Cuspidal Calogero-Moser and Lusztig families for Coxeter groups
CCUSPIDAL CALOGERO–MOSER AND LUSZTIG FAMILIESFOR COXETER GROUPS
GWYN BELLAMY AND ULRICH THIELA
BSTRACT . The goal of this paper is to compute the cuspidal Calogero–Moser families forall infinite families of finite Coxeter groups, at all parameters. We do this by first computingthe symplectic leaves of the associated Calogero–Moser space and then by classifying certain“rigid” modules. Numerical evidence suggests that there is a very close relationship betweenCalogero–Moser families and Lusztig families. Our classification shows that, additionally, thecuspidal Calogero–Moser families equal cuspidal Lusztig families for the infinite families ofCoxeter groups.
Introduction
Based on the relationship between Dunkl operators, the Knizhnik–Zamolodchikov connection,and Hecke algebras, it became apparent very soon after the introduction of rational Chered-nik algebras by Etingof and Ginzburg [16] that there is a very close connection betweenthese algebras and cyclotomic Hecke algebras [11]. This connection is encoded in the Knizh-nik–Zamolodchikov functor, introduced in [24], and is a key tool in the representation theory ofrational Cherednik algebras at 𝑡 ̸ = 0 .In the quasi-classical limit 𝑡 = 0 the Knizhnik–Zamolodchikov functor no longer exists andno functorial connection to Hecke algebras is currently known. Astonishingly, as first noticedby Gordon and Martino [26], it seems that there is still, none the less, a close relationshipbetween rational Cherednik algebras in 𝑡 = 0 and Hecke algebras, suggesting that there may bean asymptotic Knizhnik–Zamolodchikov functor in the quasi-classical limit. The aim of thisarticle is to add weight to this expectation by comparing cuspidal Calogero–Moser familieswith cuspidal Lusztig families. Families
Etingof and Ginzburg [16] defined, for any finite reflection group ( h , 𝑊 ) and a function c : Ref ( 𝑊 ) → C from the set of reflections of 𝑊 to the complex numbers which is invariantunder 𝑊 -conjugation, the rational Cherednik algebra H c ( 𝑊 ) at 𝑡 = 0 . The spectrum of thecentre of this algebra is an affine Poisson deformation X c ( 𝑊 ) of the symplectic singularity ( h × h * ) /𝑊 , called the Calogero–Moser space. This theory exists in particular for finite Coxetergroups 𝑊 . In this case, one can also attach to 𝑊 the Hecke algebra ℋ 𝐿 ( 𝑊 ) depending on aweight function 𝐿 : 𝑊 → R . The space of weight functions 𝐿 and the space of real valued c -functions is the same so that one can relate invariants coming from Hecke algebras with thosecoming from rational Cherednik algebras.Gordon [25] has defined the notion of Calogero–Moser c -families of Irr ( 𝑊 ) , which on thegeometric side correspond to the C * -fixed points of the Calogero–Moser space X c ( 𝑊 ) . Workof several people, in particular Gordon and Martino, has shown that: Date: May 26, 2016 (first version: May 3, 2015)Final version to appear in J. AlgebraG
WYN B ELLAMY , School of Mathematics and Statistics, University Gardens, University of Glasgow, Glasgow,G12 8QW, UK, [email protected] U LRICH T HIEL , Universität Stuttgart, Fachbereich Mathematik, Institut für Algebra und Zahlentheorie, Lehrstuhlfür Algebra, Pfaffenwaldring 57, 70569 Stuttgart, Germany, [email protected] a r X i v : . [ m a t h . R T ] J un C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Fact. If 𝑊 is a Coxeter group of type 𝐴, 𝐵, 𝐷 , 𝐼 ( 𝑚 ) or 𝐻 , then the Lusztig c -families equalthe Calogero–Moser c -families for all c : Ref ( 𝑊 ) → R .We refer to §2D for more details. It is conjectured by Gordon–Martino [26] that this is indeedtrue for all finite Coxeter groups; see also Bonnafé–Rouquier [9]. There is so far no conceptualexplanation for this connection. Bonnafé and Rouquier [9] furthermore constructed analogsof constructible characters and cells on the Calogero–Moser side, and collected evidencesupporting their conjecture that these notions coincide with Lusztig’s notions; see also [8]. Cuspidal families
The key to defining constructible representations and Lusztig families for Hecke algebras isLusztig’s truncated induction, also called j -induction. This leads to the concept of cuspidal Lusztig families, which are those that cannot be described as being j -induced from a family fora proper parabolic subgroup. Cuspidal families play a key role in describing certain unipotentrepresentations for the corresponding finite groups of Lie type. In [2] the first author alsointroduced the notion of cuspidal Calogero–Moser families. This time the definition is geometric:a family is cuspidal if the support of every module in the family is a zero-dimensional symplecticleaf of the Calogero–Moser space. In this article we determine the cuspidal Calogero–Moserfamilies for the Coxeter groups of type 𝐴, 𝐵, 𝐷 and 𝐼 ( 𝑚 ) . Our main result states (see §3): Theorem A. If 𝑊 is of type 𝐴, 𝐵, 𝐷 or 𝐼 ( 𝑚 ) , then the cuspidal Lusztig c -families equal the cuspidal Calogero–Moser c -families for all c : Ref ( 𝑊 ) → R .The proof follows from a case-by-case analysis in sections §5 to §8 using theoretical methodswe develop in section §4. Based on this theorem we make the following conjecture. Conjecture B.
For any finite Coxeter group the cuspidal Lusztig c -families equal the cuspidalCalogero–Moser c -families for all real parameters c .Because of Theorem A this conjecture remains open only for the six exceptional Coxetergroups 𝐻 , 𝐻 , 𝐹 , 𝐸 , 𝐸 , 𝐸 . Rigid representations
The main ingredient for calculating the cuspidal Calogero–Moser families, and hence confirmingTheorem A, is the notion of a rigid module: a H c ( 𝑊 ) -module is said to be rigid if it isirreducible as a 𝑊 -module. These have already played a role in the representation theory ofrational Cherednik algebras at 𝑡 ̸ = 0 , see e.g. [6] or [17], and at 𝑡 = 0 they were studied by thesecond author in [41]. The terminology comes from the theory of module varieties. Namely,for any 𝑑 < | 𝑊 | , we show in Lemma 4.9 that the set 𝑋 of rigid modules in Rep 𝑑 ( H c ( 𝑊 )) ,the variety parameterizing representations of dimension 𝑑 , is open. Therefore, though thesemodules often appear in families with respect to the parameter c , the module structure (forfixed parameter c ) on a rigid module cannot be deformed to a continuous family. This is thefirst clue that there is a strong connection between rigid representations and zero-dimensionalleaves of X c ( 𝑊 ) (and hence to cuspidal Calogero-Moser families).In this article we classify the rigid modules for all non-exceptional Coxeter groups and allparameters. The importance of these modules is explained by our second main result which weprove in §4: Theorem C.
Let 𝑊 be an arbitrary finite complex reflection group. If the simple module 𝐿 c ( 𝜆 ) ,where 𝜆 ∈ Irr ( 𝑊 ) , is a rigid H c ( 𝑊 ) -module, then the Calogero–Moser c -family to which itbelongs is cuspidal. WYN B ELLAMY AND U LRICH T HIEL Rigid modules are easily computed, and using Theorem C this allows us to identify certaincuspidal families. Remarkably, for the non-exceptional Coxeter groups we can show thatthe cuspidal Calogero–Moser families are precisely those containing the rigid modules. Thecuspidal Lusztig families are similarly characterized.
Remark.
