Modular Representations For The Restricted Cartan Type Lie Algebras

Let be a finite-dimensional restricted Lie algebra over an algebraically closed field F and let x = HornF(F). A ?- module M has character x provided that Dpm - D[p]m = x(D)pm for D 6 ?and M. Not every module has a character, but at least every simple module has one [SF, Theorem 2.5, p. 207]. If M is a module with x =0 then we call M a restricted -module; if x ≠0, then M is called a nonrestricted module.Fix n ≥ 1, and let W = W(n, 1) be the restricted Witt algebra over an algebraically closed field F of characteristic p ≥ 5. Let x W = HomF(W,F) Let W = be the standard grading on W and put Wi = then in [H4] height ht(x) of the character x was defined by:ht(x) = min{i ≥ -l|x(Wi) = 0}.In 1941, Chang [Ch] worked with the smallest Witt algebra W(l, 1) and determined all the simple modules with arbitrary characters. Later, Strade [St] gave proofs of many of Chang's results in a different approach. Koreshkov [K] studied the next smallest Witt algebra W(2,1).Recently, Holmes [H4] worked with the general Witt algebra W(n, 1) and gave a uniform treatment of the three cases ht(x) = -1,0,1. He classified all simple modules and also obtained their dimension formulas. The simple modules for W(n, 1) with exceptional weights and ht(x) ≤ 0 were constructed by means of a general de Rham complex in [H4]. Since the other three types of Lie algebras, namely, special Lie algebras, Hamiltonian Lie algebras and contact Lie algebras, are all Lie subalgebras of Witt algebras and all graded, we can define the height of a character similarly. The purpose of this paper is to classify and find dimensions of the simple modules with exceptional weights and character height at most zero for these three Lie algebras. The definition of exceptional weights of each of these three Lie algebras is given in the beginning of Section 1, 2 and 3.This, together with [HZ], and the earlier work of Holmes [H4], completes the classification of the simple modules with height at most one for the restricted Cartan type Lie algebras.The paper is arranged as follows. In Section 1, we consider the restricted special Lie algebra S. Section 2 deals with the restricted Hamiltonian Lie algebra H. In Section 3 we mainly construct simple modules for the restricted contact Lie algebra K with htx = 0, - 1. Section 4 presents the dualityTypeset byof these modules. In Section 5 we proves the simplicity of these modules. Finally, in Section 6 we assemble the results and state the main theorems. Section 7 is the furthor study of the uniform basis for the contact algebra case, where for the case ht(x) = 0, we give an alternate determination for the simplicity by the uniform basis.