# A Gelfand Model for Weyl Groups of TypeD2n

Authors: José O. Araujo; Luis C. Maiarú; Mauro Natale
Journal: ISRN Algebra
 (2012)

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## Abstract

A Gelfand model for a finite groupG is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.


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## Body

## 1. Introduction

Gelfand models for a finite group are complex representations whose character is the sum of all irreducible characters of the given group. In this sense, Bernstein et al. have presented Gelfand models for semisimple compact Lie groups, see [1]. Since then, Gelfand models have been developed in several articles; see [2–12], among these there are two types of models that can be associated with reflection groups: the involution model and the polynomial model.Parallel works, made by Klyachko, on one side, and by Inglis, Richardson, and Saxl, on the other, showed an identity that describes a Gelfand model associated with the symmetric group. The identity is given by(1.1)χCk↑Sn=∑χλ,
where Ck is the centralizer of an involution in 𝔖n with exactly k fixed points, χCk is a linear character of Ck, and χλ is an irreducible character of 𝔖n associated with the partition λ of n with exactly k odd terms. From this identity, it follows immediately that
(1.2)∑kχCk↑Sn=∑λχλ,
where the centralizers Ck are in correspondence with the conjugacy classes of involutions in 𝔖n.Later on, this type of models was called aninvolution model by Baddeley [6]. He also proved that if H is a finite group that admits an involution model, then so does the semidirect product Hn×s𝔖n.Baddeley’s result implies the existence of involution models for classic Weyl groups, with the exception of the group of typeD2n. An involution model for a Weyl group of type An is presented in [8] by Inglis et al. and for a Weyl group of type Bn an involution model is shown in [6, 13]. In [6], Baddeley presents an involution model for a Weyl group of type D2n+1, and in [14] it is proved that there is no involution model for a Weyl group of type D2n with n≥2. In [6], it is mentioned that is not difficult to prove that there is an involution model for a Weyl group of type G2 and that it has been checked using computers the non existence of involution models for exceptional Weyl groups of type F4, E6, E7, and E8. In [15], Vinroot does some research about involution models for irreducible non crystallographic Coxeter groups. He proves the existence of an involution model for groups of type I2n(n≥3,n≠6) and H3 and presents a conceptual demonstration of the no existence of an involution model for the group of type H4.More recently, in [16] the generalized involution modelhas been studied in order to include some cases of unitary reflection groups.A reflection groupG comes equipped with a canonical representation called the geometric representationof G. The geometric representation induces a natural action of G on the space of polynomial functions.Chevalley [17], Shephard and Todd [18], Steinberg [19], and others studied the corresponding action on the space ℋG of G-harmonic polynomials proving that ℋG is isomorphic to the regular representation of G, and thus ℋG contains a Gelfand model for G. On the other hand, Macdonald found irreducible representations of a Weyl group associated with the root systems of the reflection subgroups that can be naturally realized in the G-harmonic polynomial space. These representations are known as Madonald representations see [20].More recently, Araujo and Aguado in [21] have associated with each finite subgroup G⊂GLn(ℂ) a subspace 𝒩G of the algebra of polynomials ℂ[x1,…,xn], defined as zeros of certain G-invariant differential operators, and have shown 𝒩G contains a Gelfand model of G. This space, called the polynomial model, is a Gelfand model for some Weyl groups. In [3–5], it was proved that 𝒩G is a Gelfand model for Weyl groups of type An, Bn and D2n+1. Garge and Oesterlé in [22], using the computation of fake degrees of the irreducible characters of a Coxeter group G, determined that 𝒩G is a Gelfand model of G if, and only if, G has not irreducible factors of type D2n, E7, or E8. The fake degrees have been determined due to works of Steinberg [23], when G is of type An, Lusztig [24], when G is of type Bn or Dn, Beynon and Lusztig [25], when G is an exceptional Weyl group, Alvis and Lusztig [26], when G is of type H4, and Macdonald, when G is of type F4 (unpublished). The remaining cases are not difficult.For the case of Weyl groups of typeD2n, neither the polynomial model nor the involution model provides a Gelfand model.In this paper the construction of a Gelfand model for a Weyl group of typeD2n will be presented. It will be built upon a light modification of the polynomial model.


