Calogero–Moser cells of dihedral groups at equal parameters

We prove that Calogero–Moser cells coincide with Kazhdan–Lusztig cells for dihedral groups in the equal parameter case. Cellules de Calogero-Moser des groupes diédraux à paramètres égaux Résumé Nous montrons que les cellules de Calogero-Moser coïncident avec les cellules de Kazhdan-Lusztig pour les groupes diédraux dans le cas des paramètres égaux.


Introduction
Calogero-Moser cells have been defined by Rouquier and the first author for any finite complex reflection group and any parameter, based on ramification theory for Calogero-Moser spaces [4,5].It is conjectured that, for Coxeter groups, Calogero-Moser cells coincide with Kazhdan-Lusztig cells [4, Conj.3.1 and 3.2], [5, Conj.LR and L], which were defined by Kazhdan-Lusztig [7] in the equal parameter case and by Lusztig [8] in the general case.The aim of this paper is to prove this conjecture for dihedral groups in the equal parameter case.
For Calogero-Moser left cells, an alternative (and partially conjectural) definition is proposed in [5,Theo. 13.3.2],based on Gaudin operators.This definition is recalled in Section 5.This is the point of view we adopt in this paper: in the relatively small case of dihedral groups, an explicit diagonalization of these operators is possible, and the computation of Calogero-Moser left cells becomes easy.

Setup
Let  be a finite dimensional Euclidean real vector space, whose positive definite symmetric bilinear form is denoted by ( • , • ), and let  be a finite subgroup of the orthogonal group O() generated by reflections.For  ∈ , we denote by  * the element of the dual space  * defined by  * () = (, ) for all  ∈ .The map  →  * ,  ↦ →  * is a -equivariant isomorphism of vector spaces.
Then Φ = −Φ, and we fix a subset Δ of Φ of cardinality dim RΦ such that every element of Φ belongs to ∈Δ R ⩾0  or to ∈Δ R ⩽0 .We set  = {  |  ∈ Δ}, so that (, ) is a finite Coxeter system.We set so that Φ = Φ + ∪ Φ − , where ∪ means disjoint union.We set We denote by  0 the longest element of  (with respect to the length function ℓ :  → Z ⩾0 defined by the choice of ).Then  0 is an involution and  0 ( 0 ) = − 0 . (2.1) We set Then ℭ is the fundamental chamber of  associated with , and  0 ∈ ℭ. Recall that its closure is a fundamental domain for the action of  on .We denote by Reg  the character afforded by the regular representation and Irr() denotes the set of irreducible characters of .We denote by 1  the trivial character of  and we set  :  →  2 = {±1},  ↦ → det().We denote by C R the vector space of maps  :

Recollection about Kazhdan-Lusztig cells
Let  ∈ C R .To the datum (, , ) are associated three partitions of  into Kazhdan-Lusztig left, right, and two-sided -cells (see for instance [2,Chap. 6]).To each Kazhdan-Lusztig left -cell  is associated a Kazhdan-Lusztig -cellular character that is denoted by  ,KL  .Then where  runs over the set of Kazhdan-Lusztig left -cells.
On the other hand, to each Kazhdan-Lusztig two-sided -cell Γ of  is associated a subset Irr ,KL Γ () called the Kazhdan-Lusztig -family associated with Γ.They form a partition of Irr(): where Γ runs over the set of Kazhdan In the above statement (g),  •  denotes the element of C R defined by ( • )  = ()  .

Gaudin operators
For  ∈ ,  ∈  reg , and  ′ ∈ , we define an endomorphism  ,, ′  of the underlying vector space of the group algebra R by the following formula [5, §13.2]:By the work of Mukhin-Tarasov-Varchenko [9], [10,Coro. 7.4], this conjecture is known to hold in type .Here is a weaker form of this conjecture.Conjecture 4.2.If  ∈ C R and ,  ′ ∈ R >0 , then the family D ,   0 ,  ′  0 has simple spectrum.
We will prove in this paper that this weaker form holds if  is dihedral and  is constant (which is the so-called "equal parameter case").
Example 4.3.The matrix of the endomorphism  0,, ′  in the canonical basis of R is diagonal, and so its spectrum can be easily computed.We get In particular, D 0,, ′ has simple spectrum if and only if  ′ ∈  reg .
We conclude this subsection by some relations between Gaudin operators.For  ∈ , we denote by   (resp.  ) the automorphism of the R-vector space R defined by left (resp.right) multiplication by  (resp. −1 ).If  :  →  2 is a linear character, we denote by  • the automorphism of the R-algebra R defined by  • () = () for all  ∈ .The following formulas are straightforward:

Calogero-Moser cellular characters
The operator  , 0 ,0  commutes with left multiplication by R.So each subspace  , 0 ,0  inherits a structure of R-module: we denote by    the character afforded by this R-module.We define the Calogero-Moser -cellular characters to be the characters of the form    for some  ∈ S , 0 ,0 .Note that we may have In particular, every irreducible character of  occurs in some Calogero-Moser -cellular character.
Remark 5.1.The family D , 0 ,0 does not have a simple spectrum in general.Indeed, if  is not abelian, then an irreducible character of degree > 1 occurs in some cellular character    , which shows that dim  , 0 ,0  ⩾ 2.

Left cells
In order to define Calogero-Moser left cells, we need to work under the following hypothesis.
Hypothesis.In this subsection, and only in this subsection, we assume that Conjecture 4.2 holds. and Therefore, for  ∈ [0, 1), the family D  ( ), 1 ,  ( )  2 has simple spectrum (indeed, if () = 0, then this follows from Example 4.3 and, if () > 0, then and so this follows from the fact that we assume that Conjecture 4.2 holds).So this spectrum varies continuously according to the parameter .But, for  = 0, we have S 3. This means that, for each  ∈ , there exists a unique continuous map and the family (  )  ∈ satisfies that whenever  ≠  ′ .However, it may happen that   (1) =   ′ (1) even if  ≠  ′ .This leads to the following definition.  (1(where  is some, or any, element of ): it is called the Calogero-Moser -cellular character associated with .
Remark 5.3.A simple choice would be to take  1 =  2 =  0 , () = , and  () = 1 − .But we want to work with this slightly more general setting for more flexibility.Indeed, one could wonder whether the notion of Calogero-Moser left -cell depends on the choices of  1 ,  2 , , .In fact, it does not, because the topological space C R × R >0 × R >0 is simply connected.
For instance, this shows that, if  ∈ R >0 , then Calogero-Moser left -cells coincide with Calogero-Moser left -cells, and their associated cellular characters agree.
If we assumed moreover that Conjecture 4.1 holds, then we could have added some more flexibility, by taking  1 ,  2 in ℭ × ℭ and replacing the path  ↦ →  () 2 by any path The formula (5.1) can be rewritten as follows: where  runs over the set of Calogero-Moser left -cells.=  ,KL  .A very weak evidence for this conjecture is the comparison between (3.1) and (5.3).Note also that it holds for  = 0, as easily shown in [5,Coro. 17.2.3].A somewhat strong evidence for this conjecture is that it holds in type , by the work of Brochier-Gordon-White [6].The aim of this paper is to deal with the far easier (but still non-trivial) case of dihedral groups whenever  is constant.The following list of properties of Calogero-Moser left cells shows that Conjecture 5.4 is compatible with Proposition 3.1.This means that  ∈ S  , 0 , (1− )  0 if and only if − ∈ S  , 0 , (1− )  0 .Since Finally, if  ∈ S , 0 ,0 , then  , 0 ,0 ).This proves that (c).By using (d) and rectifying the signs if necessary thanks to a linear character, we may, and we will, assume that   > 0 for all  ∈ Ref().We have Let  denote the diagonal endomorphism  0, 0 , 0 − 0 and let  denote the Gaudin operator The matrix  is a real matrix with non-negative coefficients, which is primitive (because  is generated by Ref ()).Let  = ( 0 ,  0 ) +1.Then +  Id R is a diagonal matrix with positive coefficients.Therefore, if  > 0, the matrix   , 0 , (1− )  0 − 0 +  Id R is a real matrix with non-negative coefficients which is primitive.By the Perron-Frobenius theorem, its spectral radius   is an eigenvalue of   , 0 , (1− )  0 − 0 +  Id R , with multiplicity 1.Therefore,   varies continuously as  varies.

Two-sided cells
Until now, there is no alternative definition of Calogero-Moser two-sided -cells in terms of Gaudin operators or something related: the only available definition is based on the ramification theory of the Calogero-Moser space [5,Part III].This depends on the choice of some prime ideal in some Galois closure of some ring extension.This choice can be adapted to the choice of the two continuous functions  and  and we will follow this choice.Moreover, to each Calogero-Moser two-sided -cell Γ is associated a subset Irr ,CM Γ () of Irr(), which is called a Calogero-Moser -family.They form a partition of Irr(): This conjecture holds in type  (see [6]).In this paper, we prove it for dihedral groups whenever  is constant.

Dihedral groups, equal parameters
Hypothesis and notation.From now on, and until the end of this paper, we assume that  = R 2 , endowed with its canonical Euclidean structure, and we denote by ( 1 ,  2 ) its canonical basis (which is an orthonormal basis).We fix a natural number  ⩾ 3 and, if  ∈ Z, we set and   =    .We assume that Δ = { 0 ,  −1 }, so that  = { 0 ,  −1 }.For simplification, we set  =  0 and  ′ =  −1 .
Then  = ⟨,  ′ ⟩ is the dihedral group of order 2, and  is the order of  ′ .Moreover, Moreover, We aim to prove Conjectures 4.2, 5.

Elements, characters
Recall that  =  0 and  ′ =  −1 .For  ⩾ 0, we set Let  + = ⟨ ′ ⟩.It is a normal cyclic subgroup of order  of .We fix a primitive -th root of unity  and, for  ∈ Z, we denote by   :  + → C × the linear character such that   ( ′ ) =   .We set If  is even, we denote by   (resp.  ′ ) the linear character of  such that   () We conclude this subsection with a fact that will be useful for our purpose: if  ⩾ 0, then so the result follows from the fact that    + =  ′   + .□

Calogero-Moser cells
The main result of our paper is the following theorem.Theorem 6.2.Conjectures 4.2, 5.4, and 5.7 hold whenever  is dihedral and  is constant.
The rest of this section is devoted to the proof of this Theorem.

Preliminaries
We use the flexibility of the definition of left cells explained in Remark 5.3.We take (one can check that  0 is a positive multiple of  1 ).We also set is an orthonormal basis of .We have for all  ⩾ 0. Also (6.5) Proof of (6.5).This is easily checked by induction on  using the fact that Now, we set  =  ′ (so that  is the rotation with angle 2/).Recall that  generates  + .Also for all ,  ∈ Z.

Proof of Conjecture 4.2
Now, we set In particular, is an eigenvector of   +  for the eigenvalue − cos( /),
Again by (6.6), the coefficient of ) is equal to − sin( /), so this proves the second statement because sin( /) ≠ 0. □ After these preliminaries, we are ready to prove the theorem.So let us first prove that Conjecture 4.2 holds.Let us assume that  > 0. Let  , denote the largest eigenvalue of   +  (as in the proof of Proposition 5.5 (c)).Then it has multiplicity 1.By the proof of Proposition 5.5 (b), − , is the smallest eigenvalue of   + , and it has multiplicity 1.We denote by  max (, ) (resp. min (, )) the  , -eigenspace (resp.the − , -eigenspace) of   + .

Some pictures
We provide in Figure 7.1 some pictures of the paths described in (6.11) and (7.3), whenever  ∈ {3, 4, 5, 6, 7, 8}.In these pictures, we have identified  and  * as all along the paper.The gray points represent the roots, the gray lines represent the reflecting hyperplanes, the blue dots are the points (3 1 /5) (the reason for choosing 3 1 /5 is to have a better-looking picture), the black dots represent the spectrum of the family D 1, 1 ,0 (i.e.
and completes the proof of (a).(d).It follows from the third equality in (4.3) and the same argument as in (b).