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ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 658201, 16 pages
doi:10.5402/2012/658201
Research Article

A Gelfand Model for Weyl Groups of Type 𝐷 2 𝑛

José O. Araujo,Β Luis C. Maiarú,Β and Mauro Natale

Departmento de Matemática, Facultad de Ciencias Exactas, UNICEN, B7000 GHG, Tandil, Argentina

Received 27 March 2012; Accepted 17 April 2012

Academic Editors: H.Β Airault, D.Β Sage, A.Β Vourdas, and H.Β You

Copyright Β© 2012 José O. Araujo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A Gelfand model for a finite group 𝐺 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 𝐺 𝐿 𝑛 ( β„‚ ) , where β„‚ is the field of complex numbers, it has been proved that each 𝐺 -module over β„‚ is isomorphic to a 𝐺 -submodule in the polynomial ring β„‚ [ π‘₯ 1 , … , π‘₯ 𝑛 ] , and taking the space of zeros of certain 𝐺 -invariant operators in the Weyl algebra, a finite-dimensional 𝐺 -space 𝒩 𝐺 in β„‚ [ π‘₯ 1 , … , π‘₯ 𝑛 ] can be obtained, which contains all the simple 𝐺 -modules over β„‚ . 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 𝐷 2 𝑛 , 𝐸 7 , or 𝐸 8 . This paper presents a model of Gelfand for a Weyl group of type 𝐷 2 𝑛 whose construction is based on the same principles as the polynomial model.

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 πœ’ 𝐢 π‘˜ ↑ 𝔖 𝑛 =  πœ’ πœ† , ( 1 . 1 ) where 𝐢 π‘˜ is the centralizer of an involution in 𝔖 𝑛 with exactly π‘˜ fixed points, πœ’ 𝐢 π‘˜ is a linear character of 𝐢 π‘˜ , and πœ’ πœ† is an irreducible character of 𝔖 𝑛 associated with the partition πœ† of 𝑛 with exactly π‘˜ odd terms. From this identity, it follows immediately that  π‘˜ πœ’ 𝐢 π‘˜ ↑ 𝔖 𝑛 =  πœ† πœ’ πœ† , ( 1 . 2 ) where the centralizers 𝐢 π‘˜ are in correspondence with the conjugacy classes of involutions in 𝔖 𝑛 .

Later on, this type of models was called an involution 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 𝐻 𝑛 Γ— 𝑠 𝔖 𝑛 .

Baddeley’s result implies the existence of involution models for classic Weyl groups, with the exception of the group of type 𝐷 2 𝑛 . An involution model for a Weyl group of type 𝐴 𝑛 is presented in [8] by Inglis et al. and for a Weyl group of type 𝐡 𝑛 an involution model is shown in [6, 13]. In [6], Baddeley presents an involution model for a Weyl group of type 𝐷 2 𝑛 + 1 , and in [14] it is proved that there is no involution model for a Weyl group of type 𝐷 2 𝑛 with 𝑛 β‰₯ 2 . In [6], it is mentioned that is not difficult to prove that there is an involution model for a Weyl group of type 𝐺 2 and that it has been checked using computers the non existence of involution models for exceptional Weyl groups of type 𝐹 4 , 𝐸 6 , 𝐸 7 , and 𝐸 8 . 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 𝐼 𝑛 2 ( 𝑛 β‰₯ 3 , 𝑛 β‰  6 ) and 𝐻 3 and presents a conceptual demonstration of the no existence of an involution model for the group of type 𝐻 4 .

More recently, in [16] the generalized involution model has been studied in order to include some cases of unitary reflection groups.

A reflection group 𝐺 comes equipped with a canonical representation called the geometric representation of 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 β„‹ 𝐺 of 𝐺 -harmonic polynomials proving that β„‹ 𝐺 is isomorphic to the regular representation of 𝐺 , and thus β„‹ 𝐺 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 βŠ‚ 𝐺 𝐿 𝑛 ( β„‚ ) a subspace 𝒩 𝐺 of the algebra of polynomials β„‚ [ π‘₯ 1 , … , π‘₯ 𝑛 ] , defined as zeros of certain 𝐺 -invariant differential operators, and have shown 𝒩 𝐺 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 𝒩 𝐺 is a Gelfand model for Weyl groups of type 𝐴 𝑛 , 𝐡 𝑛 and 𝐷 2 𝑛 + 1 . Garge and Oesterlé in [22], using the computation of fake degrees of the irreducible characters of a Coxeter group 𝐺 , determined that 𝒩 𝐺 is a Gelfand model of 𝐺 if, and only if, 𝐺 has not irreducible factors of type 𝐷 2 𝑛 , 𝐸 7 , or 𝐸 8 . The fake degrees have been determined due to works of Steinberg [23], when 𝐺 is of type 𝐴 𝑛 , Lusztig [24], when 𝐺 is of type 𝐡 𝑛 or 𝐷 𝑛 , Beynon and Lusztig [25], when 𝐺 is an exceptional Weyl group, Alvis and Lusztig [26], when 𝐺 is of type 𝐻 4 , and Macdonald, when 𝐺 is of type 𝐹 4 (unpublished). The remaining cases are not difficult.

For the case of Weyl groups of type 𝐷 2 𝑛 , 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 type 𝐷 2 𝑛 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 𝐺 𝐿 𝑛 ( β„‚ ) and 𝒫 the polynomial ring β„‚ [ π‘₯ 1 , … , π‘₯ 𝑛 ] .

Let 𝐈 𝑛 = { 1 , … , 𝑛 } be the set of the first 𝑛 natural numbers and β„³ 𝑛 the set of multi-index functions: β„³ 𝑛 = ξ€½ 𝛼 ∢ 𝐈 𝑛 ⟢ β„• 0 ξ€Ύ ( 2 . 1 )

For each 𝛼 ∈ β„³ 𝑛 the following notation will be used in the rest of this paper: 𝛼 𝑖 ξ€· 𝛼 = 𝛼 ( 𝑖 ) , 𝛼 = 1 , … , 𝛼 𝑛 ξ€Έ , | 𝛼 | = 𝑛  𝑖 = 1 𝛼 𝑖 . ( 2 . 2 )

Let π’œ = β„‚ ⟨ π‘₯ 1 , … , π‘₯ 𝑛 , πœ• 1 , … , πœ• 𝑛 ⟩ be the Weyl algebra of β„‚ -linear differential operators generated by the multiplication operators π‘₯ 𝑖 and partial differential operators πœ• 𝑖 = πœ• / πœ• π‘₯ 𝑖 with 1 ≀ 𝑖 ≀ 𝑛 .