While this paper was in preparation, the preprint [14] appeared, where rigid modulesalso play a key role (though the definition there is slightly different). Based on the analogywith affine Hecke algebras, they are called "one- 𝑊 -type" modules in loc. cit. . In the preprint[14] the author gives a different notion of cuspidal Calogero–Moser families. Namely, in loc.cit. a family is said to be cuspidal if it contains a rigid module. By Theorem C, every cuspidalfamily in our sense is cuspidal in the sense of [14]. However, it is clear that for most complexreflection groups that are not of Coxeter type there exist many cuspidal families (in our sense)that are not cuspidal in the sense of loc. cit. . Moreover, as shown in loc. cit. , Conjecture B is false for the Weyl group of type 𝐸 if we use the definition of cuspidal used in loc. cit. Symplectic leaves
As previously noted, the notion of cuspidal Calogero–Moser families depends on the fact thatthe Calogero–Moser space X c ( 𝑊 ) is stratified by finitely many symplectic leaves. These leavesare naturally labeled by conjugacy classes of parabolic subgroups ( 𝑊 ′ ) of 𝑊 . There are twonatural partial orderings on the set of symplectic leaves: a geometric one given in terms ofthe closures of leaves, and another, algebraic one given in terms of inclusions of parabolicsubgroups. It is clear that the geometric ordering refines the algebraic ordering.Using results of Martino, we describe all symplectic leaves for the Coxeter groups of type 𝐴, 𝐵, 𝐷 and 𝐼 ( 𝑚 ) in terms of the conjugacy classes of parabolic subgroups. We also describethe two orderings on the set of symplectic leaves in these cases (see Theorem 6.2, Theorem 7.2and [2, Tables 1,2]). Based on this we arrive at the following conjecture. Conjecture D.
Let 𝑊 be a finite Coxeter group.(a) Each conjugacy class of parabolic subgroups ( 𝑊 ′ ) labels at most one symplectic leaf.(b) The geometric ordering on leaves equals the algebraic ordering.We note that both statements of Conjecture D may fail if 𝑊 is not a Coxeter group. Clifford Theory
Our results for Coxeter groups of type 𝐷 are deduced from the corresponding results for thegroups of type 𝐵 using the fact that 𝐷 𝑛 (cid:67) 𝐵 𝑛 . More generally, we consider a complex reflectiongroup ( h , 𝑊 ) and a normal subgroup 𝐾 (cid:67) 𝑊 such that ( h | 𝐾 , 𝐾 ) is also a reflection group.This situation is also considered in [4] and by Liboz [30].Based on a suggestion of Rouquier, we show that Γ :=
𝑊/𝐾 acts on the Calogero–Moserspace X c ( 𝐾 ) such that X c ( 𝑊 ) = X c ( 𝐾 ) / Γ . This allows us to deduce the Calogero–Moserfamilies for 𝐾 from the Calogero–Moser families for 𝑊 , generalising results of [4]. Cuspidalfamilies and rigid representations behave well under this correspondence. We also describe thesymplectic leaves in X c ( 𝐾 ) in terms of those of X c ( 𝑊 ) . Acknowledgements.
The authors would like to thank Cédric Bonnafé and Meinolf Geck formany fruitful discussions. We also thank Dan Ciubotaru for informing us about his preprint [14]and his result that for 𝐸 the cuspidal Lusztig family does not contain rigid modules. Moreover,we would like to thank Gunter Malle for commenting on a preliminary version of this article.The second author was partially supported by the DFG Schwerpunktprogramm 1489. C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS C ONTENTS §1. Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4§2. Lusztig families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6§3. Cuspidal Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9§4. Calculating cuspidal Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13§5. Type 𝐴 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21§6. Type 𝐵 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22§7. Type 𝐷 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34§8. Type 𝐼 ( 𝑚 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 §1. Calogero–Moser families We begin by recalling the definition of the main protagonists of this paper—the Calogero–Moserfamilies for complex reflection groups. They are obtained from the block structure of therestricted rational Cherednik algebra studied by Gordon [25], which is a finite-dimensionalquotient of the rational Cherednik algebra introduced by Etingof and Ginzburg [16]. §1A. Rational Cherednik algebras
Let ( h , 𝑊 ) be a finite complex reflection group. By this we mean that 𝑊 is a non-trivial finitesubgroup of GL ( h ) for some finite-dimensional complex vector space h such that 𝑊 is generatedby its set Ref ( 𝑊 ) of reflections , i.e., by those elements 𝑠 ∈ 𝑊 such that Ker ( id h − 𝑠 ) is ofcodimension one in h . Let ( · , · ) : h × h * → C be the natural pairing defined by ( 𝑦, 𝑥 ) = 𝑥 ( 𝑦 ) .For 𝑠 ∈ Ref ( 𝑊 ) we fix 𝛼 𝑠 ∈ h * to be a basis of the one-dimensional space Im ( 𝑠 − | h * and 𝛼 ∨ 𝑠 ∈ h to be a basis of the one-dimensional space Im ( 𝑠 − | h , normalised so that 𝛼 𝑠 ( 𝛼 ∨ 𝑠 ) = 2 .Our discussion will not depend on the choice of 𝛼 𝑠 and 𝛼 ∨ 𝑠 . Note that the group 𝑊 acts on Ref ( 𝑊 ) by conjugation. Choose a function c : Ref ( 𝑊 ) → C which is invariant under 𝑊 -conjugation (we say that c is 𝑊 -equivariant ) and furthermore choose a complex number 𝑡 ∈ C .The rational Cherednik algebra H 𝑡, c ( 𝑊 ) , as introduced by Etingof and Ginzburg [16], is thequotient of the skew group algebra of the tensor algebra, 𝑇 ( h ⊕ h * ) (cid:111) 𝑊 , by the ideal generatedby the relations [ 𝑥, 𝑥 ′ ] = [ 𝑦, 𝑦 ′ ] = 0 for all 𝑥, 𝑥 ′ ∈ h * and 𝑦, 𝑦 ′ ∈ h , and(1) [ 𝑦, 𝑥 ] = 𝑡 ( 𝑦, 𝑥 ) − ∑︁ 𝑠 ∈ Ref ( 𝑊 ) c ( 𝑠 )( 𝑦, 𝛼 𝑠 )( 𝛼 ∨ 𝑠 , 𝑥 ) 𝑠 , ∀ 𝑦 ∈ h , 𝑥 ∈ h * . We concentrate on the case 𝑡 = 0 and set H c := H , c . For any 𝛼 ∈ C ∖{ } , the algebras H 𝛼 c ( 𝑊 ) and H c ( 𝑊 ) are naturally isomorphic. Therefore we are free to rescale c by 𝛼 whenever thisis convenient. A fundamental result for rational Cherednik algebras, proved by Etingof andGinzburg [16, Theorem 1.3], is that the PBW property holds for all c , i.e., the natural map(2) C [ h ] ⊗ C C 𝑊 ⊗ C C [ h * ] → H c ( 𝑊 ) is an isomorphism of C -vector spaces. The rational Cherednik algebra is naturally Z -graded by deg ( 𝑥 ) = 1 for 𝑥 ∈ h * , deg ( 𝑦 ) = − for 𝑦 ∈ h , and deg ( 𝑤 ) = 0 for 𝑤 ∈ 𝑊 . We note that nosuch grading exists for general symplectic reflection algebras. §1B. Calogero–Moser space The centre Z c ( 𝑊 ) of H c ( 𝑊 ) is an affine domain. We shall denote by X c ( 𝑊 ) := Spec ( Z c ( 𝑊 )) the corresponding affine variety. It is called the (generalized) Calogero–Moser space associatedto 𝑊 at parameter c . These varieties define a flat family of deformations of ( h ⊕ h * ) /𝑊 overthe affine C -space of dimension | Ref ( 𝑊 ) /𝑊 | . The following was shown for Coxeter groupsin [16, Proposition 4.15], and the general case is due to [25, Proposition 3.6]. WYN B ELLAMY AND U LRICH T HIEL Proposition 1.1 (Etingof–Ginzburg, Gordon) . The subspace 𝐷 ( 𝑊 ) := C [ h ] 𝑊 ⊗ C C [ h * ] 𝑊 of H c ( 𝑊 ) is a central subalgebra and Z c ( 𝑊 ) is a free 𝐷 ( 𝑊 ) -module of rank | 𝑊 | .The inclusions C [ h ] 𝑊 ˓ → 𝑍 c ( 𝑊 ) and C [ h * ] 𝑊 ˓ → 𝑍 c ( 𝑊 ) define finite surjective morphisms 𝜋 c : X c ( 𝑊 ) (cid:16) h /𝑊 and 𝜛 c : X c ( 𝑊 ) (cid:16) h * /𝑊 . We write ϒ c := 𝜋 c × 𝜛 c : X c ( 𝑊 ) (cid:16) h /𝑊 × h * /𝑊 for the product morphism. It is a finite, and hence closed, surjective morphism. Note thatboth Z c ( 𝑊 ) and 𝐷 ( 𝑊 ) are graded subalgebras of H c ( 𝑊 ) . This implies that X c ( 𝑊 ) and h /𝑊 × h * /𝑊 carry a C * -action making ϒ c a C * -equivariant morphism. §1C. Restricted rational Cherednik algebras The inclusion of algebras 𝐷 ( 𝑊 ) ˓ → Z c ( 𝑊 ) allows us to define the restricted rational Chered-nik algebra H c ( 𝑊 ) as the quotient H c ( 𝑊 ) = H c ( 𝑊 ) 𝐷 ( 𝑊 ) + · 𝐻 c ( 𝑊 ) , where 𝐷 ( 𝑊 ) + denotes the ideal in 𝐷 ( 𝑊 ) of elements with zero constant term. This algebrawas originally introduced, and extensively studied, by Gordon [25]. The PBW theorem impliesthat(3) H c ( 𝑊 ) ≃ C [ h ] co 𝑊 ⊗ C C 𝑊 ⊗ C C [ h * ] co 𝑊 as C -vector spaces. Here, C [ h ] co 𝑊 = C [ h ] / ⟨ C [ h ] 𝑊 + ⟩ is the coinvariant algebra of 𝑊 and C [ h * ] co 𝑊 is defined analogously. Since 𝑊 is a reflectiongroup, the coinvariant algebra C [ h ] co 𝑊 is of dimension | 𝑊 | and is isomorphic to the regularrepresentation as a 𝑊 -module. Thus, dim H c ( 𝑊 ) = | 𝑊 | . The restricted rational Cherednikalgebra is a quotient of H c ( 𝑊 ) by an ideal generated by homogeneous elements and so it isalso a graded algebra. This combined with the triangular decomposition (3) of H c ( 𝑊 ) impliesthat the representation theory of H c ( 𝑊 ) has a rich combinatorial structure. The following isdue to Gordon [25], based on an abstract framework by Holmes and Nakano [27]. First of all,note that the skew-group algebra C [ h * ] co 𝑊 (cid:111) 𝑊 is a graded subalgebra of H c ( 𝑊 ) . Definition 1.2.
