Abstract

A Gelfand model for a finite group ๐บ is a complex representation of ๐บ, which is isomorphic to the direct sum of all irreducible representations of ๐บ. When ๐บ is isomorphic to a subgroup of ๐บ๐ฟ๐‘›(โ„‚), where โ„‚ is the field of complex numbers, it has been proved that each ๐บ-module over โ„‚ is isomorphic to a ๐บ-submodule in the polynomial ring โ„‚[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›], and taking the space of zeros of certain ๐บ-invariant operators in the Weyl algebra, a finite-dimensional ๐บ-space ๐’ฉ๐บ in โ„‚[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›] can be obtained, which contains all the simple ๐บ-modules over โ„‚. This type of representation has been named polynomial model. It has been proved that when ๐บ is a Coxeter group, the polynomial model is a Gelfand model for ๐บ if, and only if, ๐บ has not an irreducible factor of type ๐ท2๐‘›, ๐ธ7, or ๐ธ8. This paper presents a model of Gelfand for a Weyl group of type ๐ท2๐‘› 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 [2โ€“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 ๐œ’๐ถ๐‘˜โ†‘๐”–๐‘›=๎“๐œ’๐œ†,(1.1) where ๐ถ๐‘˜ is the centralizer of an involution in ๐”–๐‘› with exactly ๐‘˜ fixed points, ๐œ’๐ถ๐‘˜ is a linear character of ๐ถ๐‘˜, and ๐œ’๐œ† is an irreducible character of ๐”–๐‘› associated with the partition ๐œ† of ๐‘› with exactly ๐‘˜ odd terms. From this identity, it follows immediately that ๎“๐‘˜๐œ’๐ถ๐‘˜โ†‘๐”–๐‘›=๎“๐œ†๐œ’๐œ†,(1.2) where the centralizers ๐ถ๐‘˜ are in correspondence with the conjugacy classes of involutions in ๐”–๐‘›.

Later on, this type of models was called an involution model by Baddeley [6]. He also proved that if ๐ป is a finite group that admits an involution model, then so does the semidirect product ๐ป๐‘›ร—๐‘ ๐”–๐‘›.

Baddeleyโ€™s result implies the existence of involution models for classic Weyl groups, with the exception of the group of type ๐ท2๐‘›. An involution model for a Weyl group of type ๐ด๐‘› is presented in [8] by Inglis et al. and for a Weyl group of type ๐ต๐‘› an involution model is shown in [6, 13]. In [6], Baddeley presents an involution model for a Weyl group of type ๐ท2๐‘›+1, and in [14] it is proved that there is no involution model for a Weyl group of type ๐ท2๐‘› with ๐‘›โ‰ฅ2. In [6], it is mentioned that is not difficult to prove that there is an involution model for a Weyl group of type ๐บ2 and that it has been checked using computers the non existence of involution models for exceptional Weyl groups of type ๐น4, ๐ธ6, ๐ธ7, and ๐ธ8. 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 ๐ผ๐‘›2(๐‘›โ‰ฅ3,๐‘›โ‰ 6) and ๐ป3 and presents a conceptual demonstration of the no existence of an involution model for the group of type ๐ป4.

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

A reflection group ๐บ comes equipped with a canonical representation called the geometric representation of ๐บ. The geometric representation induces a natural action of ๐บ on the space of polynomial functions.

Chevalley [17], Shephard and Todd [18], Steinberg [19], and others studied the corresponding action on the space โ„‹๐บ of ๐บ-harmonic polynomials proving that โ„‹๐บ is isomorphic to the regular representation of ๐บ, and thus โ„‹๐บ contains a Gelfand model for ๐บ. 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 ๐บ-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 ๐บโŠ‚๐บ๐ฟ๐‘›(โ„‚) a subspace ๐’ฉ๐บ of the algebra of polynomials โ„‚[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›], defined as zeros of certain ๐บ-invariant differential operators, and have shown ๐’ฉ๐บ contains a Gelfand model of ๐บ. This space, called the polynomial model, is a Gelfand model for some Weyl groups. In [3โ€“5], it was proved that ๐’ฉ๐บ is a Gelfand model for Weyl groups of type ๐ด๐‘›, ๐ต๐‘› and ๐ท2๐‘›+1. Garge and Oesterlรฉ in [22], using the computation of fake degrees of the irreducible characters of a Coxeter group ๐บ, determined that ๐’ฉ๐บ is a Gelfand model of ๐บ if, and only if, ๐บ has not irreducible factors of type ๐ท2๐‘›, ๐ธ7, or ๐ธ8. The fake degrees have been determined due to works of Steinberg [23], when ๐บ is of type ๐ด๐‘›, Lusztig [24], when ๐บ is of type ๐ต๐‘› or ๐ท๐‘›, Beynon and Lusztig [25], when ๐บ is an exceptional Weyl group, Alvis and Lusztig [26], when ๐บ is of type ๐ป4, and Macdonald, when ๐บ is of type ๐น4 (unpublished). The remaining cases are not difficult.

For the case of Weyl groups of type ๐ท2๐‘›, 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 ๐ท2๐‘› 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.

๐บ will denote a finite subgroup of ๐บ๐ฟ๐‘›(โ„‚) and ๐’ซ the polynomial ring โ„‚[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›].

Let ๐ˆ๐‘›={1,โ€ฆ,๐‘›} be the set of the first ๐‘› natural numbers and โ„ณ๐‘› the set of multi-index functions: โ„ณ๐‘›=๎€ฝ๐›ผโˆถ๐ˆ๐‘›โŸถโ„•0๎€พ(2.1)

For each ๐›ผโˆˆโ„ณ๐‘› the following notation will be used in the rest of this paper: ๐›ผ๐‘–๎€ท๐›ผ=๐›ผ(๐‘–),๐›ผ=1,โ€ฆ,๐›ผ๐‘›๎€ธ,|๐›ผ|=๐‘›๎“๐‘–=1๐›ผ๐‘–.(2.2)

Let ๐’œ=โ„‚โŸจ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›,๐œ•1,โ€ฆ,๐œ•๐‘›โŸฉ be the Weyl algebra of โ„‚-linear differential operators generated by the multiplication operators ๐‘ฅ๐‘– and partial differential operators ๐œ•๐‘–=๐œ•/๐œ•๐‘ฅ๐‘– with 1โ‰ค๐‘–โ‰ค๐‘›.

