I(X)-Reduced Representations Of Graded Lie Algebra W(m;n) Of Cartan Type

In this paper, we discuss the representation of Cartan type Lie algebra W(m; n) in characteristic p > 2, from the viewpoint of reducing rank. When the character x is regular semisimple, we can reduce higer-rank representations to lower-rank representations. Because the graded Car-tan tpye Lie algebra L = W(m; n) is not a restricted Lie algebra when n ≠ 1, Kac-Weisfeiler's method of dividing the irreducible modules with character functions is not work. Thanks to the discuss of generalized restricted Lie algebras by Shu Bin, we have known that the generalized restricted x-reduced module category coincides with the x-reduced module category of its primitive p-envelope, where x|- is the trivial extension of x in the primitive p-envelope. So we can describe the irreducible representation of L = W(m; n) by discussing the irreducible representation of its primitive p-envelope For an index subset I {1,2, ... ,m} and its supplement I|^ {1,2, ... ,m} \ I, we have the divided powrer algebra (I|^) and the two generalized Witt algebras W(I) and W(I|^). For any Lie algebra g, we can define a loop algebra associated with a divided algebra g. We can reduce the representation of to the representation of its graded Lie subalgebra and S_I|^ normalize each loop subalgebra of (I|^) W{I). This makes it possible to consider some induced representations related to W(I) and W(I|^). From this viewpoint we can reduce the representation of L to the lower-rank representation when X is semisimple: when x is semisimple,the irreducible generalized x - reduced representation of W(m;n) coincides with the induced representation for the corresponding irreducible representation of...