Restrictions On Sl_d+1 Of A Class Of Witt Algebra Modules

Let d be a positive integer,the Witt algebra Wd is the derivation algebra of the LaurenL polynomial ring Ad=C[x1±1,x2±1,…,xd±1]in commuting variables x1,x2,…,xd.The representation theory of Witt algebra has been widely studied by many scholars.In 1986,Guangyu Shen defined a class of weight module over Wd.For any ??Cd,b?C and an sId-module V on which the identity matrix acts as the scalar b,denote Fb?(V)=V(?)Ad,then Fb?(V)becomes a natural Wd-module.The functor Fb? is also called the Larsson functor.These modules were also studied by Larsson and Rao.Rencently,some scholars generalized this construction and defined a class of Wd-module F(P;V),where P is an admissible Wd-module,and V is a gld-module.Lie algebra Wd has a natural subalgebra sldd+1,so the Wd-module F(P;V)can be viewed as an sld+1-module naturally.In this paper,we mainly consider the sld+1-module F(?;a,b)constructed by taking P = ?(?)and V=V(a,b).The main results are as follows:(1).We determine the irreducibility of the sld+1-module F(?;a,b);(2).We construct the proper submodules of the sld+1-module F(A;a,b)when it is reducible.