Documenta Math. 709 Irreducible Modules over the Virasoro Algebra

In this paper, we construct two different classes of Virasoro modules from twisting Harish-Chandra modules over the twisted Heisenberg-Virasoro algebra by an automorphism of the twisted Heisenberg-Virasoro algebra. Weight modules in the first class are some irreducible highest weight modules over the twisted HeisenbergVirasoro algebra. The non-weight modules in the first class are irreducible Whittaker modules over the Virasoro algebra. We obtain concrete bases for all irreducible Whittaker modules (instead of a quotient of modules). This generalizes known results on Whittaker modules. The second class of modules are non-weight modules which are not Whittaker modules. We determine the irreducibility and isomorphism classes of these modules. 2010 Mathematics Subject Classification: 17B10, 17B20, 17B65, 17B66, 17B68


Introduction
Throughout this paper, we will use C, C * , Z, Z + and N to denote the sets of complex numbers, nonzero complex numbers, integers, nonnegative integers and positive integers respectively.The theory of weight modules with finite-dimensional weight spaces over the Heisenberg algebra, the Virasoro algebra and the twisted Heisenberg-Virasoro algebra are fairly well developed.We refer the readers to [1], [4] [5], [11], [12], [13] and the references therein.For weight modules with infinite dimensional weight spaces, see [3], [7], [17].Recently Whittaker modules over those algebras were studied by many authors, see for example [2], [6], [10], [14], [16].Besides Whittaker modules, some new non-weight modules over the Virasoro algebra were just constructed in [8].We will use modules over the twisted Heisenberg-Virasoro algebra to study modules over the Virasoro algebra.Now we first recall the twisted Heisenberg-Virasoro algebra.The twisted Heisenberg-Virasoro algebra L is the universal central extension of the Lie algebra {f (t) d dt + g(t)|f, g ∈ C[t, t −1 ]} of differential operators of order at most one on the Laurent polynomial algebra C[t, t −1 ].More precisely, the twisted Heisenbeg-Virasoro algebra L is a Lie algebra over C with the basis {d n , t n , z 1 , z 2 , z 3 |n ∈ Z} and the Lie bracket given by The Lie algebra L has a Virasoro subalgebra Vir with basis {d i , z 1 |i ∈ Z}, and a Heisenberg subalgebra H with basis {t i , z 3 |i ∈ Z}.
Let σ be an endomorphism of L, and V be any weight module of L. We can make V into another L-module, by defining the new action of L on V as We will call the new module as the twisted module of V by σ, and denote it by V σ .To avoid any ambiguity, we will not omit the circ for the new action.The module V σ can be regarded as the Vir module by restriction to the Virasoro subalgebra.One important fact is that we can get a lot of new irreducible modules over Vir in this simple way, which include some new irreducible Whittaker modules.Since these modules are generally not weight modules, it is not trivial to determine isomorphism classes and irreducibility for these modules.
The paper is organized as follows.In section 2, we collect some known results for later use.In section 3, we construct our first class of Virasoro modules by twisting a highest weight L module (oscillator representation) with automorphisms of L, then we obtain some new irreducible Whittaker modules L ψm, ż1 , where m > 0, over the Virasoro algebra.This concrete realization allows us to give concrete bases for all irreducible Whittaker modules (not only as a quotient of modules).Our bases for irreducible Whittaker modules L ψm, ż1 with m = 1 generalize those results in [14] where it was required: ψ 1 (d 1 )ψ 1 (d 2 ) = 0, and those results in [16] where an explicit formula for the Whittaker vector was give only for ψ 1 (d 1 ) = 0 and ψ 1 (d 2 ) = 0 in terms of Jack symmetric polynomial.
In section 4, we construct our second class of Virasoro modules by twisting L modules of intermediate series with automorphisms of L. Then we determine the isomorphism classes and irreducibility of these Virasoro modules.