## 2. Polynomial Model

The notation introduced in this section will be used in the remaining of this paper.G will denote a finite subgroup of GLn(ℂ) and 𝒫 the polynomial ring ℂ[x1,…,xn].LetIn={1,…,n} be the set of the first n natural numbers and ℳn the set of multi-index functions:
(2.1)Mn={α:In⟶N0}For eachα∈ℳn the following notation will be used in the rest of this paper:
(2.2)αi=α(i),α=(α1,…,αn),|α|=∑i=1nαi.Let𝒜=ℂ〈x1,…,xn,∂1,…,∂n〉 be the Weyl algebra of ℂ-linear differential operators generated by the multiplication operators xi and partial differential operators ∂i=∂/∂xi with 1≤i≤n.It is known that eachD∈𝒜 has a unique expression as a finite sum (see [27]):
(2.3)D=∑λα,βxα∂β,
where α,β∈ℳn, λα,β∈ℂ, and
(2.4)xα=x1α1x2α2⋯xnαn,∂β=∂1β1∂2β2⋯∂nβn.
The degree of D is defined by
(2.5)deg⁡(D)=max⁡{∑i(αi-βi):λα,β≠0}.
The Weyl algebra is a graduated algebra 𝒜=⨁i∈ℤ𝒜i, where
(2.6)Ai={∑α,β∈Mnλα,βxα∂β:|α|-|β|=i}.
The action of G on 𝒫 induces an action of G on the endomorphism ring Endℂ(𝒫), which is defined by
(2.7)(g⋅D)(p)=(gDg-1)(p)(g∈G,D∈EndC(P)).
This action can be restricted to the Weyl algebra 𝒜 noting that each 𝒜i is invariant under the action of G.LetℐG be the subalgebra of G-invariant operators in 𝒜, that is,
(2.8)IG={D∈A:g⋅D=D,∀g∈G}.
Notice that ℐG is contained in the centralizer of G in Endℂ(𝒫).LetℐG- be the subspace of the Weyl algebra, formed by the G-invariant operators with negative degree
(2.9)IG-={D∈IG:deg⁡(D)<0}.Definition 2.1.
Let𝒩G be the subspace of 𝒫 defined by
(2.10)NG={p∈P:D(p)=0,∀D∈IG-}.𝒩G is named the polynomial model of G.Notice that𝒩G is a G-module.Below, some properties of𝒩G will be mentioned.Theorem 2.2.
𝒩
G is a finite-dimensional G-module, and every simple G-module has a copy in 𝒩G.Proof.
See [21, page 38].The analysis of the polynomial model for Coxeter groups has been completely solved by the following theorem.Theorem 2.3.
LetG be a finite irreducible Coxeter group, and let W be its realization as a reflection group. Then, the polynomial model 𝒩W  is a Gelfand model for G if, and only if, W is not a Weyl group of type D2n, E7, or E8.Proof.
See [22, page 7].In the following sections it will be presented a characterization of the polynomial model for the classical Weyl groups of typeAn, Bn and Dn.


### 2.1. Polynomial Model for a Weyl Group of TypeAn

LetG be a Weyl group of type An-1. It is known that G can be presented as the symmetric group 𝔖n.The symmetric group𝔖n acts on the set of multi-index functions ℳn by
(2.11)σ⋅α=α∘σ-1(σ∈Sn,α∈Mn).
This action induces a natural homomorphism from 𝔖n in Aut(𝒫) given by
(2.12)σ(∑α∈Mnλαxα)=∑α∈Mnλαxσ⋅α(λα∈C).


#### 2.1.1.𝔖n-Minimal Orbit

Let𝒪n be the orbit space of 𝔖n in ℳn. It is clear that if two multi-indexes α and β belong to the same orbit γ, then |α| and |β| take the same value, where |α| and |β| are defined by (2.2), and this value will be denoted by |γ|.Definition 2.4.
Two orbitsγ and δ will be called 𝔖n-equivalent, denoted byγ~𝔖nδ, if there exists a bijection φ:ℕ0→ℕ0 such that
(2.13)δ={φ∘α:α∈γ}.Definition 2.5.
An orbitγ will be called 𝔖n-minimal if |γ|≤|δ| for all δ∈𝒪n  such that γ~𝔖nδ.Proposition 2.6.
An orbitγ is 𝔖n-minimal if, and only if, for each α∈γ, there exists a nonnegative integer h such that (1)
Im(α)={0,1,…,h-1},(2)
|α-1(i)|≥|α-1(i+1)| for all 0≤i≤h-1(|α-1(i)| being the cardinal of the set α-1(i)).Proof.
See [4, page 1845].Definition 2.7.
For eachγ∈𝒪n, let Sγ be the subspace of 𝒫 defined by
(2.14)Sγ={∑α∈γλαxα:λα∈C}


#### 2.1.2. The SpaceSγ∂

Let∂ be the operator defined by ∂=∑i=1n∂i, where ∂i are the partial differential operators as above. For each γ∈𝒪n, let Sγ∂ be the subspace defined by
(2.15)Sγ∂={P∈Sγ:∂(P)=0}


#### 2.1.3. The Structure of𝒩𝔖n

Below the main theorem regarding𝒩𝔖n is announced without proof. For further details see [4, page 1850].Theorem 2.8.
S
γ
∂ is an irreducible G-module, and 𝒩𝔖n can be decomposed as
(2.16)NSn=⨁γ∈OSn-minimalSγ∂.
Moreover, 𝒩𝔖n is a Gelfand model of 𝔖n.


### 2.2. Polynomial Model for a Weyl Group of TypeBn

The Gelfand model for a Weyl group of typeBn will be described using the same ideas as the previous section.Let𝒞2={1,-1}⊂ℂ* be the subgroup of order two. The Weyl group ℬn, of type Bn, can be presented as the semidirect product
(2.17)Bn=C2n×sSn,
where 𝒞2n=𝒞2×⋯×𝒞2 and the semidirect product is induced by the natural action of 𝔖n on 𝒞2n:
(2.18)σ⋅(w1,…,wn)=(wσ(1),…,wσ(n))(σ∈Sn,(w1,…,wn)∈C2n).The action of𝔖n on ℳn induces a natural homomorphism from ℬn on Aut(𝒫) given by
(2.19)(w,σ)(∑α∈Mnλαxα)=∑α∈Mnλα(wx)σα(λα∈C)
with
(2.20)(wx)σα=∏i=1n(wixi)(σα)i.