It is known that each 𝐷 ∈ π’œ has a unique expression as a finite sum (see [27]):  πœ† 𝐷 = 𝛼 , 𝛽 π‘₯ 𝛼 πœ• 𝛽 , ( 2 . 3 ) where 𝛼 , 𝛽 ∈ β„³ 𝑛 , πœ† 𝛼 , 𝛽 ∈ β„‚ , and π‘₯ 𝛼 = π‘₯ 𝛼 1 1 π‘₯ 𝛼 2 2 β‹― π‘₯ 𝛼 𝑛 𝑛 , πœ• 𝛽 = πœ• 𝛽 1 1 πœ• 𝛽 2 2 β‹― πœ• 𝛽 𝑛 𝑛 . ( 2 . 4 ) The degree of 𝐷 is defined by ξƒ―  d e g ( 𝐷 ) = m a x 𝑖 ξ€· 𝛼 𝑖 βˆ’ 𝛽 𝑖 ξ€Έ ∢ πœ† 𝛼 , 𝛽 ξƒ° . β‰  0 ( 2 . 5 ) The Weyl algebra is a graduated algebra ⨁ π’œ = 𝑖 ∈ β„€ π’œ 𝑖 , where π’œ 𝑖 = ⎧ βŽͺ ⎨ βŽͺ ⎩  𝛼 , 𝛽 ∈ β„³ 𝑛 πœ† 𝛼 , 𝛽 π‘₯ 𝛼 πœ• 𝛽 | | 𝛽 | | ⎫ βŽͺ ⎬ βŽͺ ⎭ . ∢ | 𝛼 | βˆ’ = 𝑖 ( 2 . 6 ) The action of 𝐺 on 𝒫 induces an action of 𝐺 on the endomorphism ring E n d β„‚ ( 𝒫 ) , which is defined by ξ€· ( 𝑔 β‹… 𝐷 ) ( 𝑝 ) = 𝑔 𝐷 𝑔 βˆ’ 1 ξ€Έ ξ€· ( 𝑝 ) 𝑔 ∈ 𝐺 , 𝐷 ∈ E n d β„‚ ξ€Έ . ( 𝒫 ) ( 2 . 7 ) This action can be restricted to the Weyl algebra π’œ noting that each π’œ 𝑖 is invariant under the action of G .

Let ℐ 𝐺 be the subalgebra of 𝐺 -invariant operators in π’œ , that is, ℐ 𝐺 = { 𝐷 ∈ π’œ ∢ 𝑔 β‹… 𝐷 = 𝐷 , βˆ€ 𝑔 ∈ 𝐺 } . ( 2 . 8 ) Notice that ℐ 𝐺 is contained in the centralizer of 𝐺 in E n d β„‚ ( 𝒫 ) .

Let ℐ βˆ’ 𝐺 be the subspace of the Weyl algebra, formed by the 𝐺 -invariant operators with negative degree ℐ βˆ’ 𝐺 = ξ€½ 𝐷 ∈ ℐ 𝐺 ξ€Ύ . ∢ d e g ( 𝐷 ) < 0 ( 2 . 9 )

Definition 2.1. Let 𝒩 𝐺 be the subspace of 𝒫 defined by 𝒩 𝐺 = ξ€½ 𝑝 ∈ 𝒫 ∢ 𝐷 ( 𝑝 ) = 0 , βˆ€ 𝐷 ∈ ℐ βˆ’ 𝐺 ξ€Ύ . ( 2 . 1 0 ) 𝒩 𝐺 is named the polynomial model of 𝐺 .

Notice that 𝒩 𝐺 is a 𝐺 -module.

Below, some properties of 𝒩 𝐺 will be mentioned.

Theorem 2.2. 𝒩 𝐺 is a finite-dimensional 𝐺 -module, and every simple 𝐺 -module has a copy in 𝒩 𝐺 .

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. Let 𝐺 be a finite irreducible Coxeter group, and let π‘Š be its realization as a reflection group. Then, the polynomial model 𝒩 π‘Š is a Gelfand model for 𝐺 if, and only if, π‘Š is not a Weyl group of type 𝐷 2 𝑛 , 𝐸 7 , or 𝐸 8 .

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 type 𝐴 𝑛 , 𝐡 𝑛 and 𝐷 𝑛 .

2.1. Polynomial Model for a Weyl Group of Type 𝐴 𝑛

Let 𝐺 be a Weyl group of type 𝐴 𝑛 βˆ’ 1 . It is known that 𝐺 can be presented as the symmetric group 𝔖 𝑛 .

The symmetric group 𝔖 𝑛 acts on the set of multi-index functions β„³ 𝑛 by 𝜎 β‹… 𝛼 = 𝛼 ∘ 𝜎 βˆ’ 1 ξ€· 𝜎 ∈ 𝔖 𝑛 , 𝛼 ∈ β„³ 𝑛 ξ€Έ . ( 2 . 1 1 ) This action induces a natural homomorphism from 𝔖 𝑛 in A u t ( 𝒫 ) given by 𝜎 βŽ› ⎜ ⎜ ⎝  𝛼 ∈ β„³ 𝑛 πœ† 𝛼 π‘₯ 𝛼 ⎞ ⎟ ⎟ ⎠ =  𝛼 ∈ β„³ 𝑛 πœ† 𝛼 π‘₯ 𝜎 β‹… 𝛼 ξ€· πœ† 𝛼 ξ€Έ . ∈ β„‚ ( 2 . 1 2 )

2.1.1. 𝔖 𝑛 -Minimal Orbit

Let π’ͺ 𝑛 be the orbit space of 𝔖 𝑛 in β„³ 𝑛 . 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 orbits 𝛾 and 𝛿 will be called 𝔖 𝑛 -equivalent, denoted by 𝛾 ∼ 𝔖 𝑛 𝛿 , if there exists a bijection πœ‘ ∢ β„• 0 β†’ β„• 0 such that 𝛿 = { πœ‘ ∘ 𝛼 ∢ 𝛼 ∈ 𝛾 } . ( 2 . 1 3 )

Definition 2.5. An orbit 𝛾 will be called 𝔖 𝑛 -minimal if | 𝛾 | ≀ | 𝛿 | for all 𝛿 ∈ π’ͺ 𝑛 such that 𝛾 ∼ 𝔖 𝑛 𝛿 .

Proposition 2.6. An orbit 𝛾 is 𝔖 𝑛 -minimal if, and only if, for each 𝛼 ∈ 𝛾 , there exists a nonnegative integer β„Ž such that (1) I m ( 𝛼 ) = { 0 , 1 , … , β„Ž βˆ’ 1 } ,(2) | 𝛼 βˆ’ 1 ( 𝑖 ) | β‰₯ | 𝛼 βˆ’ 1 ( 𝑖 + 1 ) | for all 0 ≀ 𝑖 ≀ β„Ž βˆ’ 1 ( | 𝛼 βˆ’ 1 ( 𝑖 ) | being the cardinal of the set 𝛼 βˆ’ 1 ( 𝑖 ) ).

Proof. See [4, page 1845].

