Irreducible representations of nongraded Witt type Lie algebras

From his classification of quadratic conformal algebras corresponding to certain Hamiltonian pairs in integrable systems, Xu gave an explicit construction of seven types of nongraded infinite-dimensional simple Lie algebra related to a finite set of locally-finite derivations on certain commutative associative algebras. In this paper, we construct a family of irreducible generalized weight modules for the nongraded Witt type Lie algebras based on finite-dimensional irreducible modules of general linear Lie algebras. Our results are nongraded generalizations of Raoʹs work on irreducible representations of the derivation Lie algebra of the algebra of Laurent polynomials in several variables.