#### 2.2.1.ℬn-Minimal Orbit

Let𝒪n be the orbit space of 𝔖n on ℳn, as above.Definition 2.9.
Two orbits,γ and δ, will be called ℬn-equivalent, denoted byγ~ℬnδ, if there exists a bijection φ:ℕ0→ℕ0 such that φ(k) and k have the same parity for all k∈ℕ0,δ={φ∘α:α∈γ}.Definition 2.10.
An orbitγ will be called ℬn-minimal if |γ|≤|δ| for all δ∈𝒪n such that γ~ℬnδ.Proposition 2.11.
An orbitγ is ℬn-minimal if, and only if, for each α∈γ and each pair i,j∈ℕ0 with the same parity, one has |α-1(i)|≥|α-1(j)| with 0≤i<j (|α-1(i)| being the cardinal of the set α-1(i)).Proof.
See [3, page 365].


#### 2.2.2. The SpaceSγΔ

LetΔ be the Laplacian operator defined by Δ=∑i=1n∂i2, where ∂i are the partial differential operators mentioned above. For each γ∈𝒪n, let SγΔ be the subspace defined by
(2.21)SγΔ={P∈Sγ:Δ(P)=0}.


#### 2.2.3. The Structure of𝒩ℬn

Below the main theorem regarding𝒩ℬn is announced without proof. See references.Theorem 2.12.
S
γ
Δ is an irreducible G-module, and 𝒩ℬn can be decomposed as
(2.22)NBn=⨁γ∈OBn-minimalSγΔ.
Moreover 𝒩ℬn is a Gelfand model of ℬn.Proof.
See [3, page 371].


### 2.3. Polynomial Model for a Weyl Group of TypeDn

Let𝒟n be the Weyl group of type Dn naturally included in ℬn. Using the previous notation, for α∈ℳn the following sets are considered:
(2.23)Eα={i∈In:αiiseven},Oα={i∈In:αiisodd}.It is easy to check that the cardinals|Eα| and |Oα| are equal for all elements in the same orbit γ. Therefore, these values will be denoted by |Eγ| and |Oγ|, respectively.


#### 2.3.1.𝒟n-Minimal Orbit

Definition 2.13.
Two orbitsγ and δ will be called 𝒟n-equivalent,denoted byγ~𝒟nδ, if there exists a bijection φ:ℕ0→ℕ0 such that(1)
∀k∈ℕ0, φ(k) and k have the same parity or φ(k) and k have different parities,(2)
δ={φ∘α:α∈γ}.Definition 2.14.
An orbitγ will be called 𝒟n-minimal if |γ|≤|δ| for all δ∈𝒪n  such that γ~𝒟nδ.Proposition 2.15.
Letγ be an orbit, and then the following statements are true (1)
γ is 𝒟n-minimal if, and only if, the following statements are verified:
(a)
givenα∈γ and i<j∈ℕ0 with the same parity, then |α-1(i)|≥|α-1(j)|,(b)
|Eγ|≤|Oγ|.(2)
Letπ:ℕ0→ℕ0 be the involution given by π(2i)=2i+1 and π(2i+1)=2i. The following assertions are equivalent:
(a)
γ and π∘γ are 𝒟n-minimal orbits,(b)
γ is ℬn-minimal,(c)
π∘γ is ℬn-minimal.(3)
There are at most two𝒟n-minimal orbits equivalent to γ.(4)
Ifn is odd, there is only one 𝒟n-minimal orbit equivalent to γ.(5)
γ and π∘γ are 𝒟n-minimal orbits if, and only if, |Eγ|=|Oγ|.Proof.
See [5, page 106].Proposition 2.16.
Letn be odd, and then the following statements are true. (1)
Ifγ is 𝒟n-minimal, then
(2.24)NBn∩Sγ=NDn∩Sγ
and 𝒩𝒟n∩Sγ is a simple 𝒟n-module.(2)
𝒩𝒟n is a Gelfand model for 𝒟n.(3)
Every simpleℬn-module remains simple when it is considered as a 𝒟n-module by restriction.(4)
By considering𝒩ℬn as a 𝒟n-module by restriction, 𝒩ℬn is isomorphic to 𝒩𝒟n⊕𝒩𝒟n.Proof.
See [5, page 110].Also in [5] it has been proved that if n is even, 𝒩𝒟n is not a Gelfand model for a Weyl group of type Dn. But it does happen that if M is a simple 𝒟n-module, then 𝒩𝒟n contains a copy ofthis, and the multiplicity of M in 𝒩𝒟n is(1)
two, ifM is isomorphic to 𝒩𝒟n∩Sγ,γ being a 𝒟n-minimal orbit such that γ≠π∘γ and |Eγ|=|Oγ|; in this case, as before, π:ℕ0→ℕ0 is the involution given by π(2i)=2i+1 and π(2i+1)=2i,(2)
one, otherwise.


## 2.1. Polynomial Model for a Weyl Group of TypeAn

LetG be a Weyl group of type An-1. It is known that G can be presented as the symmetric group 𝔖n.The symmetric group𝔖n acts on the set of multi-index functions ℳn by
(2.11)σ⋅α=α∘σ-1(σ∈Sn,α∈Mn).
This action induces a natural homomorphism from 𝔖n in Aut(𝒫) given by
(2.12)σ(∑α∈Mnλαxα)=∑α∈Mnλαxσ⋅α(λα∈C).