Definition 2.7. For each 𝛾 ∈ π’ͺ 𝑛 , let 𝑆 𝛾 be the subspace of 𝒫 defined by 𝑆 𝛾 = ξƒ―  𝛼 ∈ 𝛾 πœ† 𝛼 π‘₯ 𝛼 ∢ πœ† 𝛼 ξƒ° ∈ β„‚ ( 2 . 1 4 )

2.1.2. The Space 𝑆 πœ• 𝛾

Let πœ• be the operator defined by βˆ‘ πœ• = 𝑛 𝑖 = 1 πœ• 𝑖 , where πœ• 𝑖 are the partial differential operators as above. For each 𝛾 ∈ π’ͺ 𝑛 , let 𝑆 πœ• 𝛾 be the subspace defined by 𝑆 πœ• 𝛾 = ξ€½ 𝑃 ∈ 𝑆 𝛾 ξ€Ύ ∢ πœ• ( 𝑃 ) = 0 ( 2 . 1 5 )

2.1.3. The Structure of 𝒩 𝔖 𝑛

Below the main theorem regarding 𝒩 𝔖 𝑛 is announced without proof. For further details see [4, page 1850].

Theorem 2.8. 𝑆 πœ• 𝛾 is an irreducible 𝐺 -module, and 𝒩 𝔖 𝑛 can be decomposed as 𝒩 𝔖 𝑛 =  𝛾 ∈ π’ͺ 𝔖 𝑛 - m i n i m a l 𝑆 πœ• 𝛾 . ( 2 . 1 6 ) Moreover, 𝒩 𝔖 𝑛 is a Gelfand model of 𝔖 𝑛 .

2.2. Polynomial Model for a Weyl Group of Type 𝐡 𝑛

The Gelfand model for a Weyl group of type 𝐡 𝑛 will be described using the same ideas as the previous section.

Let π’ž 2 = { 1 , βˆ’ 1 } βŠ‚ β„‚ βˆ— be the subgroup of order two. The Weyl group ℬ 𝑛 , of type 𝐡 𝑛 , can be presented as the semidirect product ℬ 𝑛 = π’ž 𝑛 2 Γ— 𝑠 𝔖 𝑛 , ( 2 . 1 7 ) where π’ž 𝑛 2 = π’ž 2 Γ— β‹― Γ— π’ž 2 and the semidirect product is induced by the natural action of 𝔖 𝑛 on π’ž 𝑛 2 : ξ€· 𝑀 𝜎 β‹… 1 , … , 𝑀 𝑛 ξ€Έ = ξ€· 𝑀 𝜎 ( 1 ) , … , 𝑀 𝜎 ( 𝑛 ) ξ€Έ ξ€· 𝜎 ∈ 𝔖 𝑛 , ξ€· 𝑀 1 , … , 𝑀 𝑛 ξ€Έ ∈ π’ž 𝑛 2 ξ€Έ . ( 2 . 1 8 )

The action of 𝔖 𝑛 on β„³ 𝑛 induces a natural homomorphism from ℬ 𝑛 on A u t ( 𝒫 ) given by βŽ› ⎜ ⎜ ⎝  ( 𝑀 , 𝜎 ) 𝛼 ∈ β„³ 𝑛 πœ† 𝛼 π‘₯ 𝛼 ⎞ ⎟ ⎟ ⎠ =  𝛼 ∈ β„³ 𝑛 πœ† 𝛼 ( 𝑀 π‘₯ ) 𝜎 𝛼 ξ€· πœ† 𝛼 ξ€Έ ∈ β„‚ ( 2 . 1 9 ) with ( 𝑀 π‘₯ ) 𝜎 𝛼 = 𝑛  𝑖 = 1 ξ€· 𝑀 𝑖 π‘₯ 𝑖 ξ€Έ ( 𝜎 𝛼 ) 𝑖 . ( 2 . 2 0 )

2.2.1. ℬ 𝑛 -Minimal Orbit

Let π’ͺ 𝑛 be the orbit space of 𝔖 𝑛 on β„³ 𝑛 , as above.

Definition 2.9. Two orbits, 𝛾 and 𝛿 , will be called ℬ 𝑛 -equivalent, denoted by 𝛾 ∼ ℬ 𝑛 𝛿 , if there exists a bijection πœ‘ ∢ β„• 0 β†’ β„• 0 such that πœ‘ ( π‘˜ ) and π‘˜ have the same parity for all π‘˜ ∈ β„• 0 , 𝛿 = { πœ‘ ∘ 𝛼 ∢ 𝛼 ∈ 𝛾 } .

Definition 2.10. An orbit 𝛾 will be called ℬ 𝑛 -minimal if | 𝛾 | ≀ | 𝛿 | for all 𝛿 ∈ π’ͺ 𝑛 such that 𝛾 ∼ ℬ 𝑛 𝛿 .

Proposition 2.11. An orbit 𝛾 is ℬ 𝑛 -minimal if, and only if, for each 𝛼 ∈ 𝛾 and each pair 𝑖 , 𝑗 ∈ β„• 0 with the same parity, one has | 𝛼 βˆ’ 1 ( 𝑖 ) | β‰₯ | 𝛼 βˆ’ 1 ( 𝑗 ) | with 0 ≀ 𝑖 < 𝑗 ( | 𝛼 βˆ’ 1 ( 𝑖 ) | being the cardinal of the set 𝛼 βˆ’ 1 ( 𝑖 ) ).

Proof. See [3, page 365].

2.2.2. The Space 𝑆 Ξ” 𝛾

Let Ξ” be the Laplacian operator defined by βˆ‘ Ξ” = 𝑛 𝑖 = 1 πœ• 2 𝑖 , where πœ• 𝑖 are the partial differential operators mentioned above. For each 𝛾 ∈ π’ͺ 𝑛 , let 𝑆 Ξ” 𝛾 be the subspace defined by 𝑆 Ξ” 𝛾 = ξ€½ 𝑃 ∈ 𝑆 𝛾 ξ€Ύ . ∢ Ξ” ( 𝑃 ) = 0 ( 2 . 2 1 )

2.2.3. The Structure of 𝒩 ℬ 𝑛

Below the main theorem regarding 𝒩 ℬ 𝑛 is announced without proof. See references.

Theorem 2.12. 𝑆 Ξ” 𝛾 is an irreducible 𝐺 -module, and 𝒩 ℬ 𝑛 can be decomposed as 𝒩 ℬ 𝑛 =  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l 𝑆 Ξ” 𝛾 . ( 2 . 2 2 ) Moreover 𝒩 ℬ 𝑛 is a Gelfand model of ℬ 𝑛 .

Proof. See [3, page 371].

2.3. Polynomial Model for a Weyl Group of Type 𝐷 𝑛

Let π’Ÿ 𝑛 be the Weyl group of type 𝐷 𝑛 naturally included in ℬ 𝑛 . Using the previous notation, for 𝛼 ∈ β„³ 𝑛 the following sets are considered: 𝐸 𝛼 = ξ€½ 𝑖 ∈ 𝐼 𝑛 ∢ 𝛼 𝑖 i s e v e n ξ€Ύ , 𝑂 𝛼 = ξ€½ 𝑖 ∈ 𝐼 𝑛 ∢ 𝛼 𝑖 i s o d d ξ€Ύ . ( 2 . 2 3 )

It is easy to check that the cardinals | 𝐸 𝛼 | and | 𝑂 𝛼 | are equal for all elements in the same orbit 𝛾 . Therefore, these values will be denoted by | 𝐸 𝛾 | and | 𝑂 𝛾 | , respectively.