It is known that each ๐ทโˆˆ๐’œ has a unique expression as a finite sum (see [27]): ๎“๐œ†๐ท=๐›ผ,๐›ฝ๐‘ฅ๐›ผ๐œ•๐›ฝ,(2.3) where ๐›ผ,๐›ฝโˆˆโ„ณ๐‘›, ๐œ†๐›ผ,๐›ฝโˆˆโ„‚, and ๐‘ฅ๐›ผ=๐‘ฅ๐›ผ11๐‘ฅ๐›ผ22โ‹ฏ๐‘ฅ๐›ผ๐‘›๐‘›,๐œ•๐›ฝ=๐œ•๐›ฝ11๐œ•๐›ฝ22โ‹ฏ๐œ•๐›ฝ๐‘›๐‘›.(2.4) The degree of ๐ท is defined by ๎ƒฏ๎“deg(๐ท)=max๐‘–๎€ท๐›ผ๐‘–โˆ’๐›ฝ๐‘–๎€ธโˆถ๐œ†๐›ผ,๐›ฝ๎ƒฐ.โ‰ 0(2.5) The Weyl algebra is a graduated algebra โจ๐’œ=๐‘–โˆˆโ„ค๐’œ๐‘–, where ๐’œ๐‘–=โŽงโŽชโŽจโŽชโŽฉ๎“๐›ผ,๐›ฝโˆˆโ„ณ๐‘›๐œ†๐›ผ,๐›ฝ๐‘ฅ๐›ผ๐œ•๐›ฝ||๐›ฝ||โŽซโŽชโŽฌโŽชโŽญ.โˆถ|๐›ผ|โˆ’=๐‘–(2.6) The action of ๐บ on ๐’ซ induces an action of ๐บ on the endomorphism ring Endโ„‚(๐’ซ), which is defined by ๎€ท(๐‘”โ‹…๐ท)(๐‘)=๐‘”๐ท๐‘”โˆ’1๎€ธ๎€ท(๐‘)๐‘”โˆˆ๐บ,๐ทโˆˆEndโ„‚๎€ธ.(๐’ซ)(2.7) This action can be restricted to the Weyl algebra ๐’œ noting that each ๐’œ๐‘– is invariant under the action of G.

Let โ„๐บ be the subalgebra of ๐บ-invariant operators in ๐’œ, that is, โ„๐บ={๐ทโˆˆ๐’œโˆถ๐‘”โ‹…๐ท=๐ท,โˆ€๐‘”โˆˆ๐บ}.(2.8) Notice that โ„๐บ is contained in the centralizer of ๐บ in Endโ„‚(๐’ซ).

Let โ„โˆ’๐บ be the subspace of the Weyl algebra, formed by the ๐บ-invariant operators with negative degree โ„โˆ’๐บ=๎€ฝ๐ทโˆˆโ„๐บ๎€พ.โˆถdeg(๐ท)<0(2.9)

Definition 2.1. Let ๐’ฉ๐บ be the subspace of ๐’ซ defined by ๐’ฉ๐บ=๎€ฝ๐‘โˆˆ๐’ซโˆถ๐ท(๐‘)=0,โˆ€๐ทโˆˆโ„โˆ’๐บ๎€พ.(2.10)๐’ฉ๐บ is named the polynomial model of ๐บ.

Notice that ๐’ฉ๐บ is a ๐บ-module.

Below, some properties of ๐’ฉ๐บ will be mentioned.

Theorem 2.2. ๐’ฉ๐บ is a finite-dimensional ๐บ-module, and every simple ๐บ-module has a copy in ๐’ฉ๐บ.

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 ๐บ be a finite irreducible Coxeter group, and let ๐‘Š be its realization as a reflection group. Then, the polynomial model ๐’ฉ๐‘Š is a Gelfand model for ๐บ if, and only if, ๐‘Š is not a Weyl group of type ๐ท2๐‘›, ๐ธ7, or ๐ธ8.

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 ๐ด๐‘›, ๐ต๐‘› and ๐ท๐‘›.

2.1. Polynomial Model for a Weyl Group of Type ๐ด๐‘›

Let ๐บ be a Weyl group of type ๐ด๐‘›โˆ’1. It is known that ๐บ can be presented as the symmetric group ๐”–๐‘›.

The symmetric group ๐”–๐‘› acts on the set of multi-index functions โ„ณ๐‘› by ๐œŽโ‹…๐›ผ=๐›ผโˆ˜๐œŽโˆ’1๎€ท๐œŽโˆˆ๐”–๐‘›,๐›ผโˆˆโ„ณ๐‘›๎€ธ.(2.11) This action induces a natural homomorphism from ๐”–๐‘› in Aut(๐’ซ) given by ๐œŽโŽ›โŽœโŽœโŽ๎“๐›ผโˆˆโ„ณ๐‘›๐œ†๐›ผ๐‘ฅ๐›ผโŽžโŽŸโŽŸโŽ =๎“๐›ผโˆˆโ„ณ๐‘›๐œ†๐›ผ๐‘ฅ๐œŽโ‹…๐›ผ๎€ท๐œ†๐›ผ๎€ธ.โˆˆโ„‚(2.12)

2.1.1. ๐”–๐‘›-Minimal Orbit

Let ๐’ช๐‘› be the orbit space of ๐”–๐‘› in โ„ณ๐‘›. It is clear that if two multi-indexes ๐›ผ and ๐›ฝ belong to the same orbit ๐›พ, then |๐›ผ| and |๐›ฝ| take the same value, where |๐›ผ| and |๐›ฝ| are defined by (2.2), and this value will be denoted by |๐›พ|.

Definition 2.4. Two orbits ๐›พ and ๐›ฟ will be called ๐”–๐‘›-equivalent, denoted by ๐›พโˆผ๐”–๐‘›๐›ฟ, if there exists a bijection ๐œ‘โˆถโ„•0โ†’โ„•0 such that ๐›ฟ={๐œ‘โˆ˜๐›ผโˆถ๐›ผโˆˆ๐›พ}.(2.13)

Definition 2.5. An orbit ๐›พ will be called ๐”–๐‘›-minimal if |๐›พ|โ‰ค|๐›ฟ| for all ๐›ฟโˆˆ๐’ช๐‘› such that ๐›พโˆผ๐”–๐‘›๐›ฟ.

