Irreducible Modules over the Virasoro Algebra

Rencai Lü, Xiangqian Guo, and Kaiming Zhao

Received: April 11, 2011 Revised: September 22, 2011

Communicated by Edward Frenkel

Abstract. 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 Heisenberg-Virasoro 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 Keywords and Phrases: Virasoro algebra, Whittaker module, irreducible module.

1 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)\frac{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

[d_{n}, d_{m}] = (m − n)d_{n+m} + δ_{n,−m} \frac{n^{3} − n}{12}z_{1},  (1.1)
[d_{n}, t^{m}] = mt^{m+n} + δ_{n,−m}(n^{2} + n)z_{2},  (1.2)
[t^{n}, t^{m}] = nδ_{n,−m}z_{3},  (1.3)
[𝕃, z_{1}] = [𝕃, z_{2}] = [𝕃, z_{3}] = 0.  (1.4)

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

x ∘ v = σ(x)v, ∀x ∈ 𝕃, v ∈ V.  (1.5)

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_{ψ_{m},ż_{1}}, 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_{ψ_{m},ż_{1}} 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.

2 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 z_{i} = 0 for i = 1, 2, 3. Set

𝕃_{+} = ∑_{n>0}(ℂd_{n} + ℂt^{n}), 𝕃_{−} = ∑_{n<0}(ℂd_{n} + ℂt^{n}),

and

𝕃_{0} = ℂd_{0} + ℂt^{0} + ℂz_{1} + ℂz_{2} + ℂz_{3}.

Then we have the triangular decomposition 𝕃 = 𝕃_{+} ⊕ 𝕃_{0} ⊕ 𝕃_{−}.

For any 𝕃-module V and (λ, λ_{H}, c_{1}, c_{2}, c_{3}) ∈ ℂ^{5}, set

V_{(λ,λ_{H},c_{1},c_{2},c_{3})} =

{v ∈ V | d_{0}v = λv, t^{0}v = λ_{H}v, and z_{i}v = c_{i}v for i = 1, 2, 3},

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,λ_{H},c_{1},c_{2},c_{3})}. 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 (λ, λ_{H}, c_{1}, c_{2}, c_{3}) ∈ ℂ^{5}, if there exists a nonzero weight vector v ∈ V_{(λ,λ_{H},c_{1},c_{2},c_{3})} such that

1) V is generated by v as an 𝕃-module;

2) 𝕃_{+}v = 0.

It is well known that, up to isomorphism, there exists a unique irreducible highest weight module V (λ, λ_{H}, c_{1}, c_{2}, c_{3}) over 𝕃 with the highest weight (λ, λ_{H}, c_{1}, c_{2}, c_{3}) ∈ ℂ^{5}. 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

d_{i}t^{k} = (a + k + bi)t^{i+k}, ∀i, k ∈ ℤ.  (2.1)

It is well-known that A_{a,b} is irreducible if and only if a ∉ ℤ or b ∉ {0, 1}. We put A′_{a,b} = A_{a,b} if A_{a,b} is irreducible; otherwise A′_{a,b} be the unique nontrivial irreducible sub-quotient of A_{a,b}.

Let us summarize some well-known results for the modules of intermediate series.

Theorem 1. Let a, b, a_{1}, b_{1} ∈ ℂ.

(1) If b ∉ {0, 1}, then A_{a,b} ≅ A_{a_{1},b_{1}} if and only if b = b_{1} and a − a_{1} ∈ ℤ;

(2) If b ∈ {0, 1} and a ∉ ℤ, then we have A_{a,b} ≅ A_{a_{1},b_{1}} if and only if b_{1} ∈ {0, 1} and a − a_{1} ∈ ℤ;

(3) If b ∈ {0, 1} and a ∈ ℤ, then we have A_{a,b} ≅ A_{a_{1},b_{1}} if and only if b_{1} = b and a_{1} ∈ ℤ;

(4) If a ∈ ℤ, then A′(a, 0) ≅ A′(a, 1) ≅ A′(0, 0).

We also need the following result from [15], and we will write it in a slightly different form for later use. For any

α = ∑_{i∈ℤ} a_{i}t^{i} ∈ ℂ[t, t^{−1}], b ∈ ℂ,  (2.2)

we have the σ = σ_{α,b} ∈ Aut(𝕃) defined as

σ(d_{n}) = d_{n} + t^{n}(α + nb) − (n + 1)a_{−n}z_{2}
−(\frac{∑_{i} a_{i}a_{−n−i}}{2} + a_{−n}nb)z_{3} + δ_{n,0}b(z_{2} + \frac{b}{2}z_{3})  (2.3)
σ(t^{n}) = t^{n} + δ_{n,0}bz_{3} − a_{−n}z_{3}, σ(z_{1}) = z_{1} − 24bz_{2} − 12b^{2}z_{3},  (2.4)
σ(z_{2}) = z_{2} + bz_{3}, σ(z_{3}) = z_{3}.  (2.5)

This can be verified directly, but one has to use the following formula

∑_{i}(m + i)a_{i}a_{−m−n−i}z_{3} = \frac{m − n}{2} ∑_{i} a_{i}a_{−m−n−i}z_{3}.

It is clear that

σ_{α,b}σ_{α_{1},b_{1}} = σ_{α+α_{1},b+b_{1}}, ∀ α, α_{1} ∈ ℂ[t, t^{−1}], b, b_{1} ∈ ℂ.  (2.6)

3 Irreducible Whittaker modules over the Virasoro algebra

Let us recall the oscillator representation of the Heisenberg-Virasoro algebra 𝕃 on the Fock space B = ℂ[x_{1}, x_{2}, …, x_{n}, …]. The action of 𝕃 is defined as (see Prop.2.3, Lemma 2.2 in [9])

t^{n} = \frac{∂}{∂x_{n}} , t^{−n} = nx_{n}, ∀ n ∈ ℕ,  (3.1)

t^{0} = 0, z_{3} = 1; z_{2} = 0, z_{1} = 1,  (3.2)
d_{k} = −\frac{1}{2} ∑_{i∈ℤ} : t^{−i} · t^{i+k} :, ∀ k ∈ ℤ.  (3.3)

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_{i_{1}}^{l_{1}} … x_{i_{k}}^{l_{k}} ∈ B, define deg(u) = ∑_{j} i_{j}l_{j}, and denote B_{i} = span{u = x_{i_{1}}^{l_{1}} … x_{i_{k}}^{l_{k}} ∈ B| deg(u) = i} for all i ∈ ℕ. Then we have the weight space decomposition B = ⊕_{i∈ℕ}B_{i}, where B_{i} has the weight −i. For any 0 ≠ f = ∑_{i=0}^{n} f_{i} with f_{i} ∈ B_{i} and f_{n} ≠ 0, denote deg(f) = n, and htm(f) = f_{n}. When α ∈ ℂ[t], 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 b ∈ ℂ, α = ∑_{i=0}^{n} a_{i}t^{i} ∈ ℂ[t], we have B^{σ_{α,b}} ≅ V (\frac{b^{2}−a_{0}^{2}}{2} , b − a_{0}, 1 − 12b^{2}, b, 1).

From now on in this section we fix b ∈ ℂ and α = ∑_{i=−m}^{m′} a_{i}t^{i} ∈ A\ℂ[t] with m > 0 and a_{−m} ≠ 0.

Lemma 2. In B^{σ_{α,b}} we have

d_{n} ∘ 1 = 0, ∀ n ≥ 2m + 1,  (3.4)
d_{n} ∘ 1 = −(\frac{∑_{i} a_{i}a_{−n−i}}{2} + a_{−n}nb), ∀ m ≤ n ≤ 2m,  (3.5)
z_{1} ∘ 1 = 1 − 12b^{2},  (3.6)
htm(d_{n} ∘ (x_{i_{1}}^{k_{1}} … x_{i_{l}}^{k_{l}})) = (m − n)a_{−m}x_{m−n}x_{i_{1}}^{k_{1}} … x_{i_{l}}^{k_{l}}, ∀ n < m.  (3.7)
htm((d_{i_{1}}d_{i_{2}} … d_{i_{k}}) ∘ 1) = ax_{m−i_{1}}x_{m−i_{2}} … x_{m−i_{k}}  (3.8)

for all i_{1}, i_{2}, …, i_{k} < m, where a = ∏_{j=1}^{k}(a_{−m}(m − i_{j})) ∈ ℂ^{∗}.

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

B = {(d_{i_{1}}d_{i_{2}} … d_{i_{k}}) ∘ 1|i_{1} ≤ i_{2} ≤ … ≤ i_{k} < m}

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.

Proof. Recall that we have assumed that α = ∑_{i=−m}^{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) = n. Say f_{n} = htm(f) with \frac{∂}{∂x_{i_{1}}} (f_{n}) ≠ 0. Then we have

g = (d_{i_{1}+m} + t^{i_{1}+m}(α + (i_{1} + m)b) · f = d_{i_{1}+m} ∘ f + af ∈ V,

where a = ^{\frac{∑_{i} a_{i}a_{−m−i_{1}−i}}{2}} ∈ ℂ. It is straightforward to compute that

htm(g) = htm(a_{−m}t^{i_{1}} · f) = a_{−m} \frac{∂}{∂x_{i_{1}}} (f_{n}) ≠ 0.

And deg(g) = n − i_{1} < n, which contradicts the choice of f. So 1 ∈ V. Thus V = B^{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 ż_{1} ∈ ℂ. Defined the one dimensional Vir_{≥m} +ℂz_{1} module ℂv by d_{i}v = ψ_{m}(d_{i})v and z_{1}v = ż_{1}v. Then we have the induced Vir module

L_{ψ_{m},ż_{1}} = Ind_{U(Vir_{≥m} +ℂz_{1})}^{U(Vir)}ℂv.

Note that L_{ψ_{m},ż_{1}} 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 the PBW theorem, L_{ψ_{m},ż_{1}} has a basis

{(d_{i_{1}}d_{i_{2}} … d_{i_{k}})w|i_{1} ≤ i_{2} ≤ … ≤ i_{k} < m}.  (3.9)

For m = 1, the irreducibility of L_{ψ_{1},ż_{1}} with ψ_{1}(d_{1})ψ_{1}(d_{2}) ≠ 0 was studied in [14] (see Proposition 4.8 and 6.1 in [14]), and an explicit formula for the Whittaker vector was give in [16] for ψ_{1}(d_{1}) ≠ 0 and ψ_{1}(d_{2}) = 0 in terms of Jack symmetric polynomial.

Theorem 5. Let b ∈ ℂ and α = ∑_{i=−m}^{m′} a_{i}t^{i} ∈ A with a_{−m} ≠ 0 and m > 0. Then B^{σ_{α,b}} ≅ L_{ψ_{m},1−12b^{2}} with ψ_{m}(d_{n}) = −(^{\frac{∑_{i} a_{i}a_{−n−i}}{2}} +a_{−n}nb), for all n ≥ m.

Proof. It follows by Lemma 2, Lemma 3 and (3.9).

Theorem 6. Suppose ż_{1} ∈ ℂ and ψ_{m}(d_{2m}) ≠ 0. Then the Whittaker module L_{ψ_{m},ż_{1}} over Vir is irreducible.

Proof. Denote ψ_{m}(d_{n}) = ḋ_{n} for n = m, …, 2m. It is easy to see that there exist b ∈ ℂ and some α(t) = ∑_{i=−m}^{0} a_{i}t^{i} ∈ ℂ[t^{−1}] with a_{−m} ≠ 0 satisfying the following equations:

\{ ż_{1} = 1 − 12b^{2}, ; ḋ_{2m} = −\frac{a_{−m}^{2}}{2} , ; ḋ_{2m−1} = −(a_{−m}a_{−m+1} + a_{−2m+1}(2m − 1)b), ; ⋮ ; ḋ_{n} = −(\frac{∑_{i} a_{i}a_{−n−i}}{2} + a_{−n}nb), ; ⋮ ; ḋ_{m} = −(\frac{∑_{i} a_{i}a_{−m−i}}{2} + a_{−m}mb).

From Theorem 5 we know that L_{ψ_{m},ż_{1}} ≅ B^{σ_{α,b}}. Using Theorem 4 we see that L_{ψ_{m},ż_{1}} is irreducible over Vir.

Now we can give the main result in this section.

Theorem 7. Suppose that ż_{1} ∈ ℂ, m ∈ ℕ, and ψ_{m} : Vir_{≥m} → ℂ is a Lie algebra homomorphism. Then the Whittaker module L_{ψ_{m},ż_{1}} 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_{ψ_{m},ż_{1}} 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[\frac{1}{2}ℤ] be the Virasoro algebra with the basis \{d_{k}, z | k ∈ \frac{1}{2}ℤ\} and subject to the relations:

[d_{n}, d_{m}] = (m − n)d_{n+m} + δ_{n,−m} \frac{n^{3} − n}{12}z_{1}, ∀ m, n ∈ \frac{1}{2}ℤ.

Then Vir[ℤ] is a subalgebra of Vir[\frac{1}{2}ℤ]. Now we define ψ_{m−1/2} on Vir[\frac{1}{2}ℤ] as

ψ_{m−1/2}(d_{k}) = ψ_{m}(d_{k}), ψ_{m−1/2}(d_{k−1/2}) = 0, ∀ k ∈ ℕ.

From Theorem 8 we know that the Whittaker module L_{ψ_{m−1/2},ż_{1}} with respect to the Whittaker pair (Vir[\frac{1}{2}ℤ], Vir_{≥m−1/2} +ℂz_{1}) is irreducible with a basis:

(d_{k_{1}}d_{k_{2}} ⋯ d_{k_{r}})(d_{p_{1}}d_{p_{2}} ⋯ d_{p_{t}})1

where k_{1}, k_{2}, ⋯, k_{r} ∈ 1/2 + ℤ with k_{1} ≤ k_{2} ≤ ⋯ ≤ k_{r} ≤ m − 1, and p_{1}, p_{2}, ⋯, p_{t} ∈ ℤ with p_{1} ≤ p_{2} ≤ ⋯ ≤ p_{t} ≤ m − 1. Clearly,

W = span{d_{p_{1}}d_{p_{2}} ⋯ d_{p_{t}}1 | p_{1}, p_{2}, ⋯, p_{t} ∈ ℤ with p_{1} ≤ p_{2} ≤ ⋯ ≤ p_{t} ≤ m − 1}

is a Vir[ℤ]-module which is a Whittaker module isomorphic to L_{ψ_{m},ż_{1}} 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

(d_{k_{1}}d_{k_{2}} ⋯ d_{k_{r}})V; k_{1}, k_{2}, ⋯, k_{r} ∈ 1/2 + ℤ with k_{1} ≤ k_{2} ≤ ⋯ ≤ k_{r} ≤ m − 1.

Then V′ is a proper subspace of L_{ψ_{m−1/2},ż_{1}}. Using PBW Theorem one can easily show that V′ is a submodules of L_{ψ_{m−1/2},ż_{1}} over Vir[\frac{1}{2}ℤ], which is a contradiction. Thus W is irreducible as a Vir[ℤ]-module. Therefore the Whittaker module L_{ψ_{m},ż_{1}} is irreducible over Vir[ℤ] if ψ_{m}(d_{2m−1}) ≠ 0.