2.3.1. π’Ÿ 𝑛 -Minimal Orbit

Definition 2.13. Two orbits 𝛾 and 𝛿 will be called π’Ÿ 𝑛 -equivalent, denoted by 𝛾 ∼ π’Ÿ 𝑛 𝛿 , if there exists a bijection πœ‘ ∢ β„• 0 β†’ β„• 0 such that(1) βˆ€ π‘˜ ∈ β„• 0 , πœ‘ ( π‘˜ ) and π‘˜ have the same parity or πœ‘ ( π‘˜ ) and π‘˜ have different parities,(2) 𝛿 = { πœ‘ ∘ 𝛼 ∢ 𝛼 ∈ 𝛾 } .

Definition 2.14. An orbit 𝛾 will be called π’Ÿ 𝑛 -minimal if | 𝛾 | ≀ | 𝛿 | for all 𝛿 ∈ π’ͺ 𝑛 such that 𝛾 ∼ π’Ÿ 𝑛 𝛿 .

Proposition 2.15. Let 𝛾 be an orbit, and then the following statements are true (1) 𝛾 is π’Ÿ 𝑛 -minimal if, and only if, the following statements are verified:(a)given 𝛼 ∈ 𝛾 and 𝑖 < 𝑗 ∈ β„• 0 with the same parity, then | 𝛼 βˆ’ 1 ( 𝑖 ) | β‰₯ | 𝛼 βˆ’ 1 ( 𝑗 ) | ,(b) | 𝐸 𝛾 | ≀ | 𝑂 𝛾 | . (2)Let πœ‹ ∢ β„• 0 β†’ β„• 0 be the involution given by πœ‹ ( 2 𝑖 ) = 2 𝑖 + 1 and πœ‹ ( 2 𝑖 + 1 ) = 2 𝑖 . The following assertions are equivalent:(a) 𝛾 and πœ‹ ∘ 𝛾 are π’Ÿ 𝑛 -minimal orbits,(b) 𝛾 is ℬ 𝑛 -minimal,(c) πœ‹ ∘ 𝛾 is ℬ 𝑛 -minimal. (3)There are at most two π’Ÿ 𝑛 -minimal orbits equivalent to 𝛾 .(4)If 𝑛 is odd, there is only one π’Ÿ 𝑛 -minimal orbit equivalent to 𝛾 .(5) 𝛾 and πœ‹ ∘ 𝛾 are π’Ÿ 𝑛 -minimal orbits if, and only if, | 𝐸 𝛾 | = | 𝑂 𝛾 | .

Proof. See [5, page 106].

Proposition 2.16. Let 𝑛 be odd, and then the following statements are true. (1)If 𝛾 is π’Ÿ 𝑛 -minimal, then 𝒩 ℬ 𝑛 ∩ 𝑆 𝛾 = 𝒩 π’Ÿ 𝑛 ∩ 𝑆 𝛾 ( 2 . 2 4 ) and 𝒩 π’Ÿ 𝑛 ∩ 𝑆 𝛾 is a simple π’Ÿ 𝑛 -module.(2) 𝒩 π’Ÿ 𝑛 is a Gelfand model for π’Ÿ 𝑛 .(3)Every simple ℬ 𝑛 -module remains simple when it is considered as a π’Ÿ 𝑛 -module by restriction.(4)By considering 𝒩 ℬ 𝑛 as a π’Ÿ 𝑛 -module by restriction, 𝒩 ℬ 𝑛 is isomorphic to 𝒩 π’Ÿ 𝑛 βŠ• 𝒩 π’Ÿ 𝑛 .

Proof. See [5, page 110].

Also in [5] it has been proved that if 𝑛 is even, 𝒩 π’Ÿ 𝑛 is not a Gelfand model for a Weyl group of type 𝐷 𝑛 . But it does happen that if 𝑀 is a simple π’Ÿ 𝑛 -module, then 𝒩 π’Ÿ 𝑛 contains a copy ofthis, and the multiplicity of 𝑀 in 𝒩 π’Ÿ 𝑛 is(1)two, if 𝑀 is isomorphic to 𝒩 π’Ÿ 𝑛 ∩ 𝑆 𝛾 , 𝛾 being a π’Ÿ 𝑛 -minimal orbit such that 𝛾 β‰  πœ‹ ∘ 𝛾 and | 𝐸 𝛾 | = | 𝑂 𝛾 | ; in this case, as before, πœ‹ ∢ β„• 0 β†’ β„• 0 is the involution given by πœ‹ ( 2 𝑖 ) = 2 𝑖 + 1 and πœ‹ ( 2 𝑖 + 1 ) = 2 𝑖 ,(2)one, otherwise.

3. Gelfand Model for a Weyl Group of Type 𝐷 2 𝑛

As before, let β„³ 𝑛 = { 𝛼 ∢ 𝐈 𝑛 β†’ β„• 0 } be the set of multi-index functions. Every 𝛼 ∈ β„³ 𝑛 has an associated vector  𝛼 ∈ β„• 𝑛 0 , which is obtained by reordering 𝛼 as follows. ξ€· 𝛼  𝛼 = 𝑖 1 , … , 𝛼 𝑖 𝑛 ξ€Έ s u c h t h a t 𝛼 𝑖 1 β‰₯ β‹― β‰₯ 𝛼 𝑖 𝑛 . ( 3 . 1 ) Thus, there is defined an order relationship βͺ― in β„³ 𝑛 given by for all 𝛼 , 𝛽 ∈ β„³ 𝑛 , 𝛼 βͺ― 𝛽 if, and only if, Μ‚ 𝛽  𝛼 = or there exists 𝑠 ( 1 ≀ 𝑠 ≀ 𝑛 ) such that  𝛼 1 = Μ‚ 𝛽 1 , … ,  𝛼 𝑠 βˆ’ 1 = Μ‚ 𝛽 𝑠 βˆ’ 1 ,  𝛼 𝑠 < Μ‚ 𝛽 𝑠 , ( 3 . 2 )  𝛼 𝑖 and Μ‚ 𝛽 𝑖 being the coordinates of the vectors  𝛼 and Μ‚ 𝛽 , respectively. Notice that this is the lexicographic order for β„• 𝑛 0 .

Proposition 3.1. Let 𝛾 ∈ π’ͺ 𝑛 and 𝛼 , 𝛽 ∈ β„³ 𝑛 , and then 𝛼 , 𝛽 ∈ 𝛾 if, and only if, Μ‚ 𝛽  𝛼 = .