Proposition 2.6. An orbit ๐›พ is ๐”–๐‘›-minimal if, and only if, for each ๐›ผโˆˆ๐›พ, there exists a nonnegative integer โ„Ž such that (1)Im(๐›ผ)={0,1,โ€ฆ,โ„Žโˆ’1},(2)|๐›ผโˆ’1(๐‘–)|โ‰ฅ|๐›ผโˆ’1(๐‘–+1)| for all 0โ‰ค๐‘–โ‰คโ„Žโˆ’1(|๐›ผโˆ’1(๐‘–)| being the cardinal of the set ๐›ผโˆ’1(๐‘–)).

Proof. See [4, page 1845].

Definition 2.7. For each ๐›พโˆˆ๐’ช๐‘›, let ๐‘†๐›พ be the subspace of ๐’ซ defined by ๐‘†๐›พ=๎ƒฏ๎“๐›ผโˆˆ๐›พ๐œ†๐›ผ๐‘ฅ๐›ผโˆถ๐œ†๐›ผ๎ƒฐโˆˆโ„‚(2.14)

2.1.2. The Space ๐‘†๐œ•๐›พ

Let ๐œ• be the operator defined by โˆ‘๐œ•=๐‘›๐‘–=1๐œ•๐‘–, where ๐œ•๐‘– are the partial differential operators as above. For each ๐›พโˆˆ๐’ช๐‘›, let ๐‘†๐œ•๐›พ be the subspace defined by ๐‘†๐œ•๐›พ=๎€ฝ๐‘ƒโˆˆ๐‘†๐›พ๎€พโˆถ๐œ•(๐‘ƒ)=0(2.15)

2.1.3. The Structure of ๐’ฉ๐”–๐‘›

Below the main theorem regarding ๐’ฉ๐”–๐‘› is announced without proof. For further details see [4, page 1850].

Theorem 2.8. ๐‘†๐œ•๐›พ is an irreducible ๐บ-module, and ๐’ฉ๐”–๐‘› can be decomposed as ๐’ฉ๐”–๐‘›=๎ถ๐›พโˆˆ๐’ช๐”–๐‘›-minimal๐‘†๐œ•๐›พ.(2.16) Moreover, ๐’ฉ๐”–๐‘› is a Gelfand model of ๐”–๐‘›.

2.2. Polynomial Model for a Weyl Group of Type ๐ต๐‘›

The Gelfand model for a Weyl group of type ๐ต๐‘› will be described using the same ideas as the previous section.

Let ๐’ž2={1,โˆ’1}โŠ‚โ„‚โˆ— be the subgroup of order two. The Weyl group โ„ฌ๐‘›, of type ๐ต๐‘›, can be presented as the semidirect product โ„ฌ๐‘›=๐’ž๐‘›2ร—๐‘ ๐”–๐‘›,(2.17) where ๐’ž๐‘›2=๐’ž2ร—โ‹ฏร—๐’ž2 and the semidirect product is induced by the natural action of ๐”–๐‘› on ๐’ž๐‘›2: ๎€ท๐‘ค๐œŽโ‹…1,โ€ฆ,๐‘ค๐‘›๎€ธ=๎€ท๐‘ค๐œŽ(1),โ€ฆ,๐‘ค๐œŽ(๐‘›)๎€ธ๎€ท๐œŽโˆˆ๐”–๐‘›,๎€ท๐‘ค1,โ€ฆ,๐‘ค๐‘›๎€ธโˆˆ๐’ž๐‘›2๎€ธ.(2.18)

The action of ๐”–๐‘› on โ„ณ๐‘› induces a natural homomorphism from โ„ฌ๐‘› on Aut(๐’ซ) given by โŽ›โŽœโŽœโŽ๎“(๐‘ค,๐œŽ)๐›ผโˆˆโ„ณ๐‘›๐œ†๐›ผ๐‘ฅ๐›ผโŽžโŽŸโŽŸโŽ =๎“๐›ผโˆˆโ„ณ๐‘›๐œ†๐›ผ(๐‘ค๐‘ฅ)๐œŽ๐›ผ๎€ท๐œ†๐›ผ๎€ธโˆˆโ„‚(2.19) with (๐‘ค๐‘ฅ)๐œŽ๐›ผ=๐‘›๎‘๐‘–=1๎€ท๐‘ค๐‘–๐‘ฅ๐‘–๎€ธ(๐œŽ๐›ผ)๐‘–.(2.20)

2.2.1. โ„ฌ๐‘›-Minimal Orbit

Let ๐’ช๐‘› be the orbit space of ๐”–๐‘› on โ„ณ๐‘›, as above.

Definition 2.9. Two orbits, ๐›พ and ๐›ฟ, will be called โ„ฌ๐‘›-equivalent, denoted by ๐›พโˆผโ„ฌ๐‘›๐›ฟ, if there exists a bijection ๐œ‘โˆถโ„•0โ†’โ„•0 such that ๐œ‘(๐‘˜) and ๐‘˜ have the same parity for all ๐‘˜โˆˆโ„•0,๐›ฟ={๐œ‘โˆ˜๐›ผโˆถ๐›ผโˆˆ๐›พ}.

Definition 2.10. An orbit ๐›พ will be called โ„ฌ๐‘›-minimal if |๐›พ|โ‰ค|๐›ฟ| for all ๐›ฟโˆˆ๐’ช๐‘› such that ๐›พโˆผโ„ฌ๐‘›๐›ฟ.

Proposition 2.11. An orbit ๐›พ is โ„ฌ๐‘›-minimal if, and only if, for each ๐›ผโˆˆ๐›พ and each pair ๐‘–,๐‘—โˆˆโ„•0 with the same parity, one has |๐›ผโˆ’1(๐‘–)|โ‰ฅ|๐›ผโˆ’1(๐‘—)| with 0โ‰ค๐‘–<๐‘— (|๐›ผโˆ’1(๐‘–)| being the cardinal of the set ๐›ผโˆ’1(๐‘–)).

Proof. See [3, page 365].