### 2.1.1.𝔖n-Minimal Orbit

Let𝒪n be the orbit space of 𝔖n in ℳn. It is clear that if two multi-indexes α and β belong to the same orbit γ, then |α| and |β| take the same value, where |α| and |β| are defined by (2.2), and this value will be denoted by |γ|.Definition 2.4.
Two orbitsγ and δ will be called 𝔖n-equivalent, denoted byγ~𝔖nδ, if there exists a bijection φ:ℕ0→ℕ0 such that
(2.13)δ={φ∘α:α∈γ}.Definition 2.5.
An orbitγ will be called 𝔖n-minimal if |γ|≤|δ| for all δ∈𝒪n  such that γ~𝔖nδ.Proposition 2.6.
An orbitγ is 𝔖n-minimal if, and only if, for each α∈γ, there exists a nonnegative integer h such that (1)
Im(α)={0,1,…,h-1},(2)
|α-1(i)|≥|α-1(i+1)| for all 0≤i≤h-1(|α-1(i)| being the cardinal of the set α-1(i)).Proof.
See [4, page 1845].Definition 2.7.
For eachγ∈𝒪n, let Sγ be the subspace of 𝒫 defined by
(2.14)Sγ={∑α∈γλαxα:λα∈C}


### 2.1.2. The SpaceSγ∂

Let∂ be the operator defined by ∂=∑i=1n∂i, where ∂i are the partial differential operators as above. For each γ∈𝒪n, let Sγ∂ be the subspace defined by
(2.15)Sγ∂={P∈Sγ:∂(P)=0}


### 2.1.3. The Structure of𝒩𝔖n

Below the main theorem regarding𝒩𝔖n is announced without proof. For further details see [4, page 1850].Theorem 2.8.
S
γ
∂ is an irreducible G-module, and 𝒩𝔖n can be decomposed as
(2.16)NSn=⨁γ∈OSn-minimalSγ∂.
Moreover, 𝒩𝔖n is a Gelfand model of 𝔖n.


## 2.1.1.𝔖n-Minimal Orbit

Let𝒪n be the orbit space of 𝔖n in ℳn. It is clear that if two multi-indexes α and β belong to the same orbit γ, then |α| and |β| take the same value, where |α| and |β| are defined by (2.2), and this value will be denoted by |γ|.Definition 2.4.
Two orbitsγ and δ will be called 𝔖n-equivalent, denoted byγ~𝔖nδ, if there exists a bijection φ:ℕ0→ℕ0 such that
(2.13)δ={φ∘α:α∈γ}.Definition 2.5.
An orbitγ will be called 𝔖n-minimal if |γ|≤|δ| for all δ∈𝒪n  such that γ~𝔖nδ.Proposition 2.6.
An orbitγ is 𝔖n-minimal if, and only if, for each α∈γ, there exists a nonnegative integer h such that (1)
Im(α)={0,1,…,h-1},(2)
|α-1(i)|≥|α-1(i+1)| for all 0≤i≤h-1(|α-1(i)| being the cardinal of the set α-1(i)).Proof.
See [4, page 1845].Definition 2.7.
For eachγ∈𝒪n, let Sγ be the subspace of 𝒫 defined by
(2.14)Sγ={∑α∈γλαxα:λα∈C}


## 2.1.2. The SpaceSγ∂

Let∂ be the operator defined by ∂=∑i=1n∂i, where ∂i are the partial differential operators as above. For each γ∈𝒪n, let Sγ∂ be the subspace defined by
(2.15)Sγ∂={P∈Sγ:∂(P)=0}


## 2.1.3. The Structure of𝒩𝔖n

Below the main theorem regarding𝒩𝔖n is announced without proof. For further details see [4, page 1850].Theorem 2.8.
S
γ
∂ is an irreducible G-module, and 𝒩𝔖n can be decomposed as
(2.16)NSn=⨁γ∈OSn-minimalSγ∂.
Moreover, 𝒩𝔖n is a Gelfand model of 𝔖n.


## 2.2. Polynomial Model for a Weyl Group of TypeBn

The Gelfand model for a Weyl group of typeBn will be described using the same ideas as the previous section.Let𝒞2={1,-1}⊂ℂ* be the subgroup of order two. The Weyl group ℬn, of type Bn, can be presented as the semidirect product
(2.17)Bn=C2n×sSn,
where 𝒞2n=𝒞2×⋯×𝒞2 and the semidirect product is induced by the natural action of 𝔖n on 𝒞2n:
(2.18)σ⋅(w1,…,wn)=(wσ(1),…,wσ(n))(σ∈Sn,(w1,…,wn)∈C2n).The action of𝔖n on ℳn induces a natural homomorphism from ℬn on Aut(𝒫) given by
(2.19)(w,σ)(∑α∈Mnλαxα)=∑α∈Mnλα(wx)σα(λα∈C)
with
(2.20)(wx)σα=∏i=1n(wixi)(σα)i.


### 2.2.1.ℬn-Minimal Orbit

Let𝒪n be the orbit space of 𝔖n on ℳn, as above.Definition 2.9.
Two orbits,γ and δ, will be called ℬn-equivalent, denoted byγ~ℬnδ, if there exists a bijection φ:ℕ0→ℕ0 such that φ(k) and k have the same parity for all k∈ℕ0,δ={φ∘α:α∈γ}.Definition 2.10.
An orbitγ will be called ℬn-minimal if |γ|≤|δ| for all δ∈𝒪n such that γ~ℬnδ.Proposition 2.11.
An orbitγ is ℬn-minimal if, and only if, for each α∈γ and each pair i,j∈ℕ0 with the same parity, one has |α-1(i)|≥|α-1(j)| with 0≤i<j (|α-1(i)| being the cardinal of the set α-1(i)).Proof.
See [3, page 365].