Proof. Let 𝛼 , 𝛽 ∈ 𝛾 , and therefore there exists 𝜎 ∈ 𝔖 𝑛 such that 𝛽 = 𝜎 𝛼 , which implies 𝛽 𝑖 = 𝛼 𝜎 βˆ’ 1 ( 𝑖 ) with 1 ≀ 𝑖 ≀ 𝑛 . Thus, it is easy to see that Μ‚ 𝛽  𝛼 = .
On the other hand, let 𝛼 , 𝛽 ∈ β„³ 𝑛 and Μ‚ 𝛽  𝛼 = , say, 𝛼 𝑖 1 = 𝛽 𝑗 1 , … , 𝛼 𝑖 𝑛 = 𝛽 𝑗 𝑛 . ( 3 . 3 ) Let 𝜎 ∈ 𝔖 𝑛 be given by 𝜎 βˆ’ 1 ξ€· 𝑖 π‘˜ ξ€Έ = 𝑗 π‘˜ ( 1 ≀ π‘˜ ≀ 𝑛 ) . ( 3 . 4 ) Then, 𝛽 = 𝜎 𝛼 , and hence both multi-indexes belong to the same orbit.

From this proposition it is clear that βͺ― induces a total orderin π’ͺ 𝑛 , which is defined by ξ€· 𝛾 βͺ― 𝛿 ⟺ 𝛼 βͺ― 𝛽 𝛾 , 𝛿 ∈ π’ͺ 𝑛 ξ€Έ . , 𝛼 ∈ 𝛾 , 𝛽 ∈ 𝛿 ( 3 . 5 ) Since the vector  𝛼 is independent of the choice 𝛼 in 𝛾 , it will be denoted by Μ‚ 𝛾 .

Proposition 3.2. Let 𝛼 ∈ β„³ 𝑛 be defined by 𝛼 𝑖 = 𝑖 βˆ’ 1 , and let 𝛾 be the orbit of 𝛼 . Then, 𝛾 is the βͺ― -maximum of the 𝔖 𝑛 -minimal orbits and Μ‚ 𝛾 = ( 𝑛 βˆ’ 1 , … , 1 , 0 ) .

Proof. From the previous considerations it is clear that 𝛾 is an 𝔖 𝑛 -minimal orbit and Μ‚ 𝛾 = ( 𝑛 βˆ’ 1 , … , 1 , 0 ) .
Now it will be proved that 𝛾 is the ⪯-maximum in the set of 𝔖 𝑛 -minimal orbits. Let 𝛿 be an orbit such that 𝛿 β‰  𝛾 and 𝛾 βͺ― 𝛿 . Then, it should exist an 𝑠 ∈ 𝐈 𝑛 satisfying Μ‚ 𝛾 𝑖 = Μ‚ 𝛿 𝑖 , βˆ€ 𝑖 < 𝑠 , Μ‚ 𝛾 𝑠 < Μ‚ 𝛿 𝑠 , ( 3 . 6 ) that is, Μ‚ 𝛿 1 Μ‚ 𝛿 = 𝑛 βˆ’ 1 β‹― 𝑠 βˆ’ 1 Μ‚ 𝛿 = 𝑛 βˆ’ ( 𝑠 βˆ’ 1 ) , 𝑠 > 𝑛 βˆ’ 𝑠 . ( 3 . 7 )
Thus, it occurs that Μ‚ 𝛿 𝑠 = 𝑛 βˆ’ ( 𝑠 βˆ’ 1 ) , and from the minimality of 𝛿 every number less than 𝑛 βˆ’ ( 𝑠 βˆ’ 1 ) must appear at least twice, which is a contradiction. And therefore 𝛾 is the maximum.

From now on, for a finite set 𝐴 , 𝔖 𝐴 will denote the symmetric group of 𝐴 , β„³ 𝐴 will denote the set of multi-index functions with domain 𝐴 , β„³ 𝐴 = ξ€½ 𝛼 ∢ 𝐀 ⟢ β„• 0 ξ€Ύ , ( 3 . 8 ) and ı will denote the function in β„³ 𝐴 given by 𝚀 ( 𝑖 ) = 1 , βˆ€ 𝑖 ∈ 𝐴 . ( 3 . 9 ) As in the case of 𝔖 𝑛 , there is a natural action from the symmetric group 𝔖 𝐴 in the set of multi-index functions β„³ 𝐴 , defined by 𝜎 β‹… 𝛼 = 𝛼 ∘ 𝜎 βˆ’ 1 ξ€· 𝜎 ∈ 𝔖 𝐴 , 𝛼 ∈ β„³ 𝐴 ξ€Έ . ( 3 . 1 0 ) It is possible to extend the concept of 𝔖 𝐴 -minimal orbit.

For each 𝛼 ∈ β„³ 𝑛 , let us consider the sets 𝐸 𝛼 = ξ€½ 𝑖 ∈ 𝐈 𝑛 ∢ 𝛼 𝑖 i s e v e n ξ€Ύ 𝑂 𝛼 = ξ€½ 𝑖 ∈ 𝐈 𝑛 ∢ 𝛼 𝑖 i s o d d ξ€Ύ ( 3 . 1 1 ) as defined in the previous section. Then, it is clear that 𝛼 can be determined from its restrictions 𝛼 𝐸 and 𝛼 𝑂 to the sets 𝐸 𝛼 and 𝑂 𝛼 , respectively. Observe that 𝛼 𝐸 ∈ β„³ 𝐸 𝛼 and 𝛼 𝑂 ∈ β„³ 𝑂 𝛼 .

Proposition 3.3. Let 𝛼 ∈ β„³ 𝑛 such that 𝛼 is ℬ 𝑛 -minimal, and then 𝛼 𝐸 / 2 is 𝔖 𝐸 𝛼 -minimal and ( 𝛼 𝑂 βˆ’ 1 ) / 2 is 𝔖 𝑂 𝛼 -minimal.

Proof. It follows from Proposition 4 in [3] and the identities | | | | ξ‚€ 𝛼 𝐸 2  βˆ’ 1 | | | | = | | 𝛼 | | ξ€· | | 𝐸 ( 𝑖 ) ( 𝑖 ) βˆ€ 𝑖 ∈ 𝛼 | | ξ€Έ , | | | | ξ‚΅ 𝛼 𝑂 βˆ’ 1 2 ξ‚Ά βˆ’ 1 | | | | = | | | | ξ€· | | 𝑂 ( 𝑗 ) 𝛼 ( 𝑗 ) βˆ€ 𝑗 ∈ 𝛼 | | ξ€Έ . ( 3 . 1 2 )

Notation 1. Let 𝒦 be the subset of β„³ 𝑛 given by ξ‚» 𝒦 = 𝛼 ∈ β„³ 𝑛 ∢ 𝛼 i s π’Ÿ 𝑛 - m i n i m a l , | | 𝐸 𝛼 | | = | | 𝑂 𝛼 | | a n d 𝛼 𝐸 2 β‰Ί 𝛼 𝑂 βˆ’ 1 2 ξ‚Ό . ( 3 . 1 3 ) It will be denoted by β„± the subset of the polynomial ring 𝒫 : ξƒ―  β„± = 𝛼 πœ† 𝛼 π‘₯ 𝛼 ∢ πœ† 𝛼 = 0 i f ξƒ° . 𝛼 ∈ 𝒦 ( 3 . 1 4 ) Note that if 𝑛 is odd, β„± is equal to 𝒫 .

Proposition 3.4. β„± is a π’Ÿ 𝑛 -submodule of the polynomial ring 𝒫 .