2.2.2. The Space ๐‘†ฮ”๐›พ

Let ฮ” be the Laplacian operator defined by โˆ‘ฮ”=๐‘›๐‘–=1๐œ•2๐‘–, where ๐œ•๐‘– are the partial differential operators mentioned above. For each ๐›พโˆˆ๐’ช๐‘›, let ๐‘†ฮ”๐›พ be the subspace defined by ๐‘†ฮ”๐›พ=๎€ฝ๐‘ƒโˆˆ๐‘†๐›พ๎€พ.โˆถฮ”(๐‘ƒ)=0(2.21)

2.2.3. The Structure of ๐’ฉโ„ฌ๐‘›

Below the main theorem regarding ๐’ฉโ„ฌ๐‘› is announced without proof. See references.

Theorem 2.12. ๐‘†ฮ”๐›พ is an irreducible ๐บ-module, and ๐’ฉโ„ฌ๐‘› can be decomposed as ๐’ฉโ„ฌ๐‘›=๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal๐‘†ฮ”๐›พ.(2.22) Moreover ๐’ฉโ„ฌ๐‘› is a Gelfand model of โ„ฌ๐‘›.

Proof. See [3, page 371].

2.3. Polynomial Model for a Weyl Group of Type ๐ท๐‘›

Let ๐’Ÿ๐‘› be the Weyl group of type ๐ท๐‘› naturally included in โ„ฌ๐‘›. Using the previous notation, for ๐›ผโˆˆโ„ณ๐‘› the following sets are considered: ๐ธ๐›ผ=๎€ฝ๐‘–โˆˆ๐ผ๐‘›โˆถ๐›ผ๐‘–iseven๎€พ,๐‘‚๐›ผ=๎€ฝ๐‘–โˆˆ๐ผ๐‘›โˆถ๐›ผ๐‘–isodd๎€พ.(2.23)

It is easy to check that the cardinals |๐ธ๐›ผ| and |๐‘‚๐›ผ| are equal for all elements in the same orbit ๐›พ. Therefore, these values will be denoted by |๐ธ๐›พ| and |๐‘‚๐›พ|, respectively.

2.3.1. ๐’Ÿ๐‘›-Minimal Orbit

Definition 2.13. Two orbits ๐›พ and ๐›ฟ will be called ๐’Ÿ๐‘›-equivalent, denoted by ๐›พโˆผ๐’Ÿ๐‘›๐›ฟ, if there exists a bijection ๐œ‘โˆถโ„•0โ†’โ„•0 such that(1)โˆ€๐‘˜โˆˆโ„•0, ๐œ‘(๐‘˜) and ๐‘˜ have the same parity or ๐œ‘(๐‘˜) and ๐‘˜ have different parities,(2)๐›ฟ={๐œ‘โˆ˜๐›ผโˆถ๐›ผโˆˆ๐›พ}.

Definition 2.14. An orbit ๐›พ will be called ๐’Ÿ๐‘›-minimal if |๐›พ|โ‰ค|๐›ฟ| for all ๐›ฟโˆˆ๐’ช๐‘› such that ๐›พโˆผ๐’Ÿ๐‘›๐›ฟ.

Proposition 2.15. Let ๐›พ be an orbit, and then the following statements are true (1)๐›พ is ๐’Ÿ๐‘›-minimal if, and only if, the following statements are verified:(a)given ๐›ผโˆˆ๐›พ and ๐‘–<๐‘—โˆˆโ„•0 with the same parity, then |๐›ผโˆ’1(๐‘–)|โ‰ฅ|๐›ผโˆ’1(๐‘—)|,(b)|๐ธ๐›พ|โ‰ค|๐‘‚๐›พ|. (2)Let ๐œ‹โˆถโ„•0โ†’โ„•0 be the involution given by ๐œ‹(2๐‘–)=2๐‘–+1 and ๐œ‹(2๐‘–+1)=2๐‘–. The following assertions are equivalent:(a)๐›พ and ๐œ‹โˆ˜๐›พ are ๐’Ÿ๐‘›-minimal orbits,(b)๐›พ is โ„ฌ๐‘›-minimal,(c)๐œ‹โˆ˜๐›พ is โ„ฌ๐‘›-minimal. (3)There are at most two ๐’Ÿ๐‘›-minimal orbits equivalent to ๐›พ.(4)If ๐‘› is odd, there is only one ๐’Ÿ๐‘›-minimal orbit equivalent to ๐›พ.(5)๐›พ and ๐œ‹โˆ˜๐›พ are ๐’Ÿ๐‘›-minimal orbits if, and only if, |๐ธ๐›พ|=|๐‘‚๐›พ|.

Proof. See [5, page 106].

Proposition 2.16. Let ๐‘› be odd, and then the following statements are true. (1)If ๐›พ is ๐’Ÿ๐‘›-minimal, then ๐’ฉโ„ฌ๐‘›โˆฉ๐‘†๐›พ=๐’ฉ๐’Ÿ๐‘›โˆฉ๐‘†๐›พ(2.24) and ๐’ฉ๐’Ÿ๐‘›โˆฉ๐‘†๐›พ is a simple ๐’Ÿ๐‘›-module.(2)๐’ฉ๐’Ÿ๐‘› is a Gelfand model for ๐’Ÿ๐‘›.(3)Every simple โ„ฌ๐‘›-module remains simple when it is considered as a ๐’Ÿ๐‘›-module by restriction.(4)By considering ๐’ฉโ„ฌ๐‘› as a ๐’Ÿ๐‘›-module by restriction, ๐’ฉโ„ฌ๐‘› is isomorphic to ๐’ฉ๐’Ÿ๐‘›โŠ•๐’ฉ๐’Ÿ๐‘›.

Proof. See [5, page 110].

Also in [5] it has been proved that if ๐‘› is even, ๐’ฉ๐’Ÿ๐‘› is not a Gelfand model for a Weyl group of type ๐ท๐‘›. But it does happen that if ๐‘€ is a simple ๐’Ÿ๐‘›-module, then ๐’ฉ๐’Ÿ๐‘› contains a copy ofthis, and the multiplicity of ๐‘€ in ๐’ฉ๐’Ÿ๐‘› is(1)two, if ๐‘€ is isomorphic to ๐’ฉ๐’Ÿ๐‘›โˆฉ๐‘†๐›พ,๐›พ being a ๐’Ÿ๐‘›-minimal orbit such that ๐›พโ‰ ๐œ‹โˆ˜๐›พ and |๐ธ๐›พ|=|๐‘‚๐›พ|; in this case, as before, ๐œ‹โˆถโ„•0โ†’โ„•0 is the involution given by ๐œ‹(2๐‘–)=2๐‘–+1 and ๐œ‹(2๐‘–+1)=2๐‘–,(2)one, otherwise.

