) . The factors on the right are the usual binomial coefficients with the convention that ( i j ) = 0 unless 0 ≤ j ≤ i. The vector space A with F -basis {x(a) | a ∈ A} becomes an associative F -algebra by defining