Preliminaries
In this section, we collect some notations and known facts for later use.For details, we refer the readers to [9], [12], [15], and the references therein.
Let us recall the definition of weight modules and highest weight modules over L.
It is well-known that L has a natural Z-gradation: deg d n =deg t n =n and deg and Then we have the triangular decomposition which we generally call the weight space of V corresponding to the weight (λ, λ H , c 1 , c 2 , c 3 ) ∈ C 5 .When t 0 , z 1 , z 2 , z 3 act as scalars on the whole space V , we shall simply write V λ instead of V (λ,λH ,c1,c2,c3) .An L-module V is called a weight module if V is the sum of all its weight spaces.
A weight L-module V is called a highest weight module with highest weight It is well known that, up to isomorphism, there exists a unique irreducible highest weight module For any a, b ∈ C, we have the Vir module A a,b , called the module of intermediate series, which has basis {t k |k ∈ Z} such that z 1 acts trivially and Let us summarize some well-known results for the modules of intermediate series.
We also need the following result from [15], and we will write it in a slightly different form for later use.
For any we have the σ = σ α,b ∈ Aut(L) defined as ) This can be verified directly, but one has to use the following formula It is clear that Documenta Mathematica 16 (2011) 709-721 Actually, B is isomorphic to the irreducible highest weight module V (0, 0, 1, 0, 1) over L as in [12].
For any homogenous polynomial u = x l1 i1 . . .x l k i k ∈ B, define deg(u) = j i j l j , and denote Then we have the weight space decomposition B = ⊕ i∈N B i , where B i has the weight −i.
we have made the Fock space B into an irreducible highest weight module B σ α,b .It is easy to verify (or from results in [1]) that: for any ) Proof.These follow from straightforward computations by using the formulas (2.3)-(2.5).
From (3.8) we know the following Lemma 3. The following set Proof.Recall that we have assumed that α = m ′ i=−m a i t i ∈ A\C[t] with m > 0 and a −m = 0. Let V be a nonzero Vir submodule of B σ α,b , and 0 = f ∈ V with lowest degree.Suppose that f ∈ C, and deg(f And deg(g Note that L ψm, ż1 is a Whittaker module with respect to the Whittaker pair (Vir, Vir ≥m +Cz 1 ), and w = 1 ⊗ v be a cyclic Whittaker vector in the sense of [2].
From Theorem 5 we know that L ψm, ż1 ∼ = B σ α,b .Using Theorem 4 we see that L ψm, ż1 is irreducible over Vir.Now we can give the main result in this section.
From Theorem 8 we know that the Whittaker module L ψ m−1/2 , ż1 with respect to the Whittaker pair (Vir[ 1 2 Z], Vir ≥m−1/2 +Cz 1 ) is irreducible with a basis: where -module which is a Whittaker module isomorphic to L ψm, ż1 with respect to the Whittaker pair (Vir[Z], Vir ≥m +Cz 1 ).
We want to show that W is irreducible as a Vir[Z]-module.To the contrary, we assume that W is not irreducible.Take a nonzero proper submodule V of W .Let V ′ be the span of the following subspaces Then V ′ is a proper subspace of L ψ m−1/2 , ż1 .Using PBW Theorem one can easily show that V ′ is a submodules of L ψ m−1/2 , ż1 over Vir[ 1 2 Z], which is a contradiction.Thus W is irreducible as a Vir[Z]-module.Therefore the Whittaker module L ψm, ż1 is irreducible over Vir We like to mention that the results in [14] and [16] are the special case of the above theorem with m = 1 and ψ 1 (d 1 )ψ 1 (d 2 ) = 0 or ψ 1 (d 1 ) = 0.
4 Irreducible modules over Vir with z 1 = 0 We can make A = C[t, t −1 ] into an L-module by defining d i = t i+1 d dt , t i acting as multiplication by t i , and z 1 , z 2 , z 3 acting as zero, i.e., The module on A is isomorphic to V (0, 0; 1) defined on Page 187 of [12] which is an irreducible module over L. We will use this module instead of the most general case V (a, b; F ), where a, b, F ∈ C, since we will essentially obtain isomorphic Virasoro modules.
In this section, the irreducibility and the isomorphism classes of such modules are completely determined.Note that if α ∈ C, then A α,b is simply a weight module of intermediate series in [12].
Proof.The statements in this Lemma are obvious.
Lemma 9.For any k ∈ Z, let Proof.For any k ∈ Z and t j ∈ A α,b , we compute Taking i = −1, 0, 1 respectively, we get (3) The Vir module A α,0 is irreducible if and only if α / ∈ Z. Now suppose that α 2 / ∈ C, then A α2,1 is a non-weight module with respect to d 0 .This forces A α1,0 to be a non-weight module with respect to d 0 .So α 1 / ∈ C. From Theorem 12 (2) and (4) we know that A α1,0 is irreducible while A α2,1 is not irreducible.So A α1,0 and A α2,1 cannot be isomorphic in this case.