Proof. It follows from the action of π’Ÿ 𝑛 given by   ( 𝑀 , 𝜎 ) 𝛼 πœ† 𝛼 π‘₯ 𝛼 ξƒͺ =  𝛼 πœ† 𝛼 ( 𝑀 π‘₯ ) 𝜎 𝛼 =  𝛼 Β± πœ† 𝛼 π‘₯ 𝜎 𝛼 ( 3 . 1 5 ) and the fact that 𝛼 ∈ 𝒦 if, and only if, 𝜎 𝛼 ∈ 𝒦 , and it results that β„± is a π’Ÿ 𝑛 -module of 𝒫 .

Proposition 3.5. β„± contains a Gelfand model for the Weyl group of type π’Ÿ 𝑛 .

Proof. It is sufficient to prove that β„± contains a submodule equivalent to the regular module π’Ÿ 𝑛 . Effectively, let us consider the polynomial: 𝑃 ξ€· π‘₯ 1 , … , π‘₯ 𝑛 ξ€Έ = ∏ 𝑛 𝑖 = 2  π‘₯ 𝑖 ξ€· π‘₯ 𝑖 ξ€Έ ∏ + 𝑖 βˆ’ 1 𝑖 βˆ’ 2 𝑗 = 1 ξ€· π‘₯ 2 𝑖 βˆ’ 𝑗 2 ξ€Έ ξ‚„ 2 𝑛 βˆ’ 1 [ ] ( 𝑛 βˆ’ 1 ) ! 2 ∏ 𝑛 𝑖 = 2 ∏ 𝑖 βˆ’ 2 𝑗 = 1 . ( 𝑖 βˆ’ 1 + 𝑗 ) ( 𝑖 βˆ’ 1 βˆ’ 𝑗 ) ( 3 . 1 6 ) Thus 𝑃 is the interpolating polynomial of the orbit of the regular vector 𝑣 = ( 0 , 1 , … , 𝑛 βˆ’ 1 ) , which satisfies 𝑃 ξ€· ( 𝑣 ) = 1 , 𝑃 ( 𝜏 𝑣 ) = 0 βˆ€ 𝜏 ∈ π’Ÿ 𝑛 ξ€Έ . , 𝜏 β‰  𝑒 ( 3 . 1 7 )
It will be proved that 𝑃 belongs to β„± . Let πœ† 𝛼 π‘₯ 𝛼 be not a null term of 𝑃 such that it is π’Ÿ 𝑛 -minimal, | 𝐸 𝛼 | = | 𝑂 𝛼 | , and 𝛼 𝐸 / 2 β‰Ί ( 𝛼 𝑂 βˆ’ 1 ) / 2 . As 𝑃 was defined in (3.16), it is easy to determine that 𝛼 1 = 0 and 𝛼 𝑗 > 0 for 1 < 𝑗 ≀ 𝑛 . Then, ( 𝛼 𝐸 / 2 ) 1 = 0 , and as 𝛼 𝐸 / 2 is 𝔖 | 𝐸 𝛼 | -minimal, it is obtained that ξ‚Š 𝛼 𝐸 2 = ξ€· | | 𝐸 𝛼 | | | | 𝐸 βˆ’ 1 , 𝛼 | | ξ€Έ , βˆ’ 2 , … , 0 ( 3 . 1 8 ) which by Proposition 3.2 is maximal, which is a contradiction.
Since β„± is a π’Ÿ 𝑛 -module, β„± contains the module generated by the orbit of 𝑃 , which is isomorphic to the regular module. Hence, β„± contains a Gelfand model.

Notation 2. Let 𝒯 be a subset of the polynomial ring 𝒫 and 𝐺 a finite subgroup of 𝐺 𝐿 𝑛 ( β„‚ ) ; we will denote by 𝒯 0 the subset of 𝒯 defined by 𝒯 0 = ξ€½ 𝑝 ∈ 𝒯 ∢ 𝐷 ( 𝑝 ) = 0 , βˆ€ 𝐷 ∈ ℐ βˆ’ 𝐺 ξ€Ύ , ( 3 . 1 9 ) ℐ βˆ’ 𝐺 being the set of differential operators invariant in the algebra of Weyl as it has been defined in (2.9).

Proposition 3.6. Let 𝐺 be a finite subgroup of 𝐺 𝐿 𝑛 ( β„‚ ) and 𝒯 a 𝐺 -module of the polynomial ring 𝒫 such that 𝒯 contains a model of 𝐺 , and then 𝒯 0 also contains a model of 𝐺 .

Proof. Let 𝑆 βŠ‚ 𝒯 be a simple 𝐺 -module and suppose that 𝑆 ΜΈ βŠ† 𝒯 0 . Then, there exists 𝐷 ∈ ℐ βˆ’ 𝐺 such that 𝐷 ( 𝑆 ) β‰  0 . Because 𝑆 is simple and 𝐷 not null, it follows that 𝐷 is injective. Thus, 𝐷 ( 𝑆 ) ≃ 𝑆 . If 𝐷 ( 𝑆 ) βŠ‚ 𝒯 0 , the proposition is proved; otherwise the procedure will be repeated. As 𝐷 is an operator of the Weyl algebra π’œ with negative degree, the procedure is finite, that is to say, there exists π‘š ∈ β„• such that 𝐷 π‘š ( 𝑆 ) βŠ‚ 𝒯 0 and 𝐷 π‘š ( 𝑆 ) ≃ 𝑆 .

Remark 3.7. An immediate consequence of this proposition is that the π’Ÿ 𝑛 -module β„± 0 =  𝑓 ∈ β„± ∢ 𝐷 ( 𝑓 ) = 0 , βˆ€ 𝐷 ∈ ℐ βˆ’ π’Ÿ 𝑛  ( 3 . 2 0 ) contains a Gelfand model because β„± is a π’Ÿ 𝑛 -module containing a Gelfand model.

Remark 3.8. Notice that if 𝑛 is odd, then β„± 0 = 𝒩 𝐷 𝑛 ; instead, if 𝑛 is even, β„± 0 βŠ† β„± ∩ 𝒩 π’Ÿ 𝑛 =  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | = | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 ( 3 . 2 1 ) that is β„± 0 βŠ†  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  ξ‚΅ ξ„ž 𝛾 𝑂 βˆ’ 1 ξ‚Ά / 2 β‰Ί 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 +  𝛾 𝐸 ξ‚΅ / 2 = ξ„ž 𝛾 𝑂 βˆ’ 1 ξ‚Ά / 2 𝑆 Ξ” 𝛾 . ( 3 . 2 2 )

Using the result established in item 4 of Proposition 2.16 for decomposing 𝒩 ℬ 𝑛 , it follows that 𝒩 ℬ 𝑛 =  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l 𝑆 Ξ” 𝛾 =  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | = | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | > | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 . ( 3 . 2 3 ) Moreover if | 𝐸 𝛾 | = | 𝑂 𝛾 |  𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | = | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 =  𝛾 𝐸 / 2 β‰Ί ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 +  𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 +  ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 β‰Ί 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 . ( 3 . 2 4 )

As a consequence of this decomposition the next lemma follows.