### 2.2.2. The SpaceSγΔ

LetΔ be the Laplacian operator defined by Δ=∑i=1n∂i2, where ∂i are the partial differential operators mentioned above. For each γ∈𝒪n, let SγΔ be the subspace defined by
(2.21)SγΔ={P∈Sγ:Δ(P)=0}.


### 2.2.3. The Structure of𝒩ℬn

Below the main theorem regarding𝒩ℬn is announced without proof. See references.Theorem 2.12.
S
γ
Δ is an irreducible G-module, and 𝒩ℬn can be decomposed as
(2.22)NBn=⨁γ∈OBn-minimalSγΔ.
Moreover 𝒩ℬn is a Gelfand model of ℬn.Proof.
See [3, page 371].


## 2.2.1.ℬn-Minimal Orbit

Let𝒪n be the orbit space of 𝔖n on ℳn, as above.Definition 2.9.
Two orbits,γ and δ, will be called ℬn-equivalent, denoted byγ~ℬnδ, if there exists a bijection φ:ℕ0→ℕ0 such that φ(k) and k have the same parity for all k∈ℕ0,δ={φ∘α:α∈γ}.Definition 2.10.
An orbitγ will be called ℬn-minimal if |γ|≤|δ| for all δ∈𝒪n such that γ~ℬnδ.Proposition 2.11.
An orbitγ is ℬn-minimal if, and only if, for each α∈γ and each pair i,j∈ℕ0 with the same parity, one has |α-1(i)|≥|α-1(j)| with 0≤i<j (|α-1(i)| being the cardinal of the set α-1(i)).Proof.
See [3, page 365].


## 2.2.2. The SpaceSγΔ

LetΔ be the Laplacian operator defined by Δ=∑i=1n∂i2, where ∂i are the partial differential operators mentioned above. For each γ∈𝒪n, let SγΔ be the subspace defined by
(2.21)SγΔ={P∈Sγ:Δ(P)=0}.


## 2.2.3. The Structure of𝒩ℬn

Below the main theorem regarding𝒩ℬn is announced without proof. See references.Theorem 2.12.
S
γ
Δ is an irreducible G-module, and 𝒩ℬn can be decomposed as
(2.22)NBn=⨁γ∈OBn-minimalSγΔ.
Moreover 𝒩ℬn is a Gelfand model of ℬn.Proof.
See [3, page 371].


## 2.3. Polynomial Model for a Weyl Group of TypeDn

Let𝒟n be the Weyl group of type Dn naturally included in ℬn. Using the previous notation, for α∈ℳn the following sets are considered:
(2.23)Eα={i∈In:αiiseven},Oα={i∈In:αiisodd}.It is easy to check that the cardinals|Eα| and |Oα| are equal for all elements in the same orbit γ. Therefore, these values will be denoted by |Eγ| and |Oγ|, respectively.


### 2.3.1.𝒟n-Minimal Orbit

Definition 2.13.
Two orbitsγ and δ will be called 𝒟n-equivalent,denoted byγ~𝒟nδ, if there exists a bijection φ:ℕ0→ℕ0 such that(1)
∀k∈ℕ0, φ(k) and k have the same parity or φ(k) and k have different parities,(2)
δ={φ∘α:α∈γ}.Definition 2.14.
An orbitγ will be called 𝒟n-minimal if |γ|≤|δ| for all δ∈𝒪n  such that γ~𝒟nδ.Proposition 2.15.
Letγ be an orbit, and then the following statements are true (1)
γ is 𝒟n-minimal if, and only if, the following statements are verified:
(a)
givenα∈γ and i<j∈ℕ0 with the same parity, then |α-1(i)|≥|α-1(j)|,(b)
|Eγ|≤|Oγ|.(2)
Letπ:ℕ0→ℕ0 be the involution given by π(2i)=2i+1 and π(2i+1)=2i. The following assertions are equivalent:
(a)
γ and π∘γ are 𝒟n-minimal orbits,(b)
γ is ℬn-minimal,(c)
π∘γ is ℬn-minimal.(3)
There are at most two𝒟n-minimal orbits equivalent to γ.(4)
Ifn is odd, there is only one 𝒟n-minimal orbit equivalent to γ.(5)
γ and π∘γ are 𝒟n-minimal orbits if, and only if, |Eγ|=|Oγ|.Proof.
See [5, page 106].Proposition 2.16.
Letn be odd, and then the following statements are true. (1)
Ifγ is 𝒟n-minimal, then
(2.24)NBn∩Sγ=NDn∩Sγ
and 𝒩𝒟n∩Sγ is a simple 𝒟n-module.(2)
𝒩𝒟n is a Gelfand model for 𝒟n.(3)
Every simpleℬn-module remains simple when it is considered as a 𝒟n-module by restriction.(4)
By considering𝒩ℬn as a 𝒟n-module by restriction, 𝒩ℬn is isomorphic to 𝒩𝒟n⊕𝒩𝒟n.Proof.
See [5, page 110].Also in [5] it has been proved that if n is even, 𝒩𝒟n is not a Gelfand model for a Weyl group of type Dn. But it does happen that if M is a simple 𝒟n-module, then 𝒩𝒟n contains a copy ofthis, and the multiplicity of M in 𝒩𝒟n is(1)
two, ifM is isomorphic to 𝒩𝒟n∩Sγ,γ being a 𝒟n-minimal orbit such that γ≠π∘γ and |Eγ|=|Oγ|; in this case, as before, π:ℕ0→ℕ0 is the involution given by π(2i)=2i+1 and π(2i+1)=2i,(2)
one, otherwise.