is a basis of B σ α,b . Theorem 4 .
For any α ∈ A\C[t] and b ∈ C, the module B σ α,b is irreducible over Vir.
σ α,b by Lemma 3. Therefore B σ α,b is irreducible as a Vir module.For any m ∈ N, denote Vir ≥m = ⊕ i≥m Cd i .Let ψ m : Vir ≥m → C be any nonzero homomorphism of Lie algebras and ż1 ∈ C. Defined the one dimensional Vir ≥m +Cz 1 module Cv by d i v = ψ m (d i )v and z 1 v = ż1 v. Then we have the induced Vir module L ψm, ż1 = Ind U(Vir) U(Vir ≥m +Cz1) Cv.

Theorem 7 .
Suppose that ż1 ∈ C, m ∈ N, and ψ m : Vir ≥m → C is a Lie algebra homomorphism.Then the Whittaker module L ψm, ż1 over Vir is irreducible if and only if ψ m (d 2m ) = 0 or ψ m (d 2m−1 ) = 0. Proof."⇒": Suppose that ψ m (d 2m ) = 0, and ψ m (d 2m−1 ) = 0. Then it is staightforward to see that L ψm, ż1 has a proper submodule generated by d m−1 •1."⇐" From Theorem 8, we need only to consider the case where ψ m (d 2m ) = 0 and ψ m (d 2m−1 ) = 0. Let Vir[ 1 2 Z] be the Virasoro algebra with the basis {d k , z | k ∈ 1 2 Z} and subject to the relations: For any α ∈ A, b ∈ C, we have the L-module A α,b = A σ α,b .The action of L on A α,b is

Proof.( 1 )
For any 0 = f = r i=s b i t i ∈ A with b s , b r = 0, define deg(f ) = (s, r) and l(f ) = r − s.Suppose that α = n i=m a i t i ∈ A \ C, with deg α = (m, n).Case 1. m < 0 < n.It is easy to see that l(f ) ≥ n − m for all f ∈ E α , hence E α = A in this case.Case 2. m ≥ 0 or n ≤ 0. Without loss of generality, we may assume that m ≥ 0. If m = 0 then n > 0 since α / ∈ C; if m > 0 then n ≥ m > 0. If a 0 / ∈ Z or m > 0, then it is easy to check that l(f ) ≥ 1 for all 0 = f ∈ E α .So E α = A. If a 0 ∈ Z and m = 0, then n > 0 and a 0 = 0.It is not hard to see that t −a0 / ∈ E α .Theorem 12. Let α ∈ A, b ∈ C. If b / ∈ {0, 1}, then A α,bis irreducible as a Vir module with action defined as in (4.1).(2)The Vir module A α,1 is irreducible if and only if α∈ C\Z.If α / ∈ C\Z, then d 0 • A α,1 = ⊕ i∈Z C(α + i)t iis the unique irreducible Vir submodule of A α,b , and Vir acts on A α,1 /(d 0 • A α,1 ) as zero.