Lemma 3.9. The dimension of 𝒩 ℬ 𝑛 is equal to βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  2 d i m 𝛾 ∈ π’ͺ ℬ 𝑛 - m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  ξ„ž ξ€· 𝛾 + 2 d i m 𝑂 ξ€Έ βˆ’ 1 / 2 β‰Ί 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  + d i m 𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 3 . 2 5 )

Proof. It results from considering the identity ξ€· 𝑆 d i m Ξ” 𝛾 ξ€Έ ξ€· 𝑆 = d i m Ξ” πœ‹ ∘ 𝛾 ξ€Έ ( 3 . 2 6 ) for each 𝛾 ∈ π’ͺ 𝑛 . This identity occurs from the fact that 𝑆 Ξ” πœ‹ and 𝑆 Ξ” πœ‹ ∘ 𝛾 are isomorphic as π’Ÿ 𝑛 -modules, see [5].

Let 𝐺 be a group; from now on we will be denote by I n v ξ€½ ( 𝐺 ) = 𝑔 ∈ 𝐺 ∢ 𝑔 2 ξ€Ύ = 𝑒 ( 3 . 2 7 ) the set of involutions of the group 𝐺 .

Lemma 3.10. Let 𝐺 be a Coxeter group and β„³ a Gelfand model for 𝐺 . Then, | | d i m ( β„³ ) = I n v | | . ( 𝐺 ) ( 3 . 2 8 )

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 of ℬ 𝑛 and the number of involutions of π’Ÿ 𝑛 will be given. This will be used in the next theorem.

Lemma 3.11. If 𝑛 is even ( 𝑛 = 2 π‘˜ ) , then 2 | | I n v ξ€· π’Ÿ 𝑛 ξ€Έ | | βˆ’ | | I n v ξ€· ℬ 𝑛 ξ€Έ | | = ( 2 π‘˜ ) ! . π‘˜ ! ( 3 . 2 9 )

Proof. If 𝜎 = ( 𝑀 , πœ‹ ) ∈ ℬ 𝑛 , with 𝑀 ∈ π’ž 𝑛 2 , and πœ‹ ∈ 𝔖 𝑛 is an involution, then the cyclic structure of 𝜎 looks like ξ€· Β± 𝑖 1 , Β± 𝑗 1 ξ€Έ β‹― ξ€· Β± 𝑖 π‘Ÿ , Β± 𝑗 π‘Ÿ ξ€Έ ξ€· Β± π‘˜ 1 ξ€Έ β‹― ξ€· Β± π‘˜ 𝑠 ξ€Έ , ( 3 . 3 0 ) where 𝐈 𝑛 = { 𝑖 1 , … 𝑖 π‘Ÿ , 𝑗 1 , … , 𝑗 π‘Ÿ , π‘˜ 1 , … , k 𝑠 } , πœ‹ = ( 𝑖 1 , 𝑗 1 ) β‹― ( 𝑖 π‘Ÿ , 𝑗 π‘Ÿ ) ( π‘˜ 1 ) β‹― ( π‘˜ 𝑠 ) is the decomposition of πœ‹ as product of disjoint cycles, and 𝑀 𝑖 𝑙 = 𝑀 𝑗 𝑙 for 1 ≀ 𝑙 ≀ π‘Ÿ . Thus, the number of involutions of ℬ 𝑛 is | | I n v ξ€· ℬ 𝑛 ξ€Έ | | = π‘˜  π‘Ÿ = 0 ∏ π‘Ÿ βˆ’ 1 𝑗 = 0 ξ€· 2 𝑛 βˆ’ 2 𝑗 ξ€Έ 2 π‘Ÿ ! π‘Ÿ 2 𝑛 βˆ’ 2 π‘Ÿ ( 3 . 3 1 ) If π‘Ÿ < π‘˜ , half of the elements belong to π’Ÿ 𝑛 and the other half to ℬ 𝑛 - π’Ÿ 𝑛 , and therefore | | I n v ξ€· π’Ÿ 𝑛 ξ€Έ | | = 1 2 π‘˜ βˆ’ 1  π‘Ÿ = 0 ∏ π‘Ÿ βˆ’ 1 𝑗 = 0 ξ€· 2 𝑛 βˆ’ 2 𝑗 ξ€Έ 2 π‘Ÿ ! π‘Ÿ 2 𝑛 βˆ’ 2 π‘Ÿ + ∏ π‘˜ βˆ’ 1 𝑗 = 0 ξ€· 2 𝑛 βˆ’ 2 𝑗 ξ€Έ 2 π‘˜ ! π‘˜ 2 𝑛 βˆ’ 2 π‘˜ . ( 3 . 3 2 ) Then, 2 | | I n v ξ€· π’Ÿ 𝑛 ξ€Έ | | βˆ’ | | I n v ξ€· ℬ 𝑛 ξ€Έ | | = ∏ π‘˜ βˆ’ 1 𝑗 = 0 ξ€· 2 𝑛 βˆ’ 2 𝑗 ξ€Έ 2 π‘˜ ! π‘˜ 2 𝑛 βˆ’ 2 π‘˜ = ( 2 π‘˜ ) ! . π‘˜ ! ( 3 . 3 3 )

Theorem 3.12. The 𝐺 -module β„± 0 is a Gelfand model for the group π’Ÿ 𝑛 .