## 2.3.1.𝒟n-Minimal Orbit

Definition 2.13.
Two orbitsγ and δ will be called 𝒟n-equivalent,denoted byγ~𝒟nδ, if there exists a bijection φ:ℕ0→ℕ0 such that(1)
∀k∈ℕ0, φ(k) and k have the same parity or φ(k) and k have different parities,(2)
δ={φ∘α:α∈γ}.Definition 2.14.
An orbitγ will be called 𝒟n-minimal if |γ|≤|δ| for all δ∈𝒪n  such that γ~𝒟nδ.Proposition 2.15.
Letγ be an orbit, and then the following statements are true (1)
γ is 𝒟n-minimal if, and only if, the following statements are verified:
(a)
givenα∈γ and i<j∈ℕ0 with the same parity, then |α-1(i)|≥|α-1(j)|,(b)
|Eγ|≤|Oγ|.(2)
Letπ:ℕ0→ℕ0 be the involution given by π(2i)=2i+1 and π(2i+1)=2i. The following assertions are equivalent:
(a)
γ and π∘γ are 𝒟n-minimal orbits,(b)
γ is ℬn-minimal,(c)
π∘γ is ℬn-minimal.(3)
There are at most two𝒟n-minimal orbits equivalent to γ.(4)
Ifn is odd, there is only one 𝒟n-minimal orbit equivalent to γ.(5)
γ and π∘γ are 𝒟n-minimal orbits if, and only if, |Eγ|=|Oγ|.Proof.
See [5, page 106].Proposition 2.16.
Letn be odd, and then the following statements are true. (1)
Ifγ is 𝒟n-minimal, then
(2.24)NBn∩Sγ=NDn∩Sγ
and 𝒩𝒟n∩Sγ is a simple 𝒟n-module.(2)
𝒩𝒟n is a Gelfand model for 𝒟n.(3)
Every simpleℬn-module remains simple when it is considered as a 𝒟n-module by restriction.(4)
By considering𝒩ℬn as a 𝒟n-module by restriction, 𝒩ℬn is isomorphic to 𝒩𝒟n⊕𝒩𝒟n.Proof.
See [5, page 110].Also in [5] it has been proved that if n is even, 𝒩𝒟n is not a Gelfand model for a Weyl group of type Dn. But it does happen that if M is a simple 𝒟n-module, then 𝒩𝒟n contains a copy ofthis, and the multiplicity of M in 𝒩𝒟n is(1)
two, ifM is isomorphic to 𝒩𝒟n∩Sγ,γ being a 𝒟n-minimal orbit such that γ≠π∘γ and |Eγ|=|Oγ|; in this case, as before, π:ℕ0→ℕ0 is the involution given by π(2i)=2i+1 and π(2i+1)=2i,(2)
one, otherwise.


