Let G be a rank n additive subgroup of C and Vir [G] the corresponding Virasoro algebra of rank n. In the present paper, irreducible weight modules with finite dimensional weight spaces over Vir[G] are completely determined. There are two different classes of them. One class consists of simple modules of intermediate series whose weight spaces are all 1-dimensional. The other is constructed by using intermediate series modules over a Virasoro subalgebra of rank n − 1. The classification of such modules over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a completely different approach.2000 Mathematics Subject Classification. 17B10, 17B65, 17B68.

Abstract. We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra sl 2 of type A(1) 1 with noncritical level. These modules can become simple Whittaker modules over sl 2 = sl 2 + Cd with the same Whittaker function and central charge. We have to modulo a central character for sl 2 to obtain simple degenerate Whittaker sl 2 -modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker sl 2 -module by a submodule generated by a scalar action of central elements of the vertex algebra V −2 (sl 2 ) is simple as sl 2 -module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for sl 2 . Quite surprisingly, with the same Whittaker function some simple degenerate sl 2 Whittaker modules at the critical level have semisimple action of d and others have free action of d. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker modules for sl 2 at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for degenerate Whittaker modules for sl 2 at noncritical level.

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For a simple module M over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras a r ) and any α ∈ C, a class of weight modules N (M, α) over the Virasoro algebra are constructed. The necessary and sufficient condition for N (M, α) to be simple is obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules Γ(α 1 , α 2 , λ 1 , λ 2 ) are defined on the polynomial algebra C[x] ⊗ C[t, t −1 ] for any α 1 , α 2 , λ 1 λ 2 ∈ C with λ 1 or λ 2 nonzero. By twisting the weight modules N (M, α) we also obtain nonweight simple Virasoro modules N (M, β) for any β ∈ C[t, t −1 ].