3. Gelfand Model for a Weyl Group of Type ๐ท2๐‘›

As before, let โ„ณ๐‘›={๐›ผโˆถ๐ˆ๐‘›โ†’โ„•0} be the set of multi-index functions. Every ๐›ผโˆˆโ„ณ๐‘› has an associated vector ๎๐›ผโˆˆโ„•๐‘›0, which is obtained by reordering ๐›ผ as follows. ๎€ท๐›ผ๎๐›ผ=๐‘–1,โ€ฆ,๐›ผ๐‘–๐‘›๎€ธsuchthat๐›ผ๐‘–1โ‰ฅโ‹ฏโ‰ฅ๐›ผ๐‘–๐‘›.(3.1) Thus, there is defined an order relationship โชฏ in โ„ณ๐‘› given by for all ๐›ผ,๐›ฝโˆˆโ„ณ๐‘›, ๐›ผโชฏ๐›ฝ if, and only if, ฬ‚๐›ฝ๎๐›ผ= or there exists ๐‘ (1โ‰ค๐‘ โ‰ค๐‘›) such that ๎๐›ผ1=ฬ‚๐›ฝ1,โ€ฆ,๎๐›ผ๐‘ โˆ’1=ฬ‚๐›ฝ๐‘ โˆ’1,๎๐›ผ๐‘ <ฬ‚๐›ฝ๐‘ ,(3.2)๎๐›ผ๐‘– and ฬ‚๐›ฝ๐‘– being the coordinates of the vectors ๎๐›ผ and ฬ‚๐›ฝ, respectively. Notice that this is the lexicographic order for โ„•๐‘›0.

Proposition 3.1. Let ๐›พโˆˆ๐’ช๐‘› and ๐›ผ,๐›ฝโˆˆโ„ณ๐‘›, and then ๐›ผ,๐›ฝโˆˆ๐›พ if, and only if, ฬ‚๐›ฝ๎๐›ผ=.

Proof. Let ๐›ผ,๐›ฝโˆˆ๐›พ, and therefore there exists ๐œŽโˆˆ๐”–๐‘› such that ๐›ฝ=๐œŽ๐›ผ, which implies ๐›ฝ๐‘–=๐›ผ๐œŽโˆ’1(๐‘–) with 1โ‰ค๐‘–โ‰ค๐‘›. Thus, it is easy to see that ฬ‚๐›ฝ๎๐›ผ=.
On the other hand, let ๐›ผ,๐›ฝโˆˆโ„ณ๐‘› and ฬ‚๐›ฝ๎๐›ผ=, say, ๐›ผ๐‘–1=๐›ฝ๐‘—1,โ€ฆ,๐›ผ๐‘–๐‘›=๐›ฝ๐‘—๐‘›.(3.3) Let ๐œŽโˆˆ๐”–๐‘› be given by ๐œŽโˆ’1๎€ท๐‘–๐‘˜๎€ธ=๐‘—๐‘˜(1โ‰ค๐‘˜โ‰ค๐‘›).(3.4) Then, ๐›ฝ=๐œŽ๐›ผ, and hence both multi-indexes belong to the same orbit.

From this proposition it is clear that โชฏ induces a total orderin ๐’ช๐‘›, which is defined by ๎€ท๐›พโชฏ๐›ฟโŸบ๐›ผโชฏ๐›ฝ๐›พ,๐›ฟโˆˆ๐’ช๐‘›๎€ธ.,๐›ผโˆˆ๐›พ,๐›ฝโˆˆ๐›ฟ(3.5) Since the vector ๎๐›ผ is independent of the choice ๐›ผ in ๐›พ, it will be denoted by ฬ‚๐›พ.

Proposition 3.2. Let ๐›ผโˆˆโ„ณ๐‘› be defined by ๐›ผ๐‘–=๐‘–โˆ’1, and let ๐›พ be the orbit of ๐›ผ. Then, ๐›พ is the โชฏ-maximum of the ๐”–๐‘›-minimal orbits and ฬ‚๐›พ=(๐‘›โˆ’1,โ€ฆ,1,0).

Proof. From the previous considerations it is clear that ๐›พ is an ๐”–๐‘›-minimal orbit and ฬ‚๐›พ=(๐‘›โˆ’1,โ€ฆ,1,0).
Now it will be proved that ๐›พ is the โชฏ-maximum in the set of ๐”–๐‘›-minimal orbits. Let ๐›ฟ be an orbit such that ๐›ฟโ‰ ๐›พ and ๐›พโชฏ๐›ฟ. Then, it should exist an ๐‘ โˆˆ๐ˆ๐‘› satisfying ฬ‚๐›พ๐‘–=ฬ‚๐›ฟ๐‘–,โˆ€๐‘–<๐‘ ,ฬ‚๐›พ๐‘ <ฬ‚๐›ฟ๐‘ ,(3.6) that is, ฬ‚๐›ฟ1ฬ‚๐›ฟ=๐‘›โˆ’1โ‹ฏ๐‘ โˆ’1ฬ‚๐›ฟ=๐‘›โˆ’(๐‘ โˆ’1),๐‘ >๐‘›โˆ’๐‘ .(3.7)
Thus, it occurs that ฬ‚๐›ฟ๐‘ =๐‘›โˆ’(๐‘ โˆ’1), and from the minimality of ๐›ฟ every number less than ๐‘›โˆ’(๐‘ โˆ’1) must appear at least twice, which is a contradiction. And therefore ๐›พ is the maximum.

From now on, for a finite set ๐ด, ๐”–๐ด will denote the symmetric group of ๐ด, โ„ณ๐ด will denote the set of multi-index functions with domain ๐ด, โ„ณ๐ด=๎€ฝ๐›ผโˆถ๐€โŸถโ„•0๎€พ,(3.8) and ฤฑ will denote the function in โ„ณ๐ด given by ๐šค(๐‘–)=1,โˆ€๐‘–โˆˆ๐ด.(3.9) As in the case of ๐”–๐‘›, there is a natural action from the symmetric group ๐”–๐ด in the set of multi-index functions โ„ณ๐ด, defined by ๐œŽโ‹…๐›ผ=๐›ผโˆ˜๐œŽโˆ’1๎€ท๐œŽโˆˆ๐”–๐ด,๐›ผโˆˆโ„ณ๐ด๎€ธ.(3.10) It is possible to extend the concept of ๐”–๐ด-minimal orbit.

For each ๐›ผโˆˆโ„ณ๐‘›, let us consider the sets ๐ธ๐›ผ=๎€ฝ๐‘–โˆˆ๐ˆ๐‘›โˆถ๐›ผ๐‘–iseven๎€พ๐‘‚๐›ผ=๎€ฝ๐‘–โˆˆ๐ˆ๐‘›โˆถ๐›ผ๐‘–isodd๎€พ(3.11) as defined in the previous section. Then, it is clear that ๐›ผ can be determined from its restrictions ๐›ผ๐ธ and ๐›ผ๐‘‚ to the sets ๐ธ๐›ผ and ๐‘‚๐›ผ, respectively. Observe that ๐›ผ๐ธโˆˆโ„ณ๐ธ๐›ผ and ๐›ผ๐‘‚โˆˆโ„ณ๐‘‚๐›ผ.

Proposition 3.3. Let ๐›ผโˆˆโ„ณ๐‘› such that ๐›ผ is โ„ฌ๐‘›-minimal, and then ๐›ผ๐ธ/2 is ๐”–๐ธ๐›ผ-minimal and (๐›ผ๐‘‚โˆ’1)/2 is ๐”–๐‘‚๐›ผ-minimal.

Proof. It follows from Proposition 4 in [3] and the identities ||||๎‚€๐›ผ๐ธ2๎‚โˆ’1||||=||๐›ผ||๎€ท||๐ธ(๐‘–)(๐‘–)โˆ€๐‘–โˆˆ๐›ผ||๎€ธ,||||๎‚ต๐›ผ๐‘‚โˆ’12๎‚ถโˆ’1||||=||||๎€ท||๐‘‚(๐‘—)๐›ผ(๐‘—)โˆ€๐‘—โˆˆ๐›ผ||๎€ธ.(3.12)

Notation 1. Let ๐’ฆ be the subset of โ„ณ๐‘› given by ๎‚ป๐’ฆ=๐›ผโˆˆโ„ณ๐‘›โˆถ๐›ผis๐’Ÿ๐‘›-minimal,||๐ธ๐›ผ||=||๐‘‚๐›ผ||and๐›ผ๐ธ2โ‰บ๐›ผ๐‘‚โˆ’12๎‚ผ.(3.13) It will be denoted by โ„ฑ the subset of the polynomial ring ๐’ซ: ๎ƒฏ๎“โ„ฑ=๐›ผ๐œ†๐›ผ๐‘ฅ๐›ผโˆถ๐œ†๐›ผ=0if๎ƒฐ.๐›ผโˆˆ๐’ฆ(3.14) Note that if ๐‘› is odd, โ„ฑ is equal to ๐’ซ.

Proposition 3.4. โ„ฑ is a ๐’Ÿ๐‘›-submodule of the polynomial ring ๐’ซ.

Proof. It follows from the action of ๐’Ÿ๐‘› given by ๎ƒฉ๎“(๐‘ค,๐œŽ)๐›ผ๐œ†๐›ผ๐‘ฅ๐›ผ๎ƒช=๎“๐›ผ๐œ†๐›ผ(๐‘ค๐‘ฅ)๐œŽ๐›ผ=๎“๐›ผยฑ๐œ†๐›ผ๐‘ฅ๐œŽ๐›ผ(3.15) and the fact that ๐›ผโˆˆ๐’ฆ if, and only if, ๐œŽ๐›ผโˆˆ๐’ฆ, and it results that โ„ฑ is a ๐’Ÿ๐‘›-module of ๐’ซ.

Proposition 3.5. โ„ฑ contains a Gelfand model for the Weyl group of type ๐’Ÿ๐‘›.

Proof. It is sufficient to prove that โ„ฑ contains a submodule equivalent to the regular module ๐’Ÿ๐‘›. Effectively, let us consider the polynomial: ๐‘ƒ๎€ท๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=โˆ๐‘›๐‘–=2๎‚ƒ๐‘ฅ๐‘–๎€ท๐‘ฅ๐‘–๎€ธโˆ+๐‘–โˆ’1๐‘–โˆ’2๐‘—=1๎€ท๐‘ฅ2๐‘–โˆ’๐‘—2๎€ธ๎‚„2๐‘›โˆ’1[](๐‘›โˆ’1)!2โˆ๐‘›๐‘–=2โˆ๐‘–โˆ’2๐‘—=1.(๐‘–โˆ’1+๐‘—)(๐‘–โˆ’1โˆ’๐‘—)(3.16) Thus ๐‘ƒ is the interpolating polynomial of the orbit of the regular vector ๐‘ฃ=(0,1,โ€ฆ,๐‘›โˆ’1), which satisfies ๐‘ƒ๎€ท(๐‘ฃ)=1,๐‘ƒ(๐œ๐‘ฃ)=0โˆ€๐œโˆˆ๐’Ÿ๐‘›๎€ธ.,๐œโ‰ ๐‘’(3.17)
It will be proved that ๐‘ƒ belongs to โ„ฑ. Let ๐œ†๐›ผ๐‘ฅ๐›ผ be not a null term of ๐‘ƒ such that it is ๐’Ÿ๐‘›-minimal, |๐ธ๐›ผ|=|๐‘‚๐›ผ|, and ๐›ผ๐ธ/2โ‰บ(๐›ผ๐‘‚โˆ’1)/2. As ๐‘ƒ was defined in (3.16), it is easy to determine that ๐›ผ1=0 and ๐›ผ๐‘—>0 for 1<๐‘—โ‰ค๐‘›. Then, (๐›ผ๐ธ/2)1=0, and as ๐›ผ๐ธ/2 is ๐”–|๐ธ๐›ผ|-minimal, it is obtained that ๎‚Š๐›ผ๐ธ2=๎€ท||๐ธ๐›ผ||||๐ธโˆ’1,๐›ผ||๎€ธ,โˆ’2,โ€ฆ,0(3.18) which by Proposition 3.2 is maximal, which is a contradiction.
Since โ„ฑ is a ๐’Ÿ๐‘›-module, โ„ฑ contains the module generated by the orbit of ๐‘ƒ, which is isomorphic to the regular module. Hence, โ„ฑ contains a Gelfand model.

Notation 2. Let ๐’ฏ be a subset of the polynomial ring ๐’ซ and ๐บ a finite subgroup of ๐บ๐ฟ๐‘›(โ„‚); we will denote by ๐’ฏ0 the subset of ๐’ฏ defined by ๐’ฏ0=๎€ฝ๐‘โˆˆ๐’ฏโˆถ๐ท(๐‘)=0,โˆ€๐ทโˆˆโ„โˆ’๐บ๎€พ,(3.19)โ„โˆ’๐บ being the set of differential operators invariant in the algebra of Weyl as it has been defined in (2.9).

Proposition 3.6. Let ๐บ be a finite subgroup of ๐บ๐ฟ๐‘›(โ„‚) and ๐’ฏ a ๐บ-module of the polynomial ring ๐’ซ such that ๐’ฏ contains a model of ๐บ, and then ๐’ฏ0 also contains a model of ๐บ.

Proof. Let ๐‘†โŠ‚๐’ฏ be a simple ๐บ-module and suppose that ๐‘†ฬธโŠ†๐’ฏ0. Then, there exists ๐ทโˆˆโ„โˆ’๐บ such that ๐ท(๐‘†)โ‰ 0. Because ๐‘† is simple and ๐ท not null, it follows that ๐ท is injective. Thus, ๐ท(๐‘†)โ‰ƒ๐‘†. If ๐ท(๐‘†)โŠ‚๐’ฏ0, the proposition is proved; otherwise the procedure will be repeated. As ๐ท is an operator of the Weyl algebra ๐’œ with negative degree, the procedure is finite, that is to say, there exists ๐‘šโˆˆโ„• such that ๐ท๐‘š(๐‘†)โŠ‚๐’ฏ0 and ๐ท๐‘š(๐‘†)โ‰ƒ๐‘†.

Remark 3.7. An immediate consequence of this proposition is that the ๐’Ÿ๐‘›-module โ„ฑ0=๎‚†๐‘“โˆˆโ„ฑโˆถ๐ท(๐‘“)=0,โˆ€๐ทโˆˆโ„โˆ’๐’Ÿ๐‘›๎‚‡(3.20) contains a Gelfand model because โ„ฑ is a ๐’Ÿ๐‘›-module containing a Gelfand model.

Remark 3.8. Notice that if ๐‘› is odd, then โ„ฑ0=๐’ฉ๐ท๐‘›; instead, if ๐‘› is even, โ„ฑ0โŠ†โ„ฑโˆฉ๐’ฉ๐’Ÿ๐‘›=๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||=||๐‘‚๐›พ||๐‘†ฮ”๐›พ(3.21) that is โ„ฑ0โŠ†๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๎‚ต๎„ž๐›พ๐‘‚โˆ’1๎‚ถ/2โ‰บ๐›พ๐ธ/2๐‘†ฮ”๐›พ+๎ถ๐›พ๐ธ๎‚ต/2=๎„ž๐›พ๐‘‚โˆ’1๎‚ถ/2๐‘†ฮ”๐›พ.(3.22)

Using the result established in item 4 of Proposition 2.16 for decomposing ๐’ฉโ„ฌ๐‘›, it follows that ๐’ฉโ„ฌ๐‘›=๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal๐‘†ฮ”๐›พ=๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||=||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||>||๐‘‚๐›พ||๐‘†ฮ”๐›พ.(3.23) Moreover if |๐ธ๐›พ|=|๐‘‚๐›พ|๎ถ๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||=||๐‘‚๐›พ||๐‘†ฮ”๐›พ=๎ถ๐›พ๐ธ/2โ‰บ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พ+๎ถ๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พ+๎ถ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2โ‰บ๐›พ๐ธ/2๐‘†ฮ”๐›พ.(3.24)

As a consequence of this decomposition the next lemma follows.

Lemma 3.9. The dimension of ๐’ฉโ„ฌ๐‘› is equal to โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ2dim๐›พโˆˆ๐’ชโ„ฌ๐‘›-minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ๎„ž๎€ท๐›พ+2dim๐‘‚๎€ธโˆ’1/2โ‰บ๐›พ๐ธ/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ+dim๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.25)

Proof. It results from considering the identity ๎€ท๐‘†dimฮ”๐›พ๎€ธ๎€ท๐‘†=dimฮ”๐œ‹โˆ˜๐›พ๎€ธ(3.26) for each ๐›พโˆˆ๐’ช๐‘›. This identity occurs from the fact that ๐‘†ฮ”๐œ‹ and ๐‘†ฮ”๐œ‹โˆ˜๐›พ are isomorphic as ๐’Ÿ๐‘›-modules, see [5].

Let ๐บ be a group; from now on we will be denote by Inv๎€ฝ(๐บ)=๐‘”โˆˆ๐บโˆถ๐‘”2๎€พ=๐‘’(3.27) the set of involutions of the group ๐บ.

Lemma 3.10. Let ๐บ be a Coxeter group and โ„ณ a Gelfand model for ๐บ. Then, ||dim(โ„ณ)=Inv||.(๐บ)(3.28)

Proof. It is a consequence from the Frรถbenius-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 โ„ฌ๐‘› and the number of involutions of ๐’Ÿ๐‘› will be given. This will be used in the next theorem.

Lemma 3.11. If ๐‘› is even (๐‘›=2๐‘˜), then 2||Inv๎€ท๐’Ÿ๐‘›๎€ธ||โˆ’||Inv๎€ทโ„ฌ๐‘›๎€ธ||=(2๐‘˜)!.๐‘˜!(3.29)

Proof. If ๐œŽ=(๐‘ค,๐œ‹)โˆˆโ„ฌ๐‘›, with ๐‘คโˆˆ๐’ž๐‘›2, and ๐œ‹โˆˆ๐”–๐‘› is an involution, then the cyclic structure of ๐œŽ looks like ๎€ทยฑ๐‘–1,ยฑ๐‘—1๎€ธโ‹ฏ๎€ทยฑ๐‘–๐‘Ÿ,ยฑ๐‘—๐‘Ÿ๎€ธ๎€ทยฑ๐‘˜1๎€ธโ‹ฏ๎€ทยฑ๐‘˜๐‘ ๎€ธ,(3.30) where ๐ˆ๐‘›={๐‘–1,โ€ฆ๐‘–๐‘Ÿ,๐‘—1,โ€ฆ,๐‘—๐‘Ÿ,๐‘˜1,โ€ฆ,k๐‘ }, ๐œ‹=(๐‘–1,๐‘—1)โ‹ฏ(๐‘–๐‘Ÿ,๐‘—๐‘Ÿ)(๐‘˜1)โ‹ฏ(๐‘˜๐‘ ) is the decomposition of ๐œ‹ as product of disjoint cycles, and ๐‘ค๐‘–๐‘™=๐‘ค๐‘—๐‘™ for 1โ‰ค๐‘™โ‰ค๐‘Ÿ. Thus, the number of involutions of โ„ฌ๐‘› is ||Inv๎€ทโ„ฌ๐‘›๎€ธ||=๐‘˜๎“๐‘Ÿ=0โˆ๐‘Ÿโˆ’1๐‘—=0๎€ท2๐‘›โˆ’2๐‘—๎€ธ2๐‘Ÿ!๐‘Ÿ2๐‘›โˆ’2๐‘Ÿ(3.31) If ๐‘Ÿ<๐‘˜, half of the elements belong to ๐’Ÿ๐‘› and the other half to โ„ฌ๐‘›-๐’Ÿ๐‘›, and therefore ||Inv๎€ท๐’Ÿ๐‘›๎€ธ||=12๐‘˜โˆ’1๎“๐‘Ÿ=0โˆ๐‘Ÿโˆ’1๐‘—=0๎€ท2๐‘›โˆ’2๐‘—๎€ธ2๐‘Ÿ!๐‘Ÿ2๐‘›โˆ’2๐‘Ÿ+โˆ๐‘˜โˆ’1๐‘—=0๎€ท2๐‘›โˆ’2๐‘—๎€ธ2๐‘˜!๐‘˜2๐‘›โˆ’2๐‘˜.(3.32) Then, 2||Inv๎€ท๐’Ÿ๐‘›๎€ธ||โˆ’||Inv๎€ทโ„ฌ๐‘›๎€ธ||=โˆ๐‘˜โˆ’1๐‘—=0๎€ท2๐‘›โˆ’2๐‘—๎€ธ2๐‘˜!๐‘˜2๐‘›โˆ’2๐‘˜=(2๐‘˜)!.๐‘˜!(3.33)

Theorem 3.12. The ๐บ-module โ„ฑ0 is a Gelfand model for the group ๐’Ÿ๐‘›.

Proof. As it has been mentioned above, when ๐‘› is odd, โ„ฑ0 is equal to ๐’ฉ๐’Ÿ๐‘›, and in [5] it has been proved that ๐’ฉ๐’Ÿ๐‘› is a Gelfand model for the group ๐’Ÿ๐‘›.
When ๐‘› is even, from the fact โ„ฑ0 contains a Gelfand model, only it is necessary to prove that dim(โ„ฑ0)โ‰ค|Inv(๐’Ÿ๐‘›)|. From identity (3.22), it results that ๎€ทโ„ฑdim0๎€ธโŽ›โŽœโŽœโŽœโŽœโŽ๎ถโ‰คdim๐›พโˆˆ๐’ชโ„ฌ๐‘›โˆ’minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2โ‰บ๐›พ๐ธ/2๐‘†ฮ”๐›พ+๎ถ๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.34) By Lemma 3.10, it follows that the dimension of the model ๐’ฉโ„ฌ๐‘› is equal to the number of involutions of the group โ„ฌ๐‘›, and thus by the Lemma 3.9 it results that ||Inv๎€ทโ„ฌ๐‘›๎€ธ||โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ=2dim๐›พโˆˆ๐’ชโ„ฌ๐‘›โˆ’minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ+2dim๐›พ๐ธ/2โ‰บ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ+dim๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ ,||Inv๎€ทโ„ฌ๐‘›๎€ธ||โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ+dim๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ=2dim๐›พโˆˆ๐’ชโ„ฌ๐‘›โˆ’minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2โชฏ๐›พ๐ธ/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ ,12โŽกโŽขโŽขโŽขโŽขโŽฃ||Inv๎€ทโ„ฌ๐‘›๎€ธ||โŽ›โŽœโŽœโŽœโŽœโŽ๎ถ+dim๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽคโŽฅโŽฅโŽฅโŽฅโŽฆโŽ›โŽœโŽœโŽœโŽœโŽ๎ถ=dim๐‘ฃโˆˆ๐’ชโ„ฌ๐‘›โˆ’minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2โชฏ๐›พ๐ธ/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ๎ถdim๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ =๎“๐›พ๐ธ/2=๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2๎€ท๐‘†dimฮ”๐›พ๎€ธ=๎“๐œ’โˆˆ๐”–๐‘›โŽ›โŽœโŽœโŽ๐‘›โŽžโŽŸโŽŸโŽ ๐œ’2๐‘›2(1)=(2๐‘›)!๐‘›!(3.35) and then 12๎‚ธ||Inv๎€ทโ„ฌ๐‘›๎€ธ||+(2๐‘›)!๎‚นโŽ›โŽœโŽœโŽœโŽœโŽ๎ถ๐‘›!=dim๐›พโˆˆ๐’ชโ„ฌ๐‘›โˆ’minimal||๐ธ๐›พ||<||๐‘‚๐›พ||๐‘†ฮ”๐›พ+๎ถ๎„ž๎€ท๐›พ๐‘‚๎€ธโˆ’1/2โชฏ๐›พ๐ธ/2๐‘†ฮ”๐›พโŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.36) On the other hand, from identity established in Lemma 3.11, it results that ||Inv๎€ท๐’Ÿ๐‘›๎€ธ||=12๎‚ธ||Inv๎€ทโ„ฌ๐‘›๎€ธ||+(2๐‘›)!๎‚น,๐‘›!(3.37) and using identities (2.2) and (3.36), it is obtained that ๎€ทโ„ฑdim0๎€ธโ‰ค||Inv๎€ท๐’Ÿ๐‘›๎€ธ||.(3.38) Therefore, it has been proved that โ„ฑ0 is a Gelfand model for ๐’Ÿ๐‘›.