## 3. Gelfand Model for a Weyl Group of TypeD2n

As before, letℳn={α:In→ℕ0} be the set of multi-index functions. Every α∈ℳn has an associated vector α^∈ℕ0n, which is obtained by reordering α as follows. 						(3.1)α^=(αi1,…,αin)suchthatαi1≥⋯≥αin.
Thus, there is defined an order relationship ⪯ in ℳn given by for all α,β∈ℳn, α⪯β if, and only if, α^=β^ or there exists s(1≤s≤n) such that
(3.2)α^1=β^1,…,α^s-1=β^s-1,α^s<β^s,α^i and β^i being the coordinates of the vectors α^ and β^, respectively. Notice that this is the lexicographic order for ℕ0n.Proposition 3.1.
Letγ∈𝒪n and α,β∈ℳn, and then α,β∈γ if, and only if,  α^=β^.Proof.
Letα,β∈γ, and therefore there exists σ∈𝔖n such that β=σα, which implies βi=ασ-1(i) with 1≤i≤n. Thus, it is easy to see that α^=β^.
On the other hand, letα,β∈ℳn and α^=β^, say,
(3.3)αi1=βj1,…,αin=βjn.
Let σ∈𝔖n be given by
(3.4)σ-1(ik)=jk(1≤k≤n).
Then, β=σα, and hence both multi-indexes belong to the same orbit.From this proposition it is clear that⪯ induces a total orderin 𝒪n, which is defined by
(3.5)γ⪯δ⟺α⪯β(γ,δ∈On,α∈γ,β∈δ).
Since the vector α^ is independent of the choice α in γ, it will be denoted by γ^.Proposition 3.2.
Letα∈ℳn be defined by αi=i-1, and let γ be the orbit of α. Then, γ is the ⪯-maximum of the 𝔖n-minimal orbits and γ^=(n-1,…,1,0).Proof.
From the previous considerations it is clear thatγ is an 𝔖n-minimal orbit and γ^=(n-1,…,1,0).
Now it will be proved thatγ is the ⪯-maximum in the set of 𝔖n-minimal orbits. Let δ be an orbit such that δ≠γ and γ⪯δ. Then, it should exist an s∈In satisfying
(3.6)γ^i=δ^i,∀i<s,γ^s<δ^s,
that is,
(3.7)δ^1=n-1⋯δ^s-1=n-(s-1),δ^s>n-s.
Thus, it occurs thatδ^s=n-(s-1), and from the minimality of δ every number less than n-(s-1) must appear at least twice, which is a contradiction. And therefore γ is the maximum.From now on, for a finite setA, 𝔖A will denote the symmetric group of A, ℳA will denote the set of multi-index functions with domain A,
(3.8)MA={α:A⟶N0},
and ı will denote the function in ℳA given by
(3.9)ı(i)=1,∀i∈A.
As in the case of 𝔖n, there is a natural action from the symmetric group 𝔖A in the set of multi-index functions ℳA, defined by
(3.10)σ⋅α=α∘σ-1(σ∈SA,α∈MA).
It is possible to extend the concept of 𝔖A-minimal orbit.For eachα∈ℳn, let us consider the sets
(3.11)Eα={i∈In:αiiseven}Oα={i∈In:αiisodd}
as defined in the previous section. Then, it is clear that α can be determined from its restrictions αE and αO to the sets Eα and Oα, respectively. Observe that αE∈ℳEα and αO∈ℳOα.Proposition 3.3.
Letα∈ℳn such that α is ℬn-minimal, and then αE/2 is 𝔖Eα-minimal and (αO-1)/2 is 𝔖Oα-minimal.Proof.
It follows from Proposition 4 in [3] and the identities
(3.12)|(αE2)-1(i)|=|α(i)|(∀i∈|Eα|),|(αO-12)-1(j)|=|α(j)|(∀j∈|Oα|).Notation 1.
Let𝒦 be the subset of ℳn given by
(3.13)K={α∈Mn:αisDn-minimal,|Eα|=|Oα|andαE2≺αO-12}.
It will be denoted by ℱ the subset of the polynomial ring 𝒫:
(3.14)F={∑α‍λαxα:λα=0ifα∈K}.
Note that if n is odd, ℱ is equal to 𝒫.Proposition 3.4.
ℱ is a 𝒟n-submodule of the polynomial ring 𝒫.Proof.
It follows from the action of𝒟n given by
(3.15)(w,σ)(∑α‍λαxα)=∑α‍λα(wx)σα=∑α‍±λαxσα
and the fact that α∈𝒦 if, and only if, σα∈𝒦, and it results that ℱ is a 𝒟n-module of 𝒫.Proposition 3.5.
ℱ contains a Gelfand model for the Weyl group of type 𝒟n.Proof.
It is sufficient to prove thatℱ contains a submodule equivalent to the regular module 𝒟n. Effectively, let us consider the polynomial:
(3.16)P(x1,…,xn)=∏i=2n[xi(xi+i-1)∏j=1i-2(xi2-j2)]2n-1[(n-1)!]2∏i=2n∏j=1i-2(i-1+j)(i-1-j).
Thus P is the interpolating polynomial of the orbit of the regular vector v=(0,1,…,n-1), which satisfies
(3.17)P(v)=1,P(τv)=0(∀τ∈Dn,τ≠e).
It will be proved thatP belongs to ℱ. Let λαxα be not a null term of P such that it is 𝒟n-minimal, |Eα|=|Oα|, and αE/2≺(αO-1)/2. As P was defined in (3.16), it is easy to determine that α1=0 and αj>0 for 1<j≤n. Then, (αE/2)1=0, and as αE/2 is 𝔖|Eα|-minimal, it is obtained that
(3.18)αE2^=(|Eα|-1,|Eα|-2,…,0),
which by Proposition 3.2 is maximal, which is a contradiction.
Sinceℱ is a 𝒟n-module, ℱ contains the module generated by the orbit of P, which is isomorphic to the regular module. Hence, ℱ contains a Gelfand model.Notation 2.
Let𝒯 be a subset of the polynomial ring 𝒫 and G a finite subgroup of GLn(ℂ); we will denote by 𝒯0 the subset of 𝒯 defined by
(3.19)T0={p∈T:D(p)=0,∀D∈IG-},ℐG- being the set of differential operators invariant in the algebra of Weyl as it has been defined in (2.9).Proposition 3.6.
LetG be a finite subgroup of GLn(ℂ) and 𝒯 a G-module of the polynomial ring 𝒫 such that 𝒯 contains a model of G, and then 𝒯0 also contains a model of G.Proof.
LetS⊂𝒯 be a simple G-module and suppose that S⊈𝒯0. Then, there exists D∈ℐG- such that D(S)≠0. Because S is simple and D not null, it follows that D is injective. Thus, D(S)≃S. If D(S)⊂𝒯0, the proposition is proved; otherwise the procedure will be repeated. As D is an operator of the Weyl algebra 𝒜 with negative degree, the procedure is finite, that is to say, there exists m∈ℕ such that Dm(S)⊂𝒯0 and Dm(S)≃S.Remark 3.7.
An immediate consequence of this proposition is that the𝒟n-module
(3.20)F0={f∈F:D(f)=0,∀D∈IDn-}
contains a Gelfand model because ℱ is a 𝒟n-module containing a Gelfand model.Remark 3.8.
Notice that ifn is odd, then ℱ0=𝒩Dn; instead, if n is even,
(3.21)F0⊆F∩NDn=⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁γ∈OBn-minimal|Eγ|=|Oγ|SγΔ
that is
(3.22)F0⊆⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁(γO-1^)/2≺γE/2SγΔ+⨁γE/2=(γO-1^)/2SγΔ.Using the result established in item 4 of Proposition2.16 for decomposing 𝒩ℬn, it follows that
(3.23)NBn=⨁γ∈OBn-minimalSγΔ=⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁γ∈OBn-minimal|Eγ|=|Oγ|SγΔ+⨁γ∈OBn-minimal|Eγ|>|Oγ|SγΔ.
Moreover if |Eγ|=|Oγ|(3.24)⨁γ∈OBn-minimal|Eγ|=|Oγ|SγΔ=⨁γE/2≺(γO-1)^/2SγΔ+⨁γE/2=(γO-1)^/2SγΔ+⨁(γO-1)^/2≺γE/2SγΔ.As a consequence of this decomposition the next lemma follows.Lemma 3.9.
The dimension of𝒩ℬn is equal to
(3.25)2dim⁡(⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ)+2dim⁡(⨁(γO-1)^/2≺γE/2SγΔ)+dim⁡(⨁γE/2=(γO-1)^/2SγΔ).Proof.
It results from considering the identity(3.26)dim⁡(SγΔ)=dim⁡(Sπ∘γΔ)
for each γ∈𝒪n. This identity occurs from the fact that SπΔ and Sπ∘γΔ are isomorphic as 𝒟n-modules, see [5].LetG be a group; from now on we will be denote by
(3.27)Inv(G)={g∈G:g2=e}
the set of involutions of the group G.Lemma 3.10.
LetG be a Coxeter group and ℳ a Gelfand model for G. Then,
(3.28)dim⁡(M)=|Inv(G)|.Proof.
It is a consequence from the Fröbenius-Schur indicator and the fact that the representations of a Coxeter group can be realized over the real numbers, see [28].With the purpose to establish the central result of this work, a relationship between the number of involutions ofℬn and the number of involutions of 𝒟n will be given. This will be used in the next theorem.Lemma 3.11.
Ifn is even (n=2k), then
(3.29)2|Inv(Dn)|-|Inv(Bn)|=(2k)!k!.Proof.
Ifσ=(w,π)∈ℬn, with w∈𝒞2n, and π∈𝔖n is an involution, then the cyclic structure of σ looks like
(3.30)(±i1,±j1)⋯(±ir,±jr)(±k1)⋯(±ks),
where In={i1,…ir,j1,…,jr,k1,…,ks}, π=(i1,j1)⋯(ir,jr)(k1)⋯(ks) is the decomposition of π as product of disjoint cycles, and wil=wjl for 1≤l≤r. Thus, the number of involutions of ℬn is
(3.31)|Inv(Bn)|=∑r=0k‍∏j=0r-1(n-2j2)r!2r2n-2r
If r<k, half of the elements belong to 𝒟n and the other half to ℬn-𝒟n, and therefore
(3.32)|Inv(Dn)|=12∑r=0k-1‍∏j=0r-1(n-2j2)r!2r2n-2r+∏j=0k-1(n-2j2)k!2k2n-2k.
Then,
(3.33)2|Inv(Dn)|-|Inv(Bn)|=∏j=0k-1(n-2j2)k!2k2n-2k=(2k)!k!.Theorem 3.12.
TheG-module ℱ0 is a Gelfand model for the group 𝒟n.Proof.
As it has been mentioned above, whenn is odd, ℱ0 is equal to  𝒩𝒟n, and in [5] it has been proved that 𝒩𝒟n is a Gelfand model for the group 𝒟n.
Whenn is even, from the fact ℱ0 contains a Gelfand model, only it is necessary to prove that dim(ℱ0)≤|Inv(𝒟n)|. From identity (3.22), it results that
(3.34)dim⁡(F0)≤dim⁡⁡(⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁(γO-1)^/2≺γE/2SγΔ+⨁γE/2=(γO-1)^/2SγΔ).
By Lemma 3.10, it follows that the dimension of the model 𝒩ℬn is equal to the number of involutions of the group ℬn, and thus by the Lemma 3.9 it results that
(3.35)|Inv(Bn)|=2dim⁡(⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ)+2dim⁡(⨁γE/2≺(γO-1)^/2SγΔ)+dim⁡(⨁γE/2=(γO-1)^/2SγΔ),|Inv(Bn)|+dim⁡(⨁γE/2=(γO-1)^/2SγΔ)=2dim⁡(⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁(γO-1)^/2⪯γE/2SγΔ),12[|Inv(Bn)|+dim⁡(⨁γE/2=(γO-1)^/2SγΔ)]=dim⁡(⨁v∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁(γO-1)^/2⪯γE/2SγΔ),dim⁡(⨁γE/2=(γO-1)^/2SγΔ)=∑γE/2=(γO-1)^/2dim⁡(SγΔ)=∑χ∈Sn(2nn)χ2(1)=(2n)!n!
and then
(3.36)12[|Inv(Bn)|+(2n)!n!]=dim⁡(⨁γ∈OBn-minimal|Eγ|<|Oγ|SγΔ+⨁(γO-1)^/2⪯γE/2SγΔ).
On the other hand, from identity established in Lemma 3.11, it results that
(3.37)|Inv(Dn)|=12[|Inv(Bn)|+(2n)!n!],
and using identities (2.2) and (3.36), it is obtained that
(3.38)dim⁡(F0)≤|Inv(Dn)|.
Therefore, it has been proved that ℱ0 is a Gelfand model for 𝒟n.



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