Schur duality for the Cartan type Lie algebra

. We decompose tensor products of the defining representation of a Cartan type Lie algebra W (n) in the case where the number of tensoring does not exceed the rank of the Lie algebra. As a result, we get a kind of Schur duality between W (n) and a finite dimensional non-semisimple algebra, which is the semi-group ring of the transformation semigroup Tm . Introduction Cartan type Lie algebras are Lie subalgebras of algebraic vector fields on a flat affine space F n , where F is a field of characteristic zero. They are Z-graded, simple Lie algebras with polynomial growth. By the result of Kac and Mathieu, Lie algebras with such properties are known to be (1) finite dimensional simple Lie algebras; (2) their loop algebras; (3) Witt algebra; and (4) Cartan type Lie algebras (see [11]). Irreducible representations of a Cartan type Lie algebra g were studied extensively by Rudakov ([15, 16]) and Kostrikin ([9]) in 1970's. If an irreducible representation admits a weight decomposition wit..