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} \frac{d}{dt}, t^{i} acting as multiplication by t^{i}, and z_{1}, z_{2}, z_{3} acting as zero, i.e.,

d_{i} · t^{j} = jt^{i+j}, t^{i} · t^{j} = t^{i+j}, z_{k} · A = 0, ∀ k = 1, 2, 3.

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. 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

d_{n} ∘ t^{i} = (α + i + nb)t^{n+i}, ∀ i, n ∈ ℤ,  (4.1)
t^{j} ∘ t^{i} = t^{i+j}, z_{k} ∘ A = 0, ∀ i, j ∈ ℤ, k = 1, 2, 3.  (4.2)

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].

Lemma 8. (1) Let V be an irreducible 𝕃 module, then V^{σ_{α,b}} is irreducible for any α ∈ A, b ∈ ℂ.

(2) A_{α,b} is irreducible as an 𝕃 module for any α ∈ A, b ∈ ℂ.

Proof. The statements in this Lemma are obvious.

Lemma 9. For any k ∈ ℤ, let

w_{k} = −\frac{1}{2}d_{k−1}d_{1} − \frac{1}{2}d_{k+1}d_{−1} + d_{k}d_{0} ∈ U(Vir).  (4.3)

Then w_{k} ∘ g = b(b − 1)t^{k}g, for all k ∈ ℤ and g ∈ A_{α,b}.

Proof. For any k ∈ ℤ and t^{j} ∈ A_{α,b}, we compute

(d_{k−i}d_{i}) ∘ t^{j} = d_{k−i} ∘ (α + j + ib)t^{i+j} = (α + j + ib)(α + i + j + (k − i)b)t^{k+j}.

Taking i = −1, 0, 1 respectively, we get

w_{k} ∘ t^{j} = [−\frac{1}{2}(α + j + b)(α + 1 + j + (k − 1)b)

−\frac{1}{2}(α+j −b)(α−1+j +(k +1)b)+(α+j)(α+j +kb)]t^{k+j}

= b(b − 1)t^{k+j}.

Thus w_{k} ∘ g = b(b − 1)t^{k}g for all k ∈ ℤ and g ∈ A_{α,b}.

Corollary 10. Suppose that b ∉ {0, 1}, 0 ≠ g ∈ A_{α,b}. Then span_{ℂ}{w_{i} ∘ g|i ∈ ℤ} = A_{α,b} if and only if g = ct^{i} for some c ∈ ℂ^{∗} and i ∈ ℤ.

Proof. From Lemma 9, span_{ℂ}{w_{i} ∘ g|i ∈ ℤ} = Ag. It is clear that Ag = A if and only if g = ct^{i} for some c ∈ ℂ^{∗} and i ∈ ℤ.

Lemma 11. If α ∈ A \ ℂ, then E_{α} = span_{ℂ}{αt^{i} + it^{i}|i ∈ ℤ} ≠ A.

Proof. For any 0 ≠ f = ∑_{i=s}^{r} b_{i}t^{i} ∈ A with b_{s}, b_{r} ≠ 0, define deg(f) = (s, r) and l(f) = r − s. Suppose that

α = ∑_{i=m}^{n} 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^{−a_{0}} ∉ E_{α}.

Theorem 12. Let α ∈ A, b ∈ ℂ.

(1) If b ∉ {0, 1}, then A_{α,b} is 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^{i}

is the unique irreducible Vir submodule of A_{α,b}, and Vir acts on A_{α,1}/(d_{0}∘ A_{α,1}) as zero.

(3) The Vir module A_{α,0} is irreducible if and only if α ∉ ℤ.

(4) We have d_{0} ∘ A_{α,1} ≅ A_{α,0} as Vir-module if α ∉ ℤ.

Proof. (1). Suppose that b ≠ 0, 1. Let M be a nonzero Vir submodule of A_{α,b}. Then by lemma 9, we have t^{i} · M ⊂ M for all i ∈ ℤ. Hence M is also an 𝕃 submodule of A_{α,b}. Thus M = A_{α,b} by lemma 8. So A_{α,b} is irreducible as Vir module in this case.

(2). Note that d_{j} ∘ g = d_{0} ∘ (t^{j}g) for all g ∈ A_{α,1}, i ∈ ℤ. So d_{0} ∘ A_{α,1} is a Vir submodule of A_{α,1}, and Vir acts trivially on A_{α,1}/(d_{0} ∘ A_{α,1}). For any nonzero Vir submodule M of A_{α,1}. Let AM = span{t^{i}g|i ∈ ℤ, g ∈ M}. Since d_{0} ∘ t^{i}g = d_{i} ∘ g ∈ M for all g ∈ M, then

d_{0} ∘ AM ⊂ M.  (4.4)

Noting that d_{j} ∘ (t^{i}g) = d_{i+j} ∘ g we see that AM is a Vir submodule. Then AM is also an 𝕃 module, and AM = A by lemma 8. Combining with (4.4), we have d_{0} ∘ A_{α,1} ⊂ M, i.e., d_{0} ∘ A_{α,1} is the unique minimum nonzero submodule of A_{α,1}. Note that d_{0} ∘ A_{α,1} = span_{ℂ}{(α + i)t^{i}|i ∈ ℤ}. Using Lemma 11 we see that A_{α,1} is not irreducible if α ∉ ℂ. For α ∈ ℂ it is well-known that A_{α,1} is not irreducible if and only if α ∈ ℤ. This proves (2).

(3) and (4). It is straightforward to check that η : A_{α,0} → d_{0} ∘ A_{α,1} with η(g) = d_{0} ∘ g is a Vir-module epimorphism. From (2), we know that A_{α,0} is irreducible if and only if η is injective. Note that if d_{0} ∘ g = 0 for some 0 ≠ g ∈ A_{α,1}, then t\frac{d}{dt}(g) + αg = 0. By comparing the terms of highest and lowest degree in t respectively, we have α ∈ ℂ. Using well-known results on A_{α,0} for α ∈ ℂ, we see that the statements in (3) and (4) are true. So we have proved the theorem.

Lemma 13. If α_{1} − α_{2} ∈ ℤ, then A_{α_{1},b} ≅ A_{α_{2},b} as Vir-modules.

Proof. Suppose that α_{1} = α_{2} + k for some k ∈ ℤ. Then it is straightforward to check that η : A_{α_{1},b} → A_{α_{2},b} with η(t^{i}) = t^{i+k} is a Vir-module isomorphism.

Lemma 14. For any α ∉ ℤ there exist finitely many nonzero v_{α}^{(i)} ∈ U(Vir) for i ∈ ℤ such that for any k ∈ ℤ, the element

u_{α,k} = ∑_{i} d_{i+k}v_{α}^{(i)} ∈ U(Vir)  (4.5)

satisfies that u_{α,k} ∘ g = t^{k}u_{α,0} ∘ g in A_{α,0} for all g ∈ A_{α,0}.

Proof. Note that A_{α,0} is irreducible as a Vir module. Since d_{0} ∘ 1 = α ≠ 0, there exists u_{α} ∈ U(Vir) such that u_{α} ∘ d_{0} ∘ 1 = 1. Let u_{α,0} = u_{α} · d_{0} ∈ U(Vir). Then we can write u_{α,0} as in (4.5) with k = 0, and we have the definition for u_{α,k}. Note that d_{k+i} ∘ g = t^{k}(d_{i} ∘ g) for all g ∈ A_{α,0} and k ∈ ℤ. Then we can easily verify that u_{α,k} ∘ g = t^{k}u_{α,0} ∘ g in A_{α,0} for all g ∈ A_{α,0}.

Theorem 15. Let α_{1}, α_{2} ∈ A, b_{1}, b_{2} ∈ ℂ.

(1) If b_{1} ∉ {0, 1}, then A_{α_{1},b_{1}} ≅ A_{α_{2},b_{2}} as Vir modules if and only if α_{1}−α_{2} ∈ ℤ and b_{1} = b_{2}.

(2) If b ∈ {0, 1}, then A_{α_{1},b} ≅ A_{α_{2},b} as Vir modules if and only if α_{1}−α_{2} ∈ ℤ.

(3) A_{α_{1},0} ≅ A_{α_{2},1} as Vir modules if and only if α_{1} − α_{2} ∈ ℤ and α_{1} ∈ ℂ\ℤ.

Proof. From Theorem 1, Theorem 12, and Lemma 13, we see that all the sufficient conditions in (1)-(3) are satisfied. So we need only to prove the necessity of the conditions in (1)-(3).

(1). From Lemma 9, we have b_{1}(b_{1}−1) = b_{2}(b_{2}−1) ≠ 0. So b_{2} ∉ {0, 1}. Suppose that σ : A_{α_{1},b_{1}} → A_{α_{2},b_{2}} is a Vir module isomorphism. By Lemma 9 again, we know that {w_{k} ∘ 1 | k ∈ ℤ} is a basis for A_{α_{1},b_{1}}. Then {w_{k} ∘ σ(1) | k ∈ ℤ} is a basis for A_{α_{2},b_{2}}. By Corollary 10, we know that σ(1) = ct^{i_{0}} for some c ∈ ℂ^{∗} and i_{0} ∈ ℤ. Computing σ(d_{j} ∘1) = d_{j} ∘σ(1), we have (i_{0} +α_{2} −α_{1})+(b_{2} −b_{1})j = 0, for all j ∈ ℤ. Thus α_{1} = α_{2} + i_{0} and b_{1} = b_{2}.

(2). Case 1. b = 0. If α_{1} ∈ ℤ, then we have α_{2} ∈ ℤ by Theorem 12 (3). Now suppose that α_{1}, α_{2} ∉ ℤ, and that σ : A_{α_{1},0} → A_{α_{2},0} is a Vir-module isomorphism. Take u_{α_{1},k} as in Lemma 14. Then u_{α_{1},k} ∘ 1 = t^{k}, for all k ∈ ℤ. Note that

σ(t^{k}) = σ(u_{α_{1},k} ∘ 1) = u_{α_{1},k} ∘ σ(1)
= t^{k}(u_{α_{1},0} ∘ σ(1)) = t^{k}σ(u_{α_{1},0} ∘ 1) = t^{k}σ(1) ∈ A_{α_{2},0}.

Then

A_{α_{2},0} = σ(A_{α_{1},0}) = span_{ℂ}{σ(u_{α_{1},k} ∘ 1)|k ∈ ℤ}
= span_{ℂ}{t^{k}σ(1)|k ∈ ℤ} = Aσ(1).

So σ(1) = ct^{i_{0}} for some c ∈ ℂ^{∗}, i_{0} ∈ ℤ, and

σ(t^{i}) = ct^{i+i_{0}}, ∀ i ∈ ℤ.

By a similar computation as in the last step in (1), we deduce that α_{1} = α_{2} +i_{0} and the result follows in this case. Case 2. b = 1. Suppose that A_{α_{1},1} ≅ A_{α_{2},1} as Vir modules. If α_{1} ∈ ℤ, then A_{α_{1},1} is a weight module with respect to d_{0}. This forces A_{α_{2},1} to be a weight module with respect to d_{0}. So α_{2} ∈ ℂ. Theorem 14 (2) ensures that α_{1} − α_{2} ∈ ℤ. Now suppose that α_{1}, α_{2} ∉ ℤ. From Theorem 14 (4) we know that A_{α_{1},0} ≅ d_{0} ∘ A_{α_{1},1} ≅ d_{0} ∘ A_{α_{2},1} ≅ A_{α_{2},0} as irreducible Vir modules. Thus α_{1} − α_{2} ∈ ℤ by Case 1.

(3). If α_{2} ∈ ℂ, then A_{α_{2},1} is a weight module with respect to d_{0}. This forces A_{α_{1},0} to be a weight module with respect to d_{0}. So α_{1} ∈ ℂ. From Theorem 1 we know that α_{1} − α_{2} ∈ ℤ and α_{1} ∈ ℂ\ℤ.

Now suppose that α_{2} ∉ ℂ, then A_{α_{2},1} is a non-weight module with respect to d_{0}. This forces A_{α_{1},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_{α_{1},0} is irreducible while A_{α_{2},1} is not irreducible. So A_{α_{1},0} and A_{α_{2},1} cannot be isomorphic in this case.

Acknowledgement. We are grateful to the referee for giving nice suggestions. X.G. is partially supported by NSF of China (11101380 and 11026155), R.L. is is partially supported by NSF of China, and K.Z. is partially supported by NSERC and NSF of China.

References

[1] E. Arbarello, C. De Concini, V. G. Kac and C. Procesi, Moduli Spaces of Curves and Representation Theory, Commun. Math. Phys., 117 (1988), 1-36.

[2] P. Batra and V. Mazorchuk, Blocks and modules for Whittaker pairs, J. Pure Appl. Algebra, 215 (2011), no.7, 1552-1568.

[3] V. Bekkert, G. Benkart and V. Futorny, Weight modules for Weyl algebras, Kac-Moody Lie algebras and related topics, Contemp. Math., vol.343, 2004, 17-42.

[4] Y. Billig, Representations of the twisted Heisenberg-Virasoro algebra at level zero, Canad. Math. Bull., 46 (2003), no.4, 529-533.

[5] V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85(2) (1986), 317-335.

[6] K. Christodoulopoulou, Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras, J. Algebra, 320 (2008), 2871-2890.

[7] C. Conley and C. Martin, A family of irreducible representations of the Witt Lie algebra with infinite-dimensional weight spaces, Compos. Math., 128(2)(2001), 153-175.

[8] X. Guo, R. Lu and K. Zhao, Fraction modules of the Virasoro algebra, preprint, 2010.

[9] V. Kac and A. Raina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, World Sci., Singapore, 1987.

[10] D. Liu, Y. Wu and L. Zhu, Whittaker modules for the twisted Heisenberg-Virasoro algebra, J. Math. Phys., 51(1) (2010), 023524, 12pp.

[11] D. Liu and C. Jiang, Harish-Chandra modules over the twisted Heisenberg-Virasoro algebra, J. Math. Phys, 49(1) (2008), 012901, 13pp.

[12] R. Lu and K. Zhao, Classification of irreducible weight modules over the twisted Heisenberg-Virasoro algebras, Comm. Contemp. Math., Vol. 12, No.2 (2010) 183-205.

[13] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math., 107 (1992), no. 2, 225-234.

[14] M. Ondrus and E. Wiesner, Whittaker Modules for the Virasoro Algebra, J. Algebra Appl., 8(2009), no.3, 363-377.

[15] R. Shen and C. Jiang, The Derivation algebra and automorphism group of the twisted Heisenberg-Virasoro algebra, Commun. Alg., 34 (2006), 2547-2558.

[16] S. Yanagida, Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial, J. Algebra, 333 (2011), 273-294.

[17] H. Zhang, A class of representations over the Virasoro algebra, J. Algebra, 190 (1997), 1-10.

R. Lü Department of Mathematics Soochow University Suzhou 215006, Jiangsu P. R. China rlu@suda.edu.cn X. Gu Department of Mathematics Zhengzhou University Zhengzhou 450001, Henan P. R. China guoxq@amss.ac.cn

K. Zhao Department of Mathematics Wilfrid Laurier University Waterloo, ON, Canada N2L 3C5 and College of Mathematics and Information Science Hebei Normal (Teachers) University Shijiazhuang Hebei, 050016 P. R. China. kzhao@wlu.ca 722

Documenta Mathematica 16 (2011)