Proof. As it has been mentioned above, when 𝑛 is odd, β„± 0 is equal to 𝒩 π’Ÿ 𝑛 , and in [5] it has been proved that 𝒩 π’Ÿ 𝑛 is a Gelfand model for the group π’Ÿ 𝑛 .
When 𝑛 is even, from the fact β„± 0 contains a Gelfand model, only it is necessary to prove that d i m ( β„± 0 ) ≀ | I n v ( π’Ÿ 𝑛 ) | . From identity (3.22), it results that ξ€· β„± d i m 0 ξ€Έ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  ≀ d i m 𝛾 ∈ π’ͺ ℬ 𝑛 βˆ’ m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 β‰Ί 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 +  𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 3 . 3 4 ) By Lemma 3.10, it follows that the dimension of the model 𝒩 ℬ 𝑛 is equal to the number of involutions of the group ℬ 𝑛 , and thus by the Lemma 3.9 it results that | | I n v ξ€· ℬ 𝑛 ξ€Έ | | βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  = 2 d i m 𝛾 ∈ π’ͺ ℬ 𝑛 βˆ’ m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  + 2 d i m 𝛾 𝐸 / 2 β‰Ί ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  + d i m 𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , | | I n v ξ€· ℬ 𝑛 ξ€Έ | | βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  + d i m 𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  = 2 d i m 𝛾 ∈ π’ͺ ℬ 𝑛 βˆ’ m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 βͺ― 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , 1 2 ⎑ ⎒ ⎒ ⎒ ⎒ ⎣ | | I n v ξ€· ℬ 𝑛 ξ€Έ | | βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  + d i m 𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  = d i m 𝑣 ∈ π’ͺ ℬ 𝑛 βˆ’ m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 βͺ― 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  d i m 𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ =  𝛾 𝐸 / 2 = ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 ξ€· 𝑆 d i m Ξ” 𝛾 ξ€Έ =  πœ’ ∈ 𝔖 𝑛 βŽ› ⎜ ⎜ ⎝ 𝑛 ⎞ ⎟ ⎟ ⎠ πœ’ 2 𝑛 2 ( 1 ) = ( 2 𝑛 ) ! 𝑛 ! ( 3 . 3 5 ) and then 1 2 ξ‚Έ | | I n v ξ€· ℬ 𝑛 ξ€Έ | | + ( 2 𝑛 ) ! ξ‚Ή βŽ› ⎜ ⎜ ⎜ ⎜ ⎝  𝑛 ! = d i m 𝛾 ∈ π’ͺ ℬ 𝑛 βˆ’ m i n i m a l | | 𝐸 𝛾 | | < | | 𝑂 𝛾 | | 𝑆 Ξ” 𝛾 +  ξ„ž ξ€· 𝛾 𝑂 ξ€Έ βˆ’ 1 / 2 βͺ― 𝛾 𝐸 / 2 𝑆 Ξ” 𝛾 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 3 . 3 6 ) On the other hand, from identity established in Lemma 3.11, it results that | | I n v ξ€· π’Ÿ 𝑛 ξ€Έ | | = 1 2 ξ‚Έ | | I n v ξ€· ℬ 𝑛 ξ€Έ | | + ( 2 𝑛 ) ! ξ‚Ή , 𝑛 ! ( 3 . 3 7 ) and using identities (2.2) and (3.36), it is obtained that ξ€· β„± d i m 0 ξ€Έ ≀ | | I n v ξ€· π’Ÿ 𝑛 ξ€Έ | | . ( 3 . 3 8 ) Therefore, it has been proved that β„± 0 is a Gelfand model for π’Ÿ 𝑛 .

References

  1. I. Bernstein, I. Gelfand, and S. Gelfand, β€œModels of representations of Lie groups,” Selecta Mathematica Sovietica, vol. 1, no. 2, pp. 121–132, 1981.
  2. R. Adin, A. Postnikov, and Y. Roichman, β€œA Gelfand model for wreath products,” Israel Journal of Mathematics, vol. 179, pp. 381–402, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. J. O. Araujo, β€œA Gel'fand model for a Weyl group of type Bn,” Contributions to Algebra and Geometry, vol. 44, no. 2, pp. 359–373, 2003.
  4. J. O. Araujo and J. L. Aguado, β€œA Gel'fand model for the symmetric group,” Communications in Algebra, vol. 29, no. 4, pp. 1841–1851, 2001. View at Publisher Β· View at Google Scholar
  5. J. O. Araujo and J. J. Bigeón, β€œA Gelfand model for a Weyl group of type Dn and the branching rules Dn↪Bn,” Journal of Algebra, vol. 294, no. 1, pp. 97–116, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. R. Baddeley, β€œModels and involution models for wreath products and certain Weyl groups,” Journal of the London Mathematical Society, vol. 44, no. 1, pp. 55–74, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. R. Howlett and C. Zworestine, On Klyachkos Model for the Representations of Finite Linear Groups, Springer, Berlin, Germany, 2000.
  8. N. F. J. Inglis, Richardson, and J. Saxl, β€œAn explicit model for the complex representations of Sn,” Archiv der Mathematik, vol. 54, no. 3, pp. 258–259, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. N. F. J. Inglis and J. Saxl, β€œAn explicit model for the complex representations of the finite general linear groups,” Archiv der Mathematik, vol. 57, no. 5, pp. 424–431, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. A. A. Klyachko, β€œModels for the complex representations of the groups GL(n, q),” Matematicheskiĭ Sbornik, vol. 48, no. 2, pp. 365–379, 1984. View at Publisher Β· View at Google Scholar
  11. V. Kodiyalam and D. N. Verma, β€œA natural representation model for symmetric groups,” 2006, http://arxiv.org/abs/math/0402216v1.
  12. J. Pantoja and J. Soto-Andrade, β€œFonctions sphériques et modèles de Gel'fand pour le groupe de mouvements rigides d'un espace para-euclidien sur un corps local,” Comptes Rendus des Séances de l'Académie des Sciences, vol. 302, no. 13, pp. 463–466, 1986.
  13. P. D. Ryan, β€œRepresentations of Weyl groups of type B induced from centralisers of involutions,” Bulletin of the Australian Mathematical Society, vol. 44, no. 2, pp. 337–344, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. R. Baddeley, β€œSome multiplicity-free characters of finite groups,” [Ph.D. Thesis], Cambridge, UK, 1991.
  15. C. R. Vinroot, β€œInvolution models of finite Coxeter groups,” Journal of Group Theory, vol. 11, no. 3, pp. 333–340, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. E. Marberg, β€œAutomorphisms and generalized involution models of finite complex reflection groups,” Journal of Algebra, vol. 334, no. 1, pp. 295–320, 2011. View at Publisher Β· View at Google Scholar
  17. C. Chevalley, β€œInvariants of finite groups generated by reflections,” American Journal of Mathematics, vol. 77, pp. 778–782, 1955. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. G. C. Shephard and J. A. Todd, β€œFinite unitary reflection groups,” Canadian Journal of Mathematics, vol. 6, pp. 274–304, 1954. View at Zentralblatt MATH
  19. R. Steinberg, β€œInvariants of finite reflection groups,” Canadian Journal of Mathematics, vol. 12, pp. 606–615, 1960. View at Zentralblatt MATH
  20. I. G. Macdonald, β€œSome irreducible representations of Weyl groups,” The Bulletin of the London Mathematical Society, vol. 4, pp. 148–150, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  21. J. O. Araujo and J. L. Aguado, β€œRepresentations of finite groups on polynomial rings,” in Actas V Congreso de Matemáticas “Dr. Antonio A. R. Monteiro”, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1999.
  22. S. M. Garge and J. Oesterlé, β€œOn Gelfand models for finite Coxeter groups,” Journal of Group Theory, vol. 13, no. 3, pp. 429–439, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  23. R. Steinberg, β€œA geometric approach to the representations of the full linear group over a Galois field,” Transactions of the American Mathematical Society, vol. 71, pp. 274–282, 1951. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  24. G. Lusztig, β€œIrreducible representations of finite classical groups,” Inventiones Mathematicae, vol. 43, no. 2, pp. 125–175, 1977. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  25. W. M. Beynon and G. Lusztig, β€œSome numerical results on the characters of exceptional Weyl groups,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 84, no. 3, pp. 417–426, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. D. Alvis and G. Lusztig, β€œThe representations and generic degrees of the Hecke algebra of type H4,” Journal fur die Reine und Angewandte Mathematik, vol. 336, pp. 201–212, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  27. J. E. Björk, Ring of Differential Operators, North-Holland, 1979.
  28. D. Bessis, β€œSur le corps de définition d'un groupe de réflexions complexe,” Communications in Algebra, vol. 25, no. 8, pp. 2703–2716, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH