Journal of Lie Theory Volume 9 (1999) 233–248 ◯^{C}1999 Heldermann Verlag

Schur duality for the Cartan type Lie algebra Wn

Kyo Nishiyama^{∗}

Communicated by G. Olshanski

Abstract. 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 𝔗_{m}.

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 with respect to Euler’s degree operators, then it is a lowest weight module or its dual except the only one case 𝔤 = W_{1}. Therefore, the description of the irreducible representations are quite easy. However, it is rather difficult to do analysis on them because representations of W_{n} are not semisimple.

In the previous paper [12] (see also [13], [14]), we found an interesting phenomenon on the tensor product representations of 𝔤 = W_{n}, which is one of the four series of Cartan type Lie algebras. Since, by definition, W_{n} is a Lie algebra of all the derivations on the polynomial ring P [z_{1}, ⋯, z_{n}] of n-variables (see Section 1.), W_{n} acts naturally on P = P [z_{1}, ⋯, z_{n}]. Form the m-fold tensor product ⊗^{m}P. Then the full commutant algebra of W_{n} in ⊗^{m}P becomes a finite dimensional algebra. So we expect simultaneous decomposition of ⊗^{m}P as a module of W_{n} and its commutant algebra.

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag

∗ Partially supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan

To be more precise, let us assume that m ≤ n. Then the full commutant algebra of W_{n} in End (⊗^{m}P) becomes the semigroup ring of the transformation semigroup 𝔗_{m}, which is a semigroup consisting of all the maps from the finite set [m] = {1, 2, ⋯, m} to itself (see, for example, [6, Chapter 1]). In this paper, we discuss on the decomposition of ⊗^{m}P as a representation of W_{n} × 𝔗_{m}. Unfortunately, the representations of W_{n} and 𝔗_{m} are not semisimple, and even worse, ⊗^{m}P admits a composition series of infinite length. However, we can still get a control on the irreducible quotients of ⊗^{m}P.

Let U be a representation of W_{n} × 𝔗_{m} and π ⊗ Σ an irreducible representation of W_{n} × 𝔗_{m}. Then we say that π ⊗ Σ has quotient multiplicity k in U if

dim Hom_{W_{n}×𝔗_{m}} (U, π ⊗ Σ) = k

holds. Further, we say that U is quotient multiplicity free if

dim Hom_{W_{n}×𝔗_{m}} (U, π ⊗ Σ) ≤ 1

for any irreducible representation π ⊗ Σ. With these terminologies, we can state our main result of this paper.

Theorem 1. [Theorem 5.1] If m ≤ n, then ⊗^{m}P is quotient multiplicity free. There is a one-to-one correspondence between the subsets of irreducible representations W_{n}^{∧} ∋ π ↔ Σ ∈ 𝔗_{m}^{∧}, which is defined by the following property:

dim Hom_{W_{n}×𝔗_{m}} (⊗^{m}P, π ⊗ Σ) = 1.

Note that the above correspondence π ↔ Σ cannot be bijective. However, it involves all the irreducible representations of the semigroup 𝔗_{m}. We give the correspondence in terms of Young diagrams or, equivalently to say, partitions of various sizes. The correspondence is an explicit one, hence we obtain a realization of irreducible representations of W_{n} as irreducible quotients of ⊗^{m}P. (See Corollary 5.2.)

Let GL_{n} act on V = ℂ^{n} as its defining representation and 𝔖_{m} act on its m-fold tensor product ⊗^{m}V by the permutations of coordinates. Then Schur duality says that there is a correspondence between the irreducible representations of GL_{n} and 𝔖_{m} via Young diagrams. Weyl became the first mathematician who pointed out the machinery works for two mutually commutant algebras which are semisimple ([18]). Our theorem is a generalization of their theory to nonsemisimple and infinite dimensional algebras.

On the other hand, our method is under strong influence of series of works of Howe ([3, 4, 5]). He developed correspondences of representations of various pairs of reductive groups, which are called Howe duality. Although the groups are reductive, their (𝔤, K)-modules, which are infinite dimensional, are not semisimple. So he needed to consider irreducible quotients to establish one-to-one correspondence. Since our theorem is proved by using the restrictions to semisimple subalgebras (or subgroups) instead of using K -types, it is the same as Howe’s in its spirit.

Recently, Benkart and Melville independently get a similar result on a certain quotient representation of ⊗^{k}P with no restriction on the power of tensor products for Lie superalgebra W (m, n). Their result is a kind of W (m, n) × 𝔖_{k} duality, which corresponds to one of our intermediate results (see Theorem 2.4). Our emphasis is on the use of the transformation semigroup 𝔗_{m} which describes the filtered structure of ⊗^{m}P.

Our theory also applies to the finite dimensional Lie superalgebra W_{n} (we use the script letter W to avoid the confusion). In this case, the whole representation space becomes finite dimensional and we expect an explicit description of composition series (cf. [17]). We also expect that it applies to representations of Cartan type Lie algebras over a field of positive characteristic. However, these are the future subjects of ours.

Acknowledgment. This work has been inspired by the explicit calculations done by H. Wang for Cartan type Lie superalgebras. The author expresses sincere thanks to him. He is also grateful to the referee for pointing out the reference [2] to him, which enriches our main result.

1. Preliminaries; Cartan-type Lie algebra W_{n}

Let V = 𝔽^{n} be an n-dimensional vector space over a field 𝔽 of characteristic 0. We fix a basis {z_{i} | 1 ≤ i ≤ n} of V and denote the dual basis in V^{∗} by {z_{i}^{∗} | 1 ≤ i ≤ n}. Let W_{n} be the Lie algebra of vector fields on V^{∗} = 𝔽^{n} with polynomial coefficients. The algebra W_{n} is considered as the algebra of all the derivations on the polynomial algebra P (V^{∗}) = 𝔽[z_{1}, z_{2}, ⋯, z_{n}] on V^{∗} (cf. [11]).

As derivations, W_{n} acts naturally on the polynomial algebra P (V^{∗}). We call P (V^{∗}) the natural representation (or defining representation) of W_{n}, and denote it by ψ. Then (ψ, P (V^{∗})) becomes a ℤ-graded W_{n}-module with the natural grading of polynomials, where the ℤ-grading of W_{n} is given by

W_{n} = ⨁_{j=−1}^{∞} W_{n}(j),

W_{n}(j) = \{∑_{i=1}^{n} f_{i}(z)\frac{∂}{∂z_{i}} | f_{i}(z) is a homogeneous polynomial

of degree j + 1} .

In this grading, the homogeneous elements of degree zero form a finite dimensional Lie algebra which is isomorphic to 𝔤𝔩_{n}(𝔽). This realization of 𝔤𝔩_{n}(𝔽) is obtained by differentiating the natural action of GL(V) on the polynomial ring P (V^{∗}) = S(V).

We briefly review the results on irreducible representations of W_{n}. Let E be an irreducible graded representation of W_{n} (n ≥ 2) which is of finite type in the sense of [9]. Then E or its contragredient has a vector killed by W_{n}(j) for arbitrary j ≥ 1 ([9]). These representations are called height 1 and studied concretely by Rudakov ([15, §13]). In particular, E is a lowest weight module or a highest weight module. In this paper, there appear only lowest weight modules. So assume that E has a lowest weight vector and put

E_{0} = {v ∈ E | W_{n}(−1)v = 0 (j ≥ 1)}.

Then E_{0} becomes a non-zero irreducible W_{n}(0) = 𝔤𝔩_{n}-module. Let λ be the highest weight of the 𝔤𝔩_{n}-module E_{0}. Since E is completely determined by λ, we write E = π_{λ}.

Now recall the natural representation (ψ, P (V^{∗})). Let us consider the m-fold tensor product representation (ψ^{⊗m}, ⊗^{m}P (V^{∗})) of W_{n}. We studied this representation in [12], in comparison with the classical Schur-Weyl’s duality for GL_{n}(ℂ) × 𝔖_{m}. In the paper, we determined the structure of the commutant algebra of W_{n} in End (⊗^{m}P (V^{∗})) in case that the number of tensor product m does not exceed n, which is rank of the Lie algebra W_{n} ([12, Theorem 2.3]). The commutant algebra is a certain semigroup ring. Let us explain it briefly.

Denote the set of positive integers less than or equal to m by [m] = {1, 2, ⋯, m}. Then the set of all the maps from [m] to itself, 𝔗_{m} = {φ : [m] → [m]}, becomes a semigroup with unit by composition of maps (see, for example, [6]). Note that the group elements (i.e., the elements which have inverses in 𝔗_{m}) are the permutations in the symmetric group 𝔖_{m}. We denote the semigroup ring of 𝔗_{m} by 𝔽[𝔗_{m}].

We identify the m-fold tensor product with the polynomial ring in n × mvariables:

⊗^{m}P (V^{∗}) ≃ 𝔽[z_{i,j} | 1 ≤ i ≤ n, 1 ≤ j ≤ m].

An element φ ∈ 𝔗_{m} acts on ⊗^{m}P (V^{∗}) as an endomorphism of algebra 𝔽[z_{i,j}] ≃ ⊗^{m}P (V^{∗}) and it acts on generators {z_{i,j}}_{i,j} as follows:

φ(z_{i,j}) = z_{i,φ(j)}.

We extend it to the semigroup ring 𝔽[𝔗_{m}] by linearity. By easy calculations, one can convince that 𝔽[𝔗_{m}] is a subalgebra in the commutant algebra. In fact, they coincide if m ≤ n.

2. Duality on the top level

Put R = 𝔽[𝔗_{m}], and consider the following filtration by two-sided ideals of R:

R_{k} = ⟨φ ∈ 𝔗_{m} | # Im φ ≤ k⟩ ( generated as a vector space),

where #N means the cardinality of a finite set N. Clearly R becomes a filtered algebra by this filtration.

Put V_{k} = R_{k} · (⊗^{m}P (V^{∗})) for k ≥ 1 and V_{0} = 𝔽. In other words, V_{k} is spanned by those polynomials in the matrix z = (z_{i,j}) that actually depends on at most k columns of z. Then (V_{k})_{0≤k≤m} becomes a natural filtration of the W_{n} × 𝔽[𝔗_{m}]-module V_{m} = ⊗^{m}P (V^{∗}). Consider the graded module

gr V = ⨁_{0≤k} V_{k}/V_{k−1} = ⨁_{0≤k} V(k) (V_{−1} = (0)).

Our aim in the present note is to decompose this module as a W_{n} × 𝔗_{m}-module.

Let us begin with decomposition of V(m) = V_{m}/V_{m−1}. If we put P (V^{∗})^{+} = P (V^{∗})/𝔽 (𝔽 being the constant functions), then clearly

V(m) ≃ ⊗^{m}P (V^{∗})^{+}

holds. However, we prefer the realization of V(m) as the quotient space of 𝔽[z_{i,j}| 1 ≤ i ≤ n, 1 ≤ j ≤ m] = ⊗^{m}P (V^{∗}).

Lemma 2.1. Assume that m ≤ n. Then, the commutant algebra of W_{n} in End V(m) is isomorphic to 𝔽[𝔖_{m}], the group ring of the symmetric group of degree m.

Proof. The proof of the lemma is essentially the same as that of Theorem 2.3 (1) in [12]. Let us recall it.

Take v = z_{1,1}z_{2,2} ⋯ z_{m,m} ∈ V(m). Then v is a cyclic vector of V(m) for the action of W_{n}. So, if E belongs to the commutant of W_{n}, then E is completely determined by the image of v under E. On the other hand, since W_{n} contains Euler’s degree operators, the image Ev is contained in the subspace of some fixed homogeneous degree.

Let us explain it more precisely. Define the degree of the monomial

∏_{1≤i≤n,1≤j≤m} z_{i,j}^{d_{i,j}} (d_{i,j} ∈ ℤ)

with respect to (z_{i,1}, ⋯, z_{i,m}) by ∑_{j=1}^{m} d_{i,j} . Then Ev has the homogeneous degree 1 with respect to (z_{i,1}, ⋯, z_{i,m}) for each i ≤ m, and 0 for m < i ≤ n.

Since the subspace of the homogeneous polynomials in V(m) of the degree described above has a basis

{∏_{i=1}^{m} z_{i,σ(i)} | σ ∈ 𝔖_{m}},

the dimension of the commutant algebra cannot exceed m!. On the other hand, it clearly contains 𝔽[𝔖_{m}] which is m!-dimensional, hence the lemma is proved.

Note that R = 𝔽[𝔗_{m}] acts on V(m) and, by definition, the action of the two-sided ideal R_{m−1} is trivial. So, the quotient algebra 𝔽[𝔗_{m}]/R_{m−1} ≃ 𝔽[𝔖_{m}] naturally acts on V(m). Therefore, the above lemma says that the image of 𝔽[𝔗_{m}] on V(m) is the full commutant of W_{n}.

Since 𝔽[𝔖_{m}] is semisimple, we decompose V(m) as a 𝔽[𝔖_{m}]-module:

V(m) ≃ \begin{matrix}∑ & ⊕ \\ D∈Y_{m} \end{matrix} Hom_{𝔖_{m}}(σ_{D}, V(m)) ⊗ σ_{D},

where Y_{m} is the set of Young diagrams of size m and σ_{D} is the corresponding irreducible representation of 𝔖_{m} (see, for example, [8], [7], ...); we prefer the terminology size to weight here, since ‘weight’ is used in the different meaning (cf. [10]). The fact that 𝔽[𝔖_{m}] is the full commutant algebra of W_{n} assures that

Hom_{𝔖_{m}}(σ_{D}, V(m))

becomes a W_{n}-module and, moreover, it is indecomposable. We define a grading on Hom_{𝔖_{m}}(σ_{D}, V(m)) by defining that f ∈ Hom_{𝔖_{m}}(σ_{D}, V(m)) is of homogeneous degree d if and only if f(σ_{D}) is contained in the homogeneous space of degree d. Then, by this grading, Hom_{𝔖_{m}}(σ_{D}, V(m)) becomes a graded W_{n}-module.

Lemma 2.2. Let D ∈ Y_{m} be a Young diagram of size m and (λ_{1}, λ_{2}, ⋯, λ_{m}) the corresponding partition of m. Then the restriction of the W_{n}-module Hom_{𝔖_{m}}(σ_{D}, V(m)) to the subalgebra W_{n}(0) ≃ 𝔤𝔩_{n} contains the irreducible representation ρ_{D} of 𝔤𝔩_{n} with highest weight (λ_{1}, λ_{2}, ⋯, λ_{m}, 0, ⋯, 0) with multiplicity one:

dim Hom_{𝔤𝔩_{n}} (ρ_{D}, Hom_{𝔖_{m}}(σ_{D}, V(m))) = 1.

Moreover, the homogeneous component of Hom_{𝔖_{m}}(σ_{D}, V(m)) of the lowest possible degree m coincides with the embedded ρ_{D}.

Proof. Let P (V^{∗})_{+} be the space of the polynomials without constant term. Then ⊗^{m}P (V^{∗})_{+} ≃ V(m) as a vector space, and we take representatives of the elements in V(m) from ⊗^{m}P (V^{∗})_{+} ⊂ ⊗^{m}P (V^{∗}). This identification preserves degrees and 𝔤𝔩_{n} × 𝔖_{m}-module structures.

Let U be the subspace of the elements of degree m in V(m). Note that m is the lowest possible degree in V(m). Since 𝔤𝔩_{n} × 𝔖_{m} preserves degree, U is a 𝔤𝔩_{n} × 𝔖_{m}-module. It holds that

U ≃ ⊗^{m}V ≃ \begin{matrix}∑ & ⊕ \\ D∈Y_{m} \end{matrix} ρ_{D} ⊗ σ_{D}

as a 𝔤𝔩_{n} × 𝔖_{m}-module, where ρ_{D} is the irreducible representation of GL(V) with highest weight (λ_{1}, ⋯, λ_{m}, 0, ⋯, 0). The first part of the above formula is obvious by definition and the second part follows from Schur duality.

By the above formula, we get

U ∩ (Hom_{𝔖_{m}}(σ_{D}, V(m)) ⊗ σ_{D}) ≃ ρ_{D} ⊗ σ_{D} ≠ 0.

Denote the above non-zero space by U_{D}. Note that U_{D} does not have GL(V)- components in common with V(m)_{D} = Hom_{𝔖_{m}}(σ_{D}, V(m)) ⊗ σ_{D} other than itself. To see this, we note that ⊗^{m}P (V^{∗}) ≃ P (V^{∗} ⊗ 𝔽^{m}) is decomposed as GL_{n} × GL_{m}- module in the following multiplicity-free manner:

P (V ⊗ 𝔽^{m}) = \begin{matrix}∑ & ⊕ \\ E \end{matrix} ρ_{E}^{(n)} ⊗ ρ_{E}^{(m)},

where E ranges over all the Young diagrams with length less than or equal to min{n, m}, and ρ_{E}^{(n)} (respectively ρ_{E}^{(m)}) is the irreducible representation of GL_{n} (respectively GL_{m}) with highest weight μ(E) = (μ_{1}, μ_{2}, ⋯) which is the partition associated with E. This decomposition is well-known and we refer to, for example, [3].

Note that, in this decomposition, the subspace

\begin{matrix}∑ & ⊕ \\ |E|=d \end{matrix} ρ_{E}^{(n)} ⊗ ρ_{E}^{(m)}

gives the full homogeneous component of ⊗^{m}P (V^{∗}) of degree d. Therefore, in our case, the representation ρ_{D} does not appear in the degree greater than m. On the other hand, m is the smallest positive degree in V(m). Hence ρ_{D} appears only in U_{D}.

The rest of the statements are clear from the above arguments.

Proposition 2.3. Let D ∈ Y_{m} be a Young diagram of size m and let (λ_{1}, λ_{2}, ⋯, λ_{m}) be the corresponding partition of m. Then the W_{n}-module Hom_{𝔖_{m}}(σ_{D}, V(m)) has the unique irreducible quotient which has the lowest weight (0, ⋯, 0, λ_{m}, λ_{m−1}, ⋯, λ_{1}).

Proof. We use the notations in the proof of Lemma 2.2.

Note that the space U generates V(m) as W_{n}-module. In fact, the vector v = z_{1,1}z_{2,2} ⋯ z_{m,m} ∈ U is a cyclic vector for V(m) (cf. the proof of Lemma 2.1). Since W_{n} commutes the action of 𝔖_{m}, U_{D} also generates V(m)_{D} = Hom_{𝔖_{m}}(σ_{D}, V(m)) ⊗ σ_{D}. By the above lemma, U_{D} does not have the GL(V)-components in common with V(m)_{D} other than itself and is irreducible under the action of 𝔤𝔩_{n} × 𝔖_{m}.

In the following in this proof, we consider W_{n} × 𝔖_{m}-module structure only unless otherwise stated. Note that 𝔖_{m} acts just according to σ_{D} on V(m)_{D}.

Let V(m)_{D} ⊃ A, B be two submodules such that both V(m)_{D}/A and V(m)_{D}/B are non-zero and irreducible. Since U_{D} is cyclic for V(m)_{D}, U_{D} and A has no GL(V)-component in common. The same statement is applicable to B. So we know A + B has no GL(V)-component in common with U_{D}. Since V(m)_{D}/A is irreducible, this means A + B ⊂ A, which concludes B ⊂ A. Similarly, we have A ⊂ B and A = B follows.

By the above argument, we know that V(m)_{D} has the largest submodule which does not contain U_{D}. Since U_{D} contains the lowest weight vector for W_{n} with weight (0, ⋯, 0, λ_{m}, λ_{m−1}, ⋯, λ_{1}), the irreducible quotient has also the same lowest weight vector.

In the following, for a Young diagram D of size m, λ = (λ_{1}, λ_{2}, ⋯) denotes the corresponding partition of D. If the length of λ is less than or equal to n, then we also denote by π_{D} an irreducible representation of W_{n} with lowest weight (λ_{n}, λ_{n−1}, ⋯, λ_{1}).

Theorem 2.4. Let P (V^{∗})^{+} = P (V^{∗})/𝔽 be an irreducible representation of W_{n} with lowest weight (0, ⋯, 0, 1). Then V(m) ≃ ⊗^{m}P (V^{∗})^{+} is a W_{n} × 𝔖_{m}-module and, under the assumption that m ≤ n, its irreducible quotients are of the form

π_{D} ⊗ σ_{D} (D ∈ Y_{m}),

where Y_{m} is the set of Young diagrams of size m. Moreover, it is quotient multiplicity-free, i.e.,

dim Hom_{W_{n}×𝔖_{m}}(V(m), π_{D} ⊗ σ_{D}) = 1 (∀D ∈ Y_{m}).

3. Irreducible representation of the semigroup 𝔗_{m}

The irreducible representations of the transformation semigroup 𝔗_{m} were studied by Hewitt and Zuckerman [2] and they got a complete classification. However, their description of the classification is combinatorial and too complicated to fit with our present situation. In this section, we study irreducible representations of the semigroup 𝔗_{m} which are obtained by induction from maximal parabolic sub-semigroups. Thanks to the above mentioned result of [2], we eventually find the representations constructed here using parabolic induction exhaust all the irreducible ones.

The arguments of Hewitt and Zuckerman essentially use restriction of the representations to the symmetric group, while we use the induced representations. Though our arguments here do not provide a new proof, we believe that they shed a new light on their classification.

Put

P_{k,m−k} = {φ ∈ 𝔗_{m} | φ([k]) ⊂ [k]} ⊂ 𝔗_{m}.

Clearly P_{k,m−k} is a sub-semigroup of 𝔗_{m}. We may call it a maximal parabolic sub-semigroup. It projects naturally onto 𝔗_{k} when we consider the first k letters [k] = {1, 2, ⋯, k}. The projection is a semigroup morphism. Let (σ, U_{σ}) be a representation of 𝔗_{k}. Then the above projection proj naturally induces a representation proj^{∗}σ of P_{k,m−k} by

proj^{∗}σ(φ) = σ(proj(φ)).  (1)

We denote this representation (proj^{∗}σ, U_{σ}) of P_{k,m−k} by the same letter σ by abuse of notation. Then we define the parabolically induced representation ind σ = ind_{P_{k,m−k}}^{𝔗_{m}}σ of 𝔗_{m} as

ind σ = ind (σ, U_{σ}) = 𝔽[𝔗_{m}] ⊗_{𝔽[P_{k,m−k}]} U_{σ}.  (2)

Proposition 3.1. Let (σ, U_{σ}) be an irreducible representation of 𝔗_{k} on which R_{k−1} acts trivially. Then the 𝔽[𝔗_{m}]-module ind_{P_{k,m−k}}^{𝔗_{m}} σ has the unique irreducible quotient.

Proof. Since R_{k−1} acts trivially, the action of σ is completely determined by 𝔖_{k} ⊂ 𝔗_{k}. Hence, as a representation of 𝔖_{k}, σ is isomorphic to σ_{D} for some Young diagram D ∈ Y_{k}. In the following, we identify (σ, U_{σ}) = (σ_{D}, U_{σ_{D}}).

Lemma 3.2. As 𝔖_{m}-modules, we have

ind_{P_{k,m−k}}^{𝔗_{m}}σ_{D} ≃ ind_{𝔖_{k}×𝔖_{m−k}}^{𝔖_{m}}(σ_{D} ⊗ 1).

Proof. We realize the representation (σ_{D}, U_{σ_{D}}) in the polynomial ring ⊗^{k}V ⊂ ⊗^{k}P (V^{∗})^{+} = ⊗^{k}P (V^{∗})/𝔽. As mentioned before, Weyl-Schur duality tells us that

⊗^{k}V ≃ \begin{matrix}∑ & ⊕ \\ E∈Y_{k} \end{matrix} ρ_{E} ⊗ σ_{E},

and we take one copy of (σ_{D}, U_{σ_{D}}) in ⊗^{k}V . Note that ⊗^{k}V , and hence U_{σ_{D}} ⊂ ⊗^{k}V , is naturally embedded into ⊗^{m}P (V^{∗})/R_{k−1} (⊗^{m}P (V^{∗})), which is a W_{n}×𝔗_{m}- module:

⊗^{k}V ∋ v_{1} ⊗ ⋯ ⊗ v_{k} ↦ v_{1} ⊗ ⋯ ⊗ v_{k} ⊗ 1 ⊗ ⋯ ⊗ 1 ∈ ⊗^{m}P (V^{∗})/R_{k−1} (⊗^{m}P (V^{∗})) .

Then, on U_{σ_{D}}, the maximal parabolic sub-semigroup P_{k,m−k} acts as described in (1) and the action of R_{k−1} vanishes. By the universality of tensor products, we get a natural map:

𝔽[𝔗_{m}] ⊗_{𝔽[P_{k,m−k}]} U_{σ_{D}} → 𝔽[𝔗_{m}] · U_{σ_{D}} ⊂ ⊗^{m}P (V^{∗})/R_{k−1} (⊗^{m}P (V^{∗})) . (3)

We shall prove the above natural map is actually an isomorphism.

Note that the right cosets of 𝔗_{m} with respect to P_{k,m−k} have representatives in 𝔖_{m}:

𝔗_{m} = ⋃_{s∈𝔖_{m}/𝔖_{k}×𝔖_{m−k}} sP_{k,m−k},

where the union is not disjoint. Using these representatives, we have

ind σ = 𝔽[𝔗_{m}] ⊗_{𝔽[P_{k,m−k}]} U_{σ} = ∑_{s∈𝔖_{m}/𝔖_{k}×𝔖_{m−k}} s ⊗ U_{σ}.

On the other hand, as a subrepresentation of the polynomial ring, we can prove easily that

𝔽[𝔗_{m}] · U_{σ} = \begin{matrix}∑ & ⊕ \\ s∈𝔖_{m}/𝔖_{k}×𝔖_{m−k} \end{matrix} sU_{σ} ⊂ ⊗^{m}P (V^{∗})/R_{k−1} (⊗^{m}P (V^{∗})) ,

and, as a 𝔖_{m}-module, this means that

𝔽[𝔗_{m}] · U_{σ_{D}} ≃ ind_{𝔖_{k}×𝔖_{m−k}}^{𝔖_{m}}(σ_{D} ⊗ 1).

Comparing the above two formulas, we can conclude that the map (3) must be bijective and the lemma follows.

Let us return to the proof of Proposition 3.1. Let λ = (λ_{1}, ⋯, λ_{k}) be a partition of k corresponding to the Young diagram D. We write λ^{#} = (λ_{1} + m − k, λ_{2}, ⋯, λ_{k}), a partition of m, and denote by D^{#} ∈ Y_{m} the corresponding Young diagram.

By Young’s rule (see, for example, [8, §2.8.2]), the induced representation

ind_{𝔖_{k}×𝔖_{m−k}}^{𝔖_{m}}(σ_{D} ⊗ 1)

is multiplicity free and contains σ_{D^{#}} . We shall show that a 𝔽[𝔗_{m}]-submodule of ind σ_{D} which contains σ_{D^{#}} must coincide with the whole module. Then, if we take all the summation of submodules which does not contain σ_{D^{#}} , it gives the unique maximal proper submodule and we have done.

Take a standard tableau B of shape D such that the first row of B consists of {1, 2, ⋯, λ_{1}}, and that the second row consists of {λ_{1} + 1, λ_{1} + 2, ⋯, λ_{1} + λ_{2}} and so on. We also take a standard tableau B^{#} of shape D^{#} which is exactly the same as B except the first row. The first row of B^{#} consists of {1, 2, ⋯, λ_{1}} ∪ {k + 1, k + 2, ⋯, m}.

Consider the Young symmetrizers c_{B} and c_{B^{#}} . Here we follow the notation in [1, §4.1] (see also [8, Theorem 3.1.10]). Put

𝔖_{λ_{1}} = 𝔖({1, 2, ⋯, λ_{1}})
𝔖_{m−k} = 𝔖({k + 1, k + 2, ⋯, m})
𝔖_{λ_{1}+(m−k)} = 𝔖({1, 2, ⋯, λ_{1}} ∪ {k + 1, k + 2, ⋯, m}),

where 𝔖({⋯}) means the permutation group of a finite set {⋯} (this is an abuse of notation, but it would not lead to misunderstandings). Then we have

c_{B^{#}} = ( ∑_{s∈𝔖_{λ_{1}+(m−k)}/𝔖_{λ_{1}}×𝔖_{m−k}} s) ( ∑_{t∈𝔖_{m−k}} t) c_{B}.

We can take u ∈ U_{σ_{D}} such that c_{B}u ≠ 0. For this u, we have

c_{B^{#}} ⊗ u = ∑_{s∈𝔖_{λ_{1}+(m−k)}/𝔖_{λ_{1}}×𝔖_{m−k}} s ⊗ ( ∑_{t∈𝔖_{m−k}} t) c_{B}u = (m − k)! ∑_{s} s ⊗ c_{B}u.

Take φ ∈ R_{k} as follows:

φ(i) = i (1 ≤ i ≤ k); φ(i) = 1 (k + 1 ≤ i ≤ m).

Then it is easy to see that the following three properties hold: for s ∈ 𝔖_{λ_{1}+(m−k)},

φs ∈ P_{k,m−k},
φs ∈ R_{k−1} if s ∉ (𝔖_{λ_{1}} × 𝔖_{m−k}) ∪ (1, m)(𝔖_{λ_{1}} × 𝔖_{m−k}),
φe = φ(1, m) = φ,

where e is the unit element and (1, m) is the transposition of 1 and m. Put u′ = (m − k)! c_{B}u. Using the above properties of φ, we calculate

φ (c_{B^{#}} ⊗ u) = ∑_{s} φs ⊗ u′ = ∑_{s} e ⊗ (φs)u′ = 2e ⊗ φu′ = 2e ⊗ u′ ≠ 0.

Since c_{B^{#}} ⊗ u belongs to U_{σ_{D^{#}}} , we get

𝔽[𝔗_{m}]U_{σ_{D^{#}}} ∋ e ⊗ u′ ≠ 0 (∃u′ ∈ U_{σ_{D}}),

hence

𝔽[𝔗_{m}]U_{σ_{D^{#}}} ⊃ 𝔽[𝔗_{m}] ⊗ U_{σ_{D}} = ind_{P_{k,m−k}}^{𝔗_{m}}σ_{D}.

Definition 3.3. We denote by Σ_{D} (D ∈ Y_{k}) the unique irreducible quotient of ind_{P_{k,m−k}}^{𝔗_{m}}σ_{D}, and call it a standard representation of 𝔗_{m} corresponding to D.

Example 3.4. We shall give an example, in which ind_{P_{k,m−k}}^{𝔗_{m}}σ_{D} is not irreducible. So it is necessary to take the irreducible quotient to obtain Σ_{D}.

Take k = m − 1 and consider the digram D which corresponds to the partition (1, ⋯, 1) = (1^{m−1}). Then σ_{D} is the sign representation of 𝔖_{m−1}. It is easy to see that

ind_{𝔖_{m−1}×𝔖_{1}}^{𝔖_{m}} σ_{(1^{m−1})} ≃ σ_{(2·1^{m−2})} ⊕ σ_{(1^{m})}.

The space σ_{(1^{m})} is killed by R_{m−1}. In fact, this is a general property of sign representation. Let U be a 𝔗_{m}-module and U_{sgn} the sign-isotypic component of U as an 𝔖_{m}-module. Then U_{sgn} is killed by R_{m−1}, hence invariant under 𝔗_{m}.

Note that Σ_{(1^{m−1})} ≄ Σ_{(2·1^{m−2})} as proved in Theorem 3.5 below, while Σ_{(1^{m−1})}|_{𝔖_{m}} ≃ Σ_{(2·1^{m−2})}|_{𝔖_{m}} ≃ σ_{(2·1^{m−2})} as 𝔖_{m}-modules.

Theorem 3.5. The standard representations

{Σ_{D} | D ∈ Y_{k} (1 ≤ k ≤ m)}

of 𝔗_{m} are mutually inequivalent, and they give a complete set of representatives of equivalence classes of irreducible representations of 𝔗_{m}.

Proof. Note that

R_{k−1} = 𝔖_{m} (P_{k,m−k} ∩ R_{k−1}) .

Therefore it is easy to see that, if D ∈ Y_{k}, R_{k−1}Σ_{D} = 0 holds. On the other hand, if we denote by φ ∈ R_{k} as in the proof of Proposition 3.1, we have shown φ · Σ_{D} ≠ 0. This implies that

Σ_{D_{1}} ≄ Σ_{D_{2}} if |D_{1}| ≠ |D_{2}|.

For D_{1}, D_{2} ∈ Y_{k}, the proof of Proposition 3.1 tells us that Σ_{D_{i}} (i = 1, 2) contains σ_{D_{i}^{#}} as a representation of 𝔖_{m}. Introducing suitable lexicographic order on

the set of Young diagram Y_{m}, we conclude that D_{i}^{#} is the largest one among {D ∈ Y_{m} | σ_{D} ⊂ Σ_{D_{i}}|_{𝔖_{m}}}. For this, see Lemma 3.2 and consult with Young’s rule. Therefore we get

Σ_{D_{1}}|_{𝔖_{m}} ≄ Σ_{D_{2}}|_{𝔖_{m}} if D_{1} ≠ D_{2}.

On the other hand, by [2, Th. 3.7 & Th. 7.4], the irreducible representations of 𝔗_{m} are completely classified. Their classification shows that {Σ_{D}}_{D} exhausts the irreducible ones.

4. Duality for general level

Let R_{k} ⊂ 𝔽[𝔗_{m}] as in §2, and put V_{k} = R_{k}(⊗^{m}P (V^{∗})). In Section 2., we clarified the structure of V(m) = V_{m}/V_{m−1} = ⊗^{m}P (V^{∗})/V_{m−1} in terms of the quotient representations. In this section, we study the structure of V(k) = V_{k}/V_{k−1} in general.

Let us consider V(k) = V_{k}/V_{k−1} as a W_{n} × 𝔗_{m}-module. We note that ⊗^{k}P (V^{∗})^{+} is contained in V(k) as the quotient image of the natural inclusion map ⊗^{k}P (V^{∗}) ↪ ⊗^{k}P (V^{∗}) ⊗ 1 ⊗ ⋯ ⊗ 1 ⊂ V_{k}. Moreover we recall that, as a W_{n} × 𝔗_{k} -module,

⊗^{k}P (V^{∗})^{+} ≃ \begin{matrix}∑ & ⊕ \\ D∈Y_{k} \end{matrix} Hom_{𝔖_{k}}(σ_{D}, ⊗^{k}P (V^{∗})^{+}) ⊗ σ_{D},

where Hom_{𝔖_{k}}(σ_{D}, ⊗^{k}P (V^{∗})^{+}) has the unique irreducible W_{n}-quotient π_{D} (Proposition 2.3).

Lemma 4.1.

ind_{P_{k,m−k}}^{𝔗_{m}} ⊗^{k} P (V^{∗})^{+} ≃ V(k)

Proof. By the universality of tensor products, we have a natural map

𝔽[𝔗_{m}] ⊗_{𝔽[P_{k,m−k}]} (⊗^{k}P (V^{∗})^{+})
≃ 𝔽[𝔗_{m}] ⊗_{𝔽[P_{k,m−k}]} (⊗^{k}P (V^{∗})^{+} ⊗ 1 ⊗ ⋯ ⊗ 1)
→ 𝔽[𝔗_{m}] · (⊗^{k}P (V^{∗})^{+} ⊗ 1 ⊗ ⋯ ⊗ 1) = V(k).

By Lemma 3.2 this map is bijective, hence gives an isomorphism.

From this lemma, we calculate as

V(k) ≃ ind_{P_{k,m−k}}^{𝔗_{m}} ⊗^{k} P (V^{∗})^{+}
≃ ind_{P_{k,m−k}}^{𝔗_{m}} \begin{matrix}∑ & ⊕ \\ D∈Y_{k} \end{matrix} Hom_{𝔖_{k}}(σ_{D}, ⊗^{k}P (V^{∗})^{+}) ⊗ σ_{D}
≃ \begin{matrix}∑ & ⊕ \\ D∈Y_{k} \end{matrix} Hom_{𝔖_{k}}(σ_{D}, ⊗^{k}P (V^{∗})^{+}) ⊗ (ind_{P_{k,m−k}}^{𝔗_{m}} σ_{D}) .

Therefore it is easy to see that the irreducible representation π_{D} ⊗ Σ_{D} is an irreducible quotient of V(k).

Theorem 4.2. Let the notations be as above. An irreducible quotient of W_{n} × 𝔗_{m}-module V(k) is of the form π_{D} ⊗ Σ_{D} (D ∈ Y_{k}). Moreover, it appears with quotient multiplicity one, i.e.,

dim Hom_{W_{n}×𝔗_{m}}(V(k), π_{D} ⊗ Σ_{D}) = 1.  (4)

Proof. In the beginning, we shall prove the multiplicity freeness (4). Let us consider V(k) as a GL(V) × 𝔖_{m}-module. Then we have proved that the multiplicity of ρ_{D} ⊗σ_{D^{#}} in V(k) is one. Since π_{D} ⊗Σ_{D} contains ρ_{D} ⊗σ_{D^{#}} exactly once, we get

dim Hom_{W_{n}×𝔗_{m}}(V(k), π_{D} ⊗ Σ_{D}) ≤ 1.

As mentioned before the theorem, the left hand side is positive, so the equality must hold.

Let U be an irreducible quotient of V(k). Then there is a proper invariant subspace V′ ⊂ V(k) such that U ≃ V(k)/V′. Let f : V(k) → U be a projection map.

We take φ ∈ R_{k} as before:

φ(i) = i (1 ≤ i ≤ k); φ(i) = 1 (k + 1 ≤ i ≤ m).

Since f is a 𝔗_{m}-morphism, we have f(φV(k)) = φf(V(k)) = φU. Note that φV(k) is a W_{n} × P_{k,m−k} -module, and is isomorphic to ⊗^{k}P (V^{∗})^{+}. Therefore φU is a W_{n} × P_{k,m−k} -module, which is a quotient of ⊗^{k}P (V^{∗})^{+}. Let us prove that the module φU is non-zero irreducible. In fact, it is easy to see that the space

φV(k) = (⊗^{k}P (v) ⊗ 1 ⊗ ⋯ ⊗ 1) / (V_{k} ∩ (⊗^{k}P (V^{∗}) ⊗ 1 ⊗ ⋯ ⊗ 1))

generates V(k) as a 𝔗_{m}-module, which means φU is cyclic for U. So, if φU = 0, we have U = 0, which is not the case. To prove irreducibility, we assume that φU has a proper W_{n} × P_{k,m−k} -submodule U′. Then 𝔽[𝔗_{m}]U′ becomes a proper W_{n} × 𝔗_{m}-submodule of U, hence zero. Let us show it. Take φv ∉ U′. If φv ∈ 𝔽[𝔗_{m}]U′, then we have

φv = φ^{2}v ∈ φ𝔽[𝔗_{m}]U′ = 𝔽[P_{k,m−k}]U′ = U′,

and this is a contradiction.

By Theorem 2.4, φU is isomorphic to π_{D} ⊗σ_{D} for some D ∈ Y_{k}. Therefore we have

ind_{P_{k,m−k}}^{𝔗_{m}} φU ≃ ind_{P_{k,m−k}}^{𝔗_{m}} (π_{D} ⊗ σ_{D}) ≃ π_{D} ⊗ ind_{P_{k,m−k}}^{𝔗_{m}} σ_{D}.

On the other hand, the universality of tensor products tells us that there is a surjective morphism

π_{D} ⊗ ind_{P_{k,m−k}}^{𝔗_{m}} σ_{D} ≃ ind_{P_{k,m−k}}^{𝔗_{m}} φU ⟶ 𝔽[𝔗_{m}]φU = U.

Since U is irreducible, it is isomorphic to π_{D} ⊗ Σ_{D} by Proposition 3.1.

5. Howe’s correspondence for W_{n} × 𝔗_{m}

Recall that we say that an irreducible representation π⊗Σ has quotient multiplicity k in a representation U if

dim Hom_{W_{n}×𝔗_{m}} (U, π ⊗ Σ) = k

holds.

Theorem 5.1. (1) The tensor product ⊗^{m}P (V^{∗}) is quotient multiplicity free. This means that if π ⊗ Σ is an irreducible representation of W_{n} × 𝔗_{m}, then

dim Hom_{W_{n}×𝔗_{m}} (⊗^{m}P (V^{∗}), π ⊗ Σ) ≤ 1.

(2) The quotient multiplicity of π ⊗ Σ is positive (hence it is one by (1)), if and only if π⊗Σ is isomorphic to π_{D}⊗Σ_{D} for some diagram D of size k (1 ≤ k ≤ m).

Proof. Consider the following exact sequence:

0 → V_{k−1} → V_{k} → V(k) → 0.

It produces a long exact sequence:

0 → Hom_{W_{n}×𝔗_{m}}(V(k), π ⊗ Σ) → Hom_{W_{n}×𝔗_{m}}(V_{k}, π ⊗ Σ)
→ Hom_{W_{n}×𝔗_{m}}(V_{k−1}, π ⊗ Σ) → · · · .  (5)

From the first three terms of the sequence, we get the following inequality

qmult(V_{k} : π ⊗ Σ) ≤ qmult(V(k) : π ⊗ Σ) + qmult(V_{k−1} : π ⊗ Σ),

where we denote dim Hom_{W_{n}×𝔗_{m}}(V, π ⊗ Σ) by qmult(V : π ⊗ Σ). Using induction on k, we get

qmult(V_{m} : π ⊗ Σ) ≤ ∑_{k=0}^{m} qmult(V(k) : π ⊗ Σ).

Assume that k > 0. By Theorem 4.2, we know qmult(V(k) : π ⊗ Σ) ≤ 1 and the equality holds if and only if π ⊗ Σ ≃ π_{D} ⊗ Σ_{D} for some D of size k. Therefore we get

qmult(V_{m} : π ⊗ Σ) ≤ 1

and the equality holds only if π ⊗ Σ ≃ π_{D} ⊗ Σ_{D} for some D or π ⊗ Σ is the trivial representation.

Let us show that the quotient multiplicity of the trivial representation is zero. Take f ∈ Hom_{W_{n}×𝔗_{m}}(V_{m}, 𝔽). We identified V_{m} ≃ 𝔽[z_{i,j}]. Note that z_{i,j} − f(z_{i,j}) is contained in the kernel of f. Since ker f is graded (because W_{n} contains Euler’s degree operators), f(z_{i,j}) is a member of ker f, hence f(z_{i,j}) = 0. However, since z_{i,j} produces a non-zero constant by differentiation, this means f(𝔽) = 0. By the similar argument as above, we conclude that f = 0.

Now we are to prove “if”-part of the statement (2). Take π_{D} ⊗ Σ_{D} with |D| = k. By Theorem 2.4, we have

dim Hom_{W_{n}×𝔗_{k}}(⊗^{k}P (V^{∗}), π_{D} ⊗ Σ_{D}^{(k)}) = 1,

where Σ_{D}^{(k)} is the standard representation of 𝔗_{k}. Take φ ∈ 𝔗_{m} such that

φ(i) = i (1 ≤ i ≤ k) ; φ(j) = k (k < j ≤ m).

Then φ maps ⊗^{m}P (V^{∗}) naturally onto ⊗^{k}P (V^{∗}), and φ is W_{n}-equivariant. Hence we have a non-zero W_{n}-equivariant map

⊗^{m}P (V^{∗}) ⟶^{φ} ⊗^{k}P (V^{∗}) ⟶ π_{D} ⊗ Σ_{D}^{(k)},

composing the above two surjective maps. This proves

Hom_{W_{n}}(⊗^{m}P (V^{∗}), π_{D}) ≠ 0.

Let us put

U_{π_{D}} = ∩{ker f | f ∈ Hom_{W_{n}}(⊗^{m}P (V^{∗}), π_{D})}.

Then it is easy to see ⊗^{m}P (V^{∗})/U_{π_{D}} is non-zero and a finite direct sum of π_{D} (cf. formula (2.5) in [5]);

⊗^{m}P (V^{∗})/U_{π_{D}} ≃ π_{D} ⊗ ω,

where ω is a certain finite dimensional vector space. Since the actions of W_{n} and 𝔗_{m} are commutative, ω inherits an action of 𝔗_{m}. Choose ω′ ⊂ ω such that ω/ω′ is irreducible. Then we get a non-zero W_{n} × 𝔗_{m}-homomorphism

⊗^{m}P (V^{∗}) ⟶^{proj.} ⊗^{m}P (V^{∗})/U_{π_{D}} ≃ π_{D} ⊗ ω ⟶^{proj.} π_{D} ⊗ (ω/ω′).

Note that the irreducible quotients of ⊗^{m}P (V^{∗}) are π_{D} ⊗ Σ_{D} for certain D by the “only if”-part of the theorem (2). Therefore it must hold that π_{D} ⊗ (ω/ω′) ≃ π_{D} ⊗ Σ_{D}, and we have done.

For an algebra A and an A-module M, we denote by R_{A}(M) the set of equivalence classes of irreducible quotients of M.

Corollary 5.2. There is a bijective correspondence between R_{W_{n}} (⊗^{m}P (V^{∗})) and R_{𝔗_{m}} (⊗^{m}P (V^{∗})) defined by the following property. Irreducible representations π ∈ W_{n}^{∧} and Σ ∈ 𝔗_{m}^{∧} are in correspondence if and only if

dim Hom_{W_{n}×𝔗_{m}} (⊗^{m}P (V^{∗}), π ⊗ Σ) = 1.

The correspondence can be given explicitly as follows:

R_{W_{n}} (⊗^{m}P (V^{∗})) = {π_{D} | D ∈ Y_{k} (1 ≤ k ≤ m)},
R_{𝔗_{m}} (⊗^{m}P (V^{∗})) = {Σ_{D} | D ∈ Y_{k} (1 ≤ k ≤ m)} = 𝔗_{m}^{∧},
and π_{D} ↔ Σ_{D}.

We call the correspondence in Corollary 5.2 Howe correspondence (cf. [4, 5]).

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Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University Kyoto 606-8501, JAPAN kyo@math.h.kyoto-u.ac.jp

Received March 10, 1998 and in final form April 27, 1998