The baby Verma module of H c ( 𝑊 ) associated to a 𝑊 -module 𝜆 is Δ c ( 𝜆 ) := H c ( 𝑊 ) ⊗ C [ h * ] co 𝑊 (cid:111) 𝑊 𝜆 , where C [ h * ] co 𝑊 + acts on 𝜆 as zero.The baby Verma module Δ c ( 𝜆 ) is naturally a graded H c ( 𝑊 ) -module, where ⊗ 𝜆 sitsin degree zero. By studying quotients of baby Verma modules, it is possible to completelyclassify the simple H c ( 𝑊 ) -modules. We denote by Irr 𝑊 the set of simple 𝑊 -modules (up toisomorphism). Similarly, we understand Irr H c ( 𝑊 ) . Proposition 1.3 (Gordon) . Let 𝜆, 𝜇 ∈ Irr 𝑊 .(1) The baby Verma module Δ c ( 𝜆 ) has a simple head. We denote it by 𝐿 c ( 𝜆 ) .(2) 𝐿 c ( 𝜆 ) is isomorphic to 𝐿 c ( 𝜇 ) if and only if 𝜆 ≃ 𝜇 .(3) The map Irr 𝑊 → Irr H c ( 𝑊 ) , 𝜆 → 𝐿 c ( 𝜆 ) , is a bijection.The bijection in the proposition allows us to transform representation theoretic informationabout H c ( 𝑊 ) into combinatorial c -dependent data about 𝑊 . The Calogero–Moser families arethe primary example of this process. C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS §1D. Calogero–Moser families
Since the algebra H c ( 𝑊 ) is finite-dimensional, it has a block decomposition H c ( 𝑊 ) = ⨁︀ 𝑘𝑖 =1 𝐵 𝑖 , with each 𝐵 𝑖 an indecomposable algebra. If 𝑏 𝑖 is the identity element of 𝐵 𝑖 thenthe identity element of H c ( 𝑊 ) is the sum 𝑏 + . . . + 𝑏 𝑘 of the 𝑏 𝑖 . For each simple H c ( 𝑊 ) -module 𝐿 , there exists a unique 𝑖 such that 𝑏 𝑖 · 𝐿 ̸ = 0 . In this case we say that 𝐿 belongs to the block 𝐵 𝑖 . By Proposition 1.3, we can (and will) identify Irr H c ( 𝑊 ) with Irr 𝑊 .Let Ω c ( 𝑊 ) be the set of equivalence classes of Irr 𝑊 under the equivalence relation 𝜆 ∼ 𝜇 ifand only if 𝐿 c ( 𝜆 ) and 𝐿 c ( 𝜇 ) belong to the same block. These equivalence classes are calledthe Calogero–Moser c -families of 𝑊 .These families have an important geometric interpretation. The image of the natural map Z c ( 𝑊 ) /𝐷 ( 𝑊 ) + · Z c ( 𝑊 ) → H c ( 𝑊 ) is clearly contained in the centre of H c ( 𝑊 ) . In general it is not equal to the centre of H c ( 𝑊 ) .However, it is a consequence of a theorem by Müller, see [13, Corollary 2.7], that the primitivecentral idempotents of H c ( 𝑊 ) , the block idempotents 𝑏 𝑖 above, are precisely the imagesof the primitive idempotents of Z c ( 𝑊 ) /𝐷 ( 𝑊 ) + · Z c ( 𝑊 ) . This shows that the natural map Irr 𝑊 → ϒ − c (0) , 𝜆 ↦→ Supp 𝐿 c ( 𝜆 ) = 𝜒 𝐿 c ( 𝜆 ) , factors through the Calogero–Moser partition.Here, ϒ − c (0) is considered as the set theoretic fibre over the origin of h /𝑊 × h * /𝑊 . Inother words, we have a natural bijection between Ω c ( 𝑊 ) and ϒ − c (0) . Now, recall that ϒ c is C * -equivariant. The only C * -fixed point of h /𝑊 × h * /𝑊 is the origin and therefore ϒ − c (0) = X c ( 𝑊 ) C * . Hence, we can identify the Calogero–Moser families Ω c ( 𝑊 ) with the C * -fixed closed points of the Calogero–Moser space X c ( 𝑊 ) .The next theorem follows from the fact that the Azumaya locus of H c ( 𝑊 ) is equal to thesmooth locus of Z c ( 𝑊 ) , which in turn follows from results by Etingof–Ginzburg [16, Theorem1.7] and Brown (see [25, Lemma 7.2]). Theorem 1.4 (Etingof–Ginzburg, Brown) . A C * -fixed closed point of X c ( 𝑊 ) is smooth if andonly if the corresponding Calogero–Moser family is a singleton, i.e. it consists only of oneirreducible character of 𝑊 . Example 1.5.
Consider the special case c = 0 . In this case X ( 𝑊 ) = ( h ⊕ h * ) /𝑊 . Thequotient morphism h ⊕ h * → ( h ⊕ h * ) /𝑊 is C * -equivariant and finite, hence X ( 𝑊 ) has onlyone C * -fixed closed point, namely the origin. In particular, there is only one Calogero–Moserfamily. §2. Lusztig families In this section we give a short summary of the other protagonist of this paper—Lusztig’sfamilies. We review some of the constructions involved in the definition of Lusztig families,such as truncated induction, as we will make use of these in the case-by-case analysis in sections§5 to §8. For more details we refer to Lusztig’s books [32, 33], and also to [22] and [20]. §2A. Hecke algebras
Throughout this section, let ( 𝑊, 𝑆 ) be a finite Coxeter system. We choose an R -valued weightfunction 𝐿 on ( 𝑊, 𝑆 ) , i.e., a function 𝐿 : 𝑊 → R satisfying 𝐿 ( 𝑤𝑤 ′ ) = 𝐿 ( 𝑤 ) + 𝐿 ( 𝑤 ′ ) forall 𝑤, 𝑤 ′ ∈ 𝑊 with ℓ ( 𝑤𝑤 ′ ) = ℓ ( 𝑤 ) + ℓ ( 𝑤 ′ ) , where ℓ is the length function of ( 𝑊, 𝑆 ) . Let 𝐴 := Z 𝑊 [ R ] be the group ring of the additive group R over the subring Z 𝑊 of C generated bythe values of the irreducible complex characters of 𝑊 . This is an integral domain and we denoteby 𝑞 𝛼 the element of 𝐴 corresponding to 𝛼 ∈ R . Note that 𝑞 𝛼 𝑞 𝛽 = 𝑞 𝛼 + 𝛽 . Set 𝑞 𝑤 := 𝑞 𝐿 ( 𝑤 ) for 𝑤 ∈ 𝑊 . Let ℋ := ℋ 𝐿 ( 𝑊, 𝑆 ) be the Hecke algebra of ( 𝑊, 𝑆 ) over 𝐴 with respect to 𝐿 . This isthe free 𝐴 -algebra with basis { 𝑇 𝑤 | 𝑤 ∈ 𝑊 } whose multiplication is uniquely determined by WYN B ELLAMY AND U LRICH T HIEL the relations(4) 𝑇 𝑠 𝑇 𝑤 = {︂ 𝑇 𝑠𝑤 if ℓ ( 𝑠𝑤 ) > ℓ ( 𝑤 ) 𝑇 𝑠𝑤 + ( 𝑞 𝑠 − 𝑞 − 𝑠 ) 𝑇 𝑤 if ℓ ( 𝑠𝑤 ) < ℓ ( 𝑤 ) for all 𝑠 ∈ 𝑆 and 𝑤 ∈ 𝑊 . It is a standard fact that the scalar extension ℋ 𝐾 of ℋ to the fractionfield 𝐾 of 𝐴 is split semisimple. It is then a consequence of Tits’s deformation theorem thatthere is a natural bijection between Irr 𝑊 and Irr ℋ 𝐾 . We write 𝐸 𝜆𝑞 for the simple ℋ 𝐾 -modulecorresponding to the simple 𝑊 -module 𝜆 under this bijection. It is also well-known that ℋ is asymmetric 𝐴 -algebra. This implies that the scalar extension ℋ 𝐾 is symmetric and so by thetheory in [23, §7] there is a Schur element s 𝜆 ∈ 𝐴 attached to every simple module 𝐸 𝜆𝑞 . Thereis a unique element a 𝜆 ∈ R ≥ satisfying 𝑞 a 𝜆 s 𝜆 ∈ Z 𝑊 [ R ≥ ] and 𝑞 a 𝜆 s 𝜆 ≡ 𝑓 𝜆 mod Z 𝑊 [ R > ] for some 𝑓 𝜆 > . This is called Lusztig’s a -invariant of 𝜆 . The Schur elements and a -invariantsare known for all Coxeter groups and all weight functions. Note that despite the notation theSchur elements s 𝜆 and the a -invariants a 𝜆 depend on 𝐿 . §2B. Truncated induction Recall that if 𝐼 ⊆ 𝑆 is any subset, then ( 𝑊 𝐼 , 𝐼 ) is naturally a Coxeter system, where 𝑊 𝐼 is thegroup generated by 𝐼 . This is called a (standard) parabolic subgroup of ( 𝑊, 𝑆 ) . The restriction 𝐿 𝐼 of our weight function 𝐿 to 𝑊 𝐼 is a weight function on ( 𝑊 𝐼 , 𝐼 ) . For any simple module 𝜇 of 𝑊 𝐼 , Lusztig defined the truncated induction (or j -induction ) as(5) j 𝑊𝑊 𝐼 𝜇 := ∑︁ 𝜆 ∈ Irr 𝑊 a 𝜆 = a 𝜇 ⟨ Ind 𝑊𝑊 𝐼 𝜇, 𝜆 ⟩ 𝜆 , where ⟨ Ind 𝑊𝑊 𝐼 𝜇, 𝜆 ⟩ denotes the multiplicity of 𝜆 in the induction of 𝜇 from 𝑊 𝐼 to 𝑊 . Keep inmind that the a -invariant a 𝜇 is computed using the restriction 𝐿 𝐼 of 𝐿 to 𝑊 𝐼 . It is shown in [19,Lemma 3.5] that for any 𝜇 ∈ Irr 𝑊 ′ there is a 𝜆 ∈ Irr 𝑊 with a 𝜆 = a 𝜇 so that the above sumis never empty. This operation extends to a morphism j 𝑊𝑊 𝐼 : K ( 𝑊 𝐼 - mod ) → K ( 𝑊 - mod ) ofGrothendieck groups. It is transitive in the sense that j 𝑊𝑊 𝐼 ∘ j 𝑊 𝐼 𝑊 𝐽 = j 𝑊𝑊 𝐽 for 𝐽 ⊆ 𝐼 ⊆ 𝑆 . §2C. Constructible characters and families Using truncated induction, Lusztig inductively defined the set
Con 𝐿 ( 𝑊 ) of 𝐿 -constructible rep-resentations of 𝑊 as follows: if 𝑊 is trivial, then Con 𝐿 ( 𝑊 ) consists of the unit representation,and otherwise Con 𝐿 ( 𝑊 ) consists of the 𝑊 -modules of the form j 𝑊𝑊 𝐼 𝐸 and ( j 𝑊𝑊 𝐼 𝐸 ) ⊗ sgn 𝑊 for all proper subsets 𝐼 (cid:40) 𝑆 and all 𝐸 ∈ Con 𝐿 𝐼 ( 𝑊 𝐼 ) . Here sgn 𝑊 is the sign representation of ( 𝑊, 𝑆 ) . A key result shown by Lusztig, [33, Proposition 22.3], says(6) for each 𝜆 ∈ Irr 𝑊 there exists 𝐸 ∈ Con 𝐿 ( 𝑊 ) such that ⟨ 𝐸, 𝜆 ⟩ ̸ = 0 . The constructible graph is the graph 𝒞 𝐿 ( 𝑊 ) with vertices Irr 𝑊 and an edge between 𝜆 and 𝜇 if and only if 𝜆 ̸ = 𝜇 and they both occur in an 𝐿 -constructible representation of 𝑊 . Theconnected components of this graph are called Lusztig’s 𝐿 -families . They define a partition of Irr 𝑊 . We denote the set of these families by Lus 𝐿 ( 𝑊 ) . Lusztig’s families are known for allfinite Coxeter groups (see [33, §22] and also §5 to §8). Example 2.1.
Consider the special case 𝐿 = 0 . The map 𝑇 𝑤 ↦→ 𝑤 extends to an algebra isomor-phism from ℋ ( 𝑊, 𝑆 ) to the group algebra 𝐴𝑊 which is compatible with the symmetrisingtraces. Hence, s 𝜆 = | 𝑊 | dim 𝜆 ∈ Z 𝑊 [ R > ] by [23, 7.2.5] and so a 𝜆 = 0 . This in turn immediatelyshows that j 𝑊𝑊 𝐼 𝜇 = Ind 𝑊𝑊 𝐼 𝜇 for any parabolic subgroup 𝑊 𝐼 of 𝑊 and 𝜇 ∈ Irr 𝑊 𝐼 . We then seethat there is only one constructible representation, namely the regular representation of 𝑊 . Inparticular, the constructible graph is connected and there is only one Lusztig family. C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS §2D. Calogero–Moser families vs. Lusztig families
It is a standard fact that 𝑊 admits a reflection representation on the complex vector space h ofdimension equal to the size of 𝑆 , namely the complexification of the geometric representation .The set Ref ( 𝑊 ) of reflections then consists precisely of all conjugates of 𝑆 in 𝑊 . Hence,to 𝑊 and a 𝑊 -equivariant function c : Ref ( 𝑊 ) → C we can attach the rational Cherednikalgebra H c ( 𝑊 ) and have the notion of Calogero–Moser c -families Ω c ( 𝑊 ) of Irr 𝑊 . It followsfrom Matsumoto’s lemma (see [22, 1.1.5]) that a weight function 𝐿 on 𝑊 is already uniquelydetermined by the values on the 𝑊 -conjugacy classes of 𝑆 , and that conversely every collectionof elements 𝑐 𝑠 ∈ R for 𝑠 ∈ 𝑆 with 𝑐 𝑠 = 𝑐 𝑡 whenever 𝑐 𝑠 and 𝑐 𝑡 are conjugate definesa unique weight function on ( 𝑊, 𝑆 ) . This shows that weight functions 𝐿 : 𝑊 → R , i.e.,parameters for Hecke algebras attached to ( 𝑊, 𝑆 ) , are nothing else than 𝑊 -equivariant functions c : Ref ( 𝑊 ) → R , i.e., R -valued parameters for rational Cherednik algebras attached to 𝑊 . Wewill thus use both notions interchangeably.Whenever we write c ≥ , resp. c > , we mean that c takes values in R ≥ , resp. R > .Similarly, we write 𝐿 ≥ , resp. 𝐿 > , if 𝐿 ( 𝑠 ) ≥ , resp. 𝐿 ( 𝑠 ) > , for all 𝑠 ∈ 𝑆 .We can twist by linear characters of 𝑊 in order to ensure that we are always in the situation c ≥ . Namely, let 𝛿 : 𝑊 → R × be a linear character. Clearly 𝛿 is uniquely defined byits values on 𝑆 , where it is ± . Conversely, for any assignment of ± to each element of 𝑆 ,such that 𝛿 ( 𝑠 ) = 𝛿 ( 𝑠 ′ ) if 𝑠 is conjugate to 𝑠 ′ , we get a well-defined linear character of 𝑊 .Then 𝑇 𝑤 ↦→ 𝛿 ( 𝑤 ) 𝑇 𝑤 defines an algebra isomorphism ℋ 𝐿 ( 𝑊, 𝑆 ) ∼ −→ ℋ 𝛿𝐿 ( 𝑊, 𝑆 ) . Given arepresentation 𝜆 of 𝑊 , 𝛿 𝜆 denotes the twist of 𝜆 by 𝛿 . It is immediate from the definition ofLusztig families that 𝜆 and 𝜇 belong to the same 𝐿 -family if and only if 𝛿 𝜆 and 𝛿 𝜇 belongto the same 𝛿𝐿 -family. Moreover, a family ℱ is 𝐿 -cuspidal (see below) if and only if 𝛿 ℱ is 𝛿𝐿 -cuspidal.Similarly, one can twist the rational Cherednik algebra by the character 𝛿 , as explained in [9,4.6B]. Again, the two representations 𝜆, 𝜇 belong to the same c -family if and only if 𝛿 𝜆 and 𝛿 𝜇 belong to the same 𝛿 c -family. Moreover, a family ℱ is c -cuspidal (see below) if and only if 𝛿 ℱ is 𝛿 c -cuspidal. Therefore, to prove Theorem A, it suffices to make the following assumption, asin [22]:We assume that 𝐿 ≥ .The following conjecture is due to Gordon–Martino [26]. Conjecture 2.2.
For any finite Coxeter group 𝑊 and any real parameter c we have Ω c ( 𝑊 ) = Lus c ( 𝑊 ) , i.e. the Calogero–Moser c -families are the same as the Lusztig c -families.We note that this conjecture was formulated in [26] for Weyl groups and weight functionstaking values in Q > . Moreover, both in [26] and [9] it was conjectured that Ω c ( 𝑊 ) coincideswith the partition of Irr 𝑊 into Kazhdan–Lusztig families . Assuming Lusztig’s conjectures P1to P15 (see [33, §14]), the Kazhdan–Lusztig families and the Lusztig families are equal (see[20, Theorem 4.3]), so that the conjecture above (which is also formulated in precisely this wayby Bonnafé [8] for parameters c > ) seems feasible.Let us record the following observation we obtain from Examples 1.5 and 2.1. Lemma 2.3.
For any 𝑊 we have Ω ( 𝑊 ) = Lus ( 𝑊 ) , i.e. Conjecture 2.2 holds for c = 0 .The work of Lusztig [32, 33], Etingof–Ginzburg [16], Gordon [25], Gordon–Martino [26],Martino [35], the first author [3, 4], and the second author [40] shows that Conjecture 2.2 holdsin many cases. WYN B ELLAMY AND U LRICH T HIEL Theorem 2.4. If 𝑊 is of type 𝐴 , 𝐵 , 𝐷 , 𝐼 ( 𝑚 ) , or 𝐻 , then Ω c ( 𝑊 ) = Lus c ( 𝑊 ) for any c : Ref ( 𝑊 ) → R .Except for type 𝐻 , which follows from [40], the proof of this theorem is also obtained herefrom §5, Corollary 6.13, Theorem 7.3, and Corollary 8.4. §2E. Cuspidal Lusztig families What is now relevant for us in this paper is that it can happen that a Lusztig family
ℱ ∈
Lus 𝐿 ( 𝑊 ) is j -induced from a parabolic subgroup 𝑊 𝐼 of 𝑊 in the sense that there is a Lusztig family ℱ ′ ∈ Lus 𝐿 𝐼 ( 𝑊 𝐼 ) such that j 𝑊𝑊 𝐼 induces a bijection between ℱ ′ and ℱ or between ℱ ′ and ℱ ⊗ sgn 𝑊 . Lusztig called a family cuspidal if it is not j -induced from a proper parabolicsubgroup of 𝑊 . Let Lus cusp 𝐿 ( 𝑊 ) ⊆ Lus 𝐿 ( 𝑊 ) be the set of cuspidal Lusztig families. Thesefamilies are the building blocks of Lusztig families and it is most important to understand them.The following useful lemma is well-known. Lemma 2.5.
For any 𝛼 ∈ R > we have Con 𝛼𝐿 ( 𝑊 ) = Con 𝐿 ( 𝑊 ) , Lus 𝛼𝐿 ( 𝑊 ) = Lus 𝐿 ( 𝑊 ) and Lus cusp 𝛼𝐿 ( 𝑊 ) = Lus cusp 𝐿 ( 𝑊 ) . Proof.
As in [22, 1.1.9] one can introduce a universal Hecke algebra ℋ over Z 𝑊 [ R 𝑛 ] , where 𝑛 is the number of 𝑊 -conjugacy classes in 𝑆 . The Hecke algebra ℋ 𝐿 for a particular weightfunction 𝐿 : 𝑆 → R is then obtained by specialisation of ℋ . The algebra ℋ admits Schurelements s 𝜆 ∈ Z 𝑊 [ R 𝑛 ] and it follows from the theory in [23, §7] that s 𝜆 specialises to theSchur element s 𝜆 of ℋ 𝐿 . From this one can deduce that the a -invariant a 𝜆 of ℋ 𝛼𝐿 is obtainedfrom the one of ℋ 𝐿 by multiplication by 𝛼 . This immediately proves the claim. (cid:4) The key fact (6) implies:
Lemma 2.6. If ℱ = { 𝜆 } is a Lusztig family such that 𝜆 ∈ Con c ( 𝑊 ) , then ℱ is not cuspidal. §3. Cuspidal Calogero–Moser families On the Calogero–Moser side we do not have anything similar to j -induction so far. However, thefirst author has introduced in [2] the notion of cuspidal Calogero–Moser families. These are alsominimal with respect to a certain condition, but this time they have a geometric interpretationvia the Poisson structure on Calogero–Moser spaces. Despite their name, the two notions ofcuspidality have, a priori, nothing in common. None the less, we will show that they coincidefor all infinite families of Coxeter groups. In this paragraph we will review the foliation ofCalogero–Moser spaces into symplectic leaves and the notion of cuspidal Calogero–Moserfamilies. §3A. Poisson structure
We consider again an arbitrary finite complex reflection group ( h , 𝑊 ) . On the vector space h ⊕ h * we have a natural 𝑊 -invariant symplectic form 𝜔 defined by 𝜔 (( 𝑦, 𝑥 ) , ( 𝑦 ′ , 𝑥 ′ )) := 𝑥 ( 𝑦 ′ ) − 𝑥 ′ ( 𝑦 ) , ∀ 𝑦, 𝑦 ′ ∈ h , 𝑥, 𝑥 ′ ∈ h * . This induces a Poisson bracket {· , ·} on C [ h ⊕ h * ] . Since the form 𝜔 is 𝑊 -invariant, the Poissonbracket is 𝑊 -invariant and restricts to the invariant ring C [ h ⊕ h * ] 𝑊 making the quotient variety ( h ⊕ h * ) /𝑊 into a Poisson variety.The Calogero–Moser space X c ( 𝑊 ) is a flat Poisson deformation of ( h ⊕ h * ) /𝑊 . The Poissonstructure on X c ( 𝑊 ) comes from the commutation in the rational Cherednik algebra at 𝑡 ̸ = 0 asfollows. Let t be an indeterminate. Clearly, H c ( 𝑊 ) = H t , c ( 𝑊 ) / t H t , c ( 𝑊 ) and therefore wecan lift elements 𝑧 , 𝑧 ∈ Z c ( 𝑊 ) to elements ̂︀ 𝑧 , ̂︀ 𝑧 ∈ H t , c ( 𝑊 ) . Now, define { 𝑧 , 𝑧 } := [ ̂︀ 𝑧 , ̂︀ 𝑧 ] t =0 , USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS where [ ̂︀ 𝑧 , ̂︀ 𝑧 ] is the commutator of ̂︀ 𝑧 and ̂︀ 𝑧 in H t , c ( 𝑊 ) and [ ̂︀ 𝑧 , ̂︀ 𝑧 ] t =0 is the projection ofthis commutator to H c ( 𝑊 ) = H t , c ( 𝑊 ) / t H t , c ( 𝑊 ) . This is indeed an element in Z c ( 𝑊 ) anddefines a Poisson structure on this ring.We recall that an ideal 𝐼 of an arbitrary Poisson algebra 𝐴 is a Poisson ideal if { 𝐼, 𝐴 } ⊆ 𝐼 ,i.e., 𝐼 is stable under the Poisson bracket { 𝑎, −} for all 𝑎 ∈ 𝐴 . The Poisson core 𝒫 ( 𝐼 ) of anideal 𝐼 of 𝐴 is the largest Poisson ideal contained in 𝐼 . By a Poisson prime (resp. maximal)ideal we mean a prime (resp. maximal) ideal which is also a Poisson ideal. The Poisson core ofany prime ideal is a Poisson prime ideal. We denote by
PSpec ( 𝐴 ) the set of all Poisson primeideals of 𝐴 and by PMax ( 𝐴 ) the set of all Poisson maximal ideals. §3B. Symplectic leaves The (analytification of the) smooth part ( X c ( 𝑊 )) sm of X c ( 𝑊 ) is a Poisson manifold and admitsa foliation into symplectic leaves; that is, a stratification into smooth connected strata suchthat the rank of the bracket is maximal along strata. The strata are the symplectic leaves ofthe manifold (see [43]). By continuing this process on the complement X c ( 𝑊 ) ∖ ( X c ) sm weend up with a decomposition of X c ( 𝑊 ) into symplectic leaves. Brown and Gordon [12] haveshown that the leaves obtained in this way are in fact algebraic , i.e., locally closed in the Zariskitopology and finite in number. The leaf of a closed point m of X c ( 𝑊 ) consists of all closedpoints n ∈ X c ( 𝑊 ) such that m and n have the same Poisson core. Furthermore, it is shownin loc. cit. that each leaf ℒ is a smooth symplectic variety, and that the closure ℒ of the leaf ℒ containing a closed point 𝜒 is the zero locus V ( 𝒫 ( m 𝜒 )) of the Poisson core of its definingmaximal ideal. This shows in particular that the closure of each symplectic leaf is an irreducibleaffine Poisson variety. Lemma 3.1.
The set of symplectic leaves of X c ( 𝑊 ) is naturally in bijection with the set PSpec ( Z c ( 𝑊 )) of Poisson prime ideals of Z c ( 𝑊 ) . Proof.
Let ℒ be a symplectic leaf. As we noted above, the closure ℒ is an irreducible affinevariety and therefore the defining ideal p ℒ = I ( ℒ ) is a prime ideal. Moreover, as ℒ = V ( 𝒫 ( m 𝜒 )) for any closed point 𝜒 of ℒ , it follows that p 𝐿 = 𝒫 ( m 𝜒 ) is a Poisson prime ideal. The map ℒ ↦→ p ℒ is injective since if p ℒ = p ℒ ′ , then ℒ = V ( p ℒ ) = V ( p ℒ ′ ) = ℒ ′ , and this implies ℒ = ℒ ′ as the symplectic leaves form a stratification of X c ( 𝑊 ) . Now, let p be an arbitraryPoisson prime ideal of Z c ( 𝑊 ) . Then ℒ p := { m ∈ Max ( Z c ( 𝑊 )) | 𝒫 ( m ) = p } is a symplecticleaf by the description of symplectic leaves due to Brown and Gordon. By construction p ℒ p = p and therefore the map ℒ ↦→ p ℒ is also surjective. (cid:4) We immediately obtain the following.
Corollary 3.2.
The set of zero-dimensional symplectic leaves of X c ( 𝑊 ) is naturally in bijectionwith the set PMax ( Z c ( 𝑊 )) of Poisson maximal ideals of Z c ( 𝑊 ) .Analogous to Lusztig–Spaltenstein induction for nilpotent adjoint orbits of a reductive group,one can show that symplectic leaves are induced from zero-dimensional leaves for parabolicsubgroups of 𝑊 . Before we discuss this we give a short recollection about parabolic subgroups. §3C. Parabolic subgroups Recall that a parabolic subgroup of 𝑊 is the pointwise stabiliser 𝑊 h ′ of a subspace h ′ of h . Bya theorem of Steinberg [38, Theorem 1.5] the pair ( h ′ , 𝑊 h ′ ) is itself a complex reflection group.Moreover, 𝑊 h ′ is the stabiliser 𝑊 𝑏 of a generic point 𝑏 of h ′ . Hence, parabolic subgroups of 𝑊 are in fact the stabilisers of points of h . WYN B ELLAMY AND U LRICH T HIEL Define the rank of a complex reflection group 𝑊 to be the dimension of a faithful reflectionrepresentation of 𝑊 of minimal dimension. Let 𝑊 ′ be a parabolic subgroup of 𝑊 . We write ( h * 𝑊 ′ ) ⊥ := { 𝑦 ∈ h | 𝑥 ( 𝑦 ) = 0 for all 𝑥 ∈ h * 𝑊 ′ } . Then h = h 𝑊 ′ ⊕ ( h * 𝑊 ′ ) ⊥ is a decomposition of h as a 𝑊 ′ -module and ( h * 𝑊 ′ ) ⊥ is a faithfulreflection representation of 𝑊 ′ of minimal rank. Hence, the rank of 𝑊 ′ is dim ( h * 𝑊 ′ ) ⊥ . We willalways consider parabolic subgroups with this minimal reflection representation. In particular,if c : Ref ( 𝑊 ) → C is a 𝑊 -equivariant function, then the restriction c ′ of c to Ref ( 𝑊 ′ ) isa 𝑊 ′ -equivariant function and we understand the rational Cherednik algebra H c ′ ( 𝑊 ′ ) to bedefined with respect to this reflection representation of 𝑊 ′ .The group 𝑊 acts on its set of parabolic subgroups by conjugation. Given a parabolicsubgroup 𝑊 ′ the corresponding conjugacy class will be denoted ( 𝑊 ′ ) . We also require thepartial ordering on conjugacy classes of parabolic subgroups of 𝑊 defined by ( 𝑊 ) ≥ ( 𝑊 ) ifand only if 𝑊 is conjugate to a subgroup of 𝑊 . The ordering is chosen in this way so that itagrees with a geometric ordering to be introduced in the next paragraph.Finally, for a given parabolic subgroup 𝑊 ′ of 𝑊 , we denote by h 𝑊 ′ reg the subset of h 𝑊 ′ consisting of those points whose stabiliser in 𝑊 is equal to 𝑊 ′ . This is a locally closed subsetof h . We denote by Ξ( 𝑊 ′ ) the quotient 𝑁 𝑊 ( 𝑊 ′ ) /𝑊 ′ , where 𝑁 𝑊 ( 𝑊 ′ ) is the normaliser of 𝑊 ′ in 𝑊 . The group Ξ( 𝑊 ′ ) acts freely on h 𝑊 ′ reg . Remark 3.3.
Suppose that ( 𝑊, 𝑆 ) is a Coxeter group. In §2D we already used the standardparabolic subgroups 𝑊 𝐼 of 𝑊 for subsets 𝐼 ⊆ 𝑆 . Let h be the (complexified) geometricrepresentation of 𝑊 so that ( h , 𝑊 ) is a complex reflection group. Then 𝑊 𝐼 is a parabolicsubgroup of 𝑊 in the sense just defined. Moreover, it follows from Steinberg’s theorem and [1,Theorem 3.1] that, up to conjugacy, the parabolic subgroups of 𝑊 are precisely the standardparabolic subgroups 𝑊 𝐼 . §3D. Parabolic subgroup attached to a symplectic leaf If 𝑊 ′ is a parabolic subgroup of 𝑊 then h 𝑊 ′ reg /𝑊 denotes the image of h 𝑊 ′ reg in h /𝑊 . Thesymplectic leaves of X c ( 𝑊 ) are natural labeled by conjugacy classes of parabolic subgroups. Theorem 3.4.
The following holds:(a) For any symplectic leaf
ℒ ⊆ X c ( 𝑊 ) there exists a unique conjugacy class 𝑊 ℒ := ( 𝑊 ′ ) of parabolic subgroups of 𝑊 such that 𝜋 c ( ℒ ) ∩ h 𝑊 ′ reg /𝑊 is dense in 𝜋 c ( ℒ ) .(b) If ℒ , ℒ ′ ⊆ 𝑋 c ( 𝑊 ) are symplectic leaves with ℒ ⊆ ℒ ′ , then 𝑊 ℒ ≤ 𝑊 ℒ ′ . Proof.
The bijection of Lemma 3.1 is denoted p ↦→ ℒ p . The proof of [2, Proposition 4.8]shows that, for each Poisson prime p , there is a unique conjugacy class ( 𝑊 ′ ) with dim h =2 rk( 𝑊 ′ ) + dim ℒ p such that dim 𝜋 c ( ℒ p ) ∩ h 𝑊 ′ reg /𝑊 = dim h 𝑊 ′ reg /𝑊. Since 𝜋 c ( ℒ ) is irreducible and dim 𝜋 c ( ℒ ) = dim h 𝑊 ′ reg /𝑊 , this implies that 𝜋 c ( ℒ p ) ∩ h 𝑊 ′ reg /𝑊 is dense in 𝜋 c ( ℒ ) . (cid:4) §3E. Cuspidal reduction I We recall the main results from [2]. For a closed point 𝜒 of X c ( 𝑊 ) with defining maximal ideal m 𝜒 of Z c ( 𝑊 ) we set H c ,𝜒 ( 𝑊 ) := H c ( 𝑊 ) / m 𝜒 · H c ( 𝑊 ) . This is a finite-dimensional C -algebra. We call it cuspidal if 𝜒 is a zero-dimensional symplecticleaf of X c ( 𝑊 ) . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Theorem 3.5.
Let ℒ be a symplectic leaf of X c ( 𝑊 ) of dimension 𝑙 and 𝜒 a point on ℒ . Thenthere exists a parabolic subgroup 𝑊 ′ of 𝑊 of rank dim h − 𝑙 and a cuspidal algebra 𝐻 c ′ ,𝜒 ′ ( 𝑊 ′ ) such that H c ,𝜒 ( 𝑊 ) ≃ Mat | 𝑊/𝑊 ′ | ( H c ′ ,𝜒 ′ ( 𝑊 ′ )) . Moreover, there exists a functor Φ 𝜒 ′ ,𝜒 : H c ′ ,𝜒 ′ ( 𝑊 ′ ) - mod ∼ −→ H c ,𝜒 ( 𝑊 ) - mod defining anequivalence of categories such that Φ 𝜒 ′ ,𝜒 ( 𝑀 ) ≃ Ind 𝑊𝑊 ′ 𝑀 ∀ 𝑀 ∈ H c ′ ,𝜒 ′ ( 𝑊 ′ ) - mod as 𝑊 -modules.Since there are only finitely many zero dimensional leaves in X c ( 𝑊 ) the above result showsthat to describe the 𝑊 -module structure of all the simple modules for a particular rationalCherednik algebra one only needs to describe the 𝑊 ′ -module structure of the cuspidal simplemodules for each parabolic subgroup 𝑊 ′ of 𝑊 . §3F. Symplectic leaves and Calogero–Moser families As explained in §1D, there is a natural bijection between the set Ω c ( 𝑊 ) of Calogero–Moserfamilies and the points in ϒ − c (0) . If m ℱ denotes the point of ϒ − c (0) corresponding to thefamily ℱ , then m ℱ lies on a unique symplectic leaf ℒ ℱ of X c ( 𝑊 ) . Using Theorem 3.4 we canattach a unique conjugacy class 𝑊 ℱ := 𝑊 ℒ ℱ of parabolic subgroups of 𝑊 to ℱ . We define apartial ordering ⪯ on the Calogero–Moser families Ω c ( 𝑊 ) by ℱ ⪯ ℱ ′ ⇐⇒ ℒ ( ℱ ) ⊆ ℒ ( ℱ ′ ) . Proposition 3.6.
The following holds for any ℱ , ℱ ′ ∈ Ω c ( 𝑊 ) :(a) ℱ ⪯ ℱ ′ and ℱ ′ ⪯ ℱ if and only if ℱ = ℱ ′ .(b) ℱ ⪯ ℱ ′ implies that 𝑊 ℱ ≤ 𝑊 ℱ ′ . Proof.
Part (a) follows from directly from the definition of ⪯ and part (b) is a consequence ofTheorem 3.4(b). (cid:4) We say that a Calogero–Moser family ℱ is cuspidal if ℒ ℱ is a zero-dimensional leaf. By Ω cusp c ( 𝑊 ) we denote the set of cuspidal Calogero–Moser c -families. It follows from Theorem3.4 that PMax ( Z c ( 𝑊 )) ⊆ ϒ − c (0) . Hence, the set of zero-dimensional symplectic leaves of X c ( 𝑊 ) is in bijection with Ω cusp c ( 𝑊 ) . Lemma 3.7.
A singleton Calogero–Moser family is not cuspidal.
Proof. If ℱ is a singleton Calogero–Moser family, then by Theorem 1.4 the correspondingpoint m ℱ of X c ( 𝑊 ) is smooth. Therefore it is contained in the unique open leaf of X c ( 𝑊 ) .Since dim X c ( 𝑊 ) > , the open leaf is not zero-dimensional and hence the family is notcuspidal. (cid:4) The following well-known lemma is analogous to Lemma 2.5.
Lemma 3.8.
For any 𝛼 ∈ C × there is a canonical algebra isomorphism H c ( 𝑊 ) ≃ −→ H 𝛼 c ( 𝑊 ) ,which induces an algebra isomorphism H c ( 𝑊 ) ≃ −→ H 𝛼 c ( 𝑊 ) and a Poisson isomorphism X c ( 𝑊 ) ≃ −→ X 𝛼 c ( 𝑊 ) . Moreover, Ω c ( 𝑊 ) = Ω 𝛼 c ( 𝑊 ) and Ω cusp c ( 𝑊 ) = Ω cusp 𝛼 c ( 𝑊 ) .We can now state the main theorem of this paper Theorem A. If 𝑊 is of type 𝐴, 𝐵, 𝐷 or 𝐼 ( 𝑚 ) , then for any parameter c ≥ the cuspidal Lusztig c -families of 𝑊 equal the cuspidal Calogero–Moser c -families of 𝑊 . WYN B ELLAMY AND U LRICH T HIEL Proof.
The Weyl groups of type 𝐴 are dealt with in §5. For type 𝐵 , see Corollary 6.26, andtype 𝐷 is dealt with in Theorem 7.3. Finally, for the dihedral groups 𝐼 ( 𝑚 ) , see §8H. (cid:4) Based on this theorem we make the following conjecture.
Conjecture B.
For any finite Coxeter group and any real parameter c the cuspidal Lusztig c -families equal the cuspidal Calogero–Moser c -families.The proof of Theorem A follows from a case-by-case analysis in sections §5 to §8 usingseveral theoretical methods we develop in the next section. We will deduce in Lemma 4.11 thatConjecture B holds for the special case c = 0 for any 𝑊 . Note that because of Lemma 2.5 andLemma 3.8 it is sufficient to prove the conjecture only up to multiplication of the parameter bypositive real numbers. §4. Calculating cuspidal Calogero–Moser families To determine the cuspidal Calogero–Moser families we develop several theoretical meth-ods—both of representation theoretic and geometric nature. On the one hand, we introduce theconcept of rigid modules here and show that these always lie in a cuspidal family. On the otherhand, we develop a Clifford theory for symplectic leaves. This allows us to deal with Weylgroups of type 𝐷 later. All this is done for complex reflection groups in general. §4A. Rigid modules The key to figuring out which Calogero–Moser families are cuspidal for Coxeter groups isthe notion of rigid H c ( 𝑊 ) -modules. We show in Theorem C below that every rigid modulebelongs to a cuspidal family. In all examples we consider it turns out that there is at most onecuspidal family. These two facts allow us to find all cuspidal families. Definition 4.1.
A simple H c ( 𝑊 ) -module 𝐿 is said to be rigid if it is irreducible as a 𝑊 -module.This notion has played an important role for rational Cherednik algebras at 𝑡 = 1 , seee.g. [6]. At 𝑡 = 0 , the second author investigated rigid modules in [41]. Recently, they alsoplayed a prominent role in the work [14] of Ciubotaru on Dirac cohomology where they werecalled one- 𝑊 -type modules. The terminology we adopt comes from the theory of modulevarieties, where it is standard. Intuitively, a rigid module is one that cannot be deformed (forfixed parameter c ) to a continuous family of representation; see Lemma 4.9. On the other hand,if a simple H c ( 𝑊 ) -module is supported on a symplectic leaf of dimension greater than zerothen one can deform the representation along the leaf. Therefore it is intuitively clear thatrigid modules should be supported at zero dimensional leaves. Showing the precise connectionbetween rigidity and cuspidality depends on the following theorem. Theorem 4.2.
Let 𝑊 be a complex reflection group. Then no irreducible 𝑊 -module is inducedfrom a proper parabolic subgroup of 𝑊 , i.e., Ind 𝑊𝑊 ′ 𝜆 is reducible for all parabolic subgroups 𝑊 ′ (cid:40) 𝑊 .In order to give the proof of Theorem 4.2, we first give some preparatory lemmata. Let 𝐺 be a finite group. Given a character 𝜒 of 𝐺 , we denote by ℓ ( 𝜒 ) the length of 𝜒 , i.e. if 𝜒 = ∑︀ 𝑛𝑖 =1 𝑛 𝑖 𝜒 𝑖 with 𝜒 𝑖 ∈ Irr ( 𝐺 ) , then ℓ ( 𝜒 ) = ∑︀ 𝑛𝑖 =1 𝑛 𝑖 . Note that ( 𝜒, 𝜒 ) = ∑︀ 𝑛𝑖 =1 𝑛 𝑖 andtherefore √︀ ( 𝜒, 𝜒 ) ≤ ℓ ( 𝜒 ) ≤ ( 𝜒, 𝜒 ) , where ( · , · ) is the scalar product of characters.We define the branching index of a subgroup 𝑃 of 𝐺 as 𝑏 𝑃 ( 𝐺 ) := min { ℓ ( 𝜓 𝐺 ) | 𝜓 ∈ Irr ( 𝑃 ) } , USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS where 𝜓 𝐺 := Ind 𝐺𝑃 𝜓 . We say that 𝑃 is branching in 𝐺 if 𝑏 𝑃 ( 𝐺 ) > , i.e., 𝜓 𝐺 is reduciblefor all 𝜓 ∈ Irr ( 𝑃 ) . We can now reformulate Theorem 4.2 as saying that all proper parabolicsubgroups of 𝑊 are branching. Lemma 4.3. If 𝐺 has a central element which is not contained in 𝑃 , then 𝑃 is branching. Proof.
Let 𝑧 ∈ 𝑍 ( 𝐺 ) ∖ 𝑃 and let 𝜓 ∈ Irr ( 𝑃 ) . Note that 𝑧 𝑃 = 𝑧𝑃 𝑧 − = 𝑃 and therefore 𝑃 ∩ 𝑧 𝑃 = 𝑃 . Similarly, we have 𝑧 𝜓 = 𝜓 . Hence, ( 𝜓, 𝑧 𝜓 ) = ( 𝜓, 𝜓 ) = 1 and therefore 𝜓 𝐺 isnot irreducible by [15, 10.25]. (cid:4) Lemma 4.4.
Let 𝑁 be a normal subgroup of 𝐺 . Let 𝑃 be a subgroup of 𝐺 with branchingindex 𝑏 𝑃 ( 𝐺 ) > [ 𝐺 : 𝑁 ] . Then 𝑃 ∩ 𝑁 is branching in 𝑁 . Proof.
Suppose that 𝑃 ∩ 𝑁 is not branching in 𝑁 . Then there exists some 𝜂 ∈ Irr ( 𝑃 ∩ 𝑁 ) with 𝜓 := 𝜂 𝑁 ∈ Irr ( 𝑁 ) . By Clifford theory for 𝑁 (cid:69) 𝐺 , see [28, Theorem 19.3], we have ( 𝜂 𝐺 , 𝜂 𝐺 ) = ( 𝜓 𝐺 , 𝜓 𝐺 ) = [ 𝐼 𝐺 ( 𝜓 ) : 𝑁 ] , where 𝐼 𝐺 ( 𝜓 ) is the inertia subgroup of 𝜓 in 𝐺 . Hence, ℓ ( 𝜂 𝐺 ) ≤ [ 𝐼 𝐺 ( 𝜓 ) : 𝑁 ] . On the other hand, by Clifford theory for 𝑁 ∩ 𝑃 (cid:69) 𝑃 we have ( 𝜂 𝑃 , 𝜂 𝑃 ) = [ 𝐼 𝑃 ( 𝜂 ) : 𝑁 ∩ 𝑃 ] . Hence, ℓ ( 𝜂 𝑃 ) ≥ √︀ [ 𝐼 𝑃 ( 𝜂 ) : 𝑁 ∩ 𝑃 ] and therefore ℓ ( 𝜂 𝐺 ) ≥ 𝑏 𝑃 ( 𝐺 ) · √︀ [ 𝐼 𝑃 ( 𝜂 ) : 𝑁 ∩ 𝑃 ] . In total, we must have 𝑏 𝑃 ( 𝐺 ) ≤ [ 𝐼 𝐺 ( 𝜓 ) : 𝑁 ] √︀ [ 𝐼 𝑃 ( 𝜂 ) : 𝑁 ∩ 𝑃 ] ≤ [ 𝐺 : 𝑁 ] . Because of our assumption on 𝑏 𝑃 ( 𝐺 ) this is a contradiction. (cid:4) Lemma 4.5.
Suppose that 𝑁 (cid:69) 𝐺 . Then a subgroup 𝑄 of 𝑁 is branching in 𝑁 if and only ifall its 𝐺 -conjugates are branching in 𝑁 . Proof.
This simply follows from the fact that
Ind 𝑁 𝑔 𝑄 ∘ Con 𝑔,𝑄 = Con 𝑔,𝑁 ∘ Ind 𝑁𝑄 and thatconjugation Con 𝑔,𝑄 with 𝑔 defines a bijection between Irr ( 𝑄 ) and Irr ( 𝑔 𝑄 ) for all 𝑔 ∈ 𝐺 . (cid:4) For the proof of Theorem 4.2 we will need the classification of complex reflection groupsdue to Shephard and Todd [37], and in particular a description of the parabolic subgroups in theinfinite series 𝐺 ( 𝑚, 𝑚, 𝑛 ) . We quickly recall the definition of these groups. Let 𝑚, 𝑝, 𝑛 ∈ N > with 𝑝 dividing 𝑚 and let 𝜁 ∈ C be a primitive 𝑚 -th root of unity. Then 𝐺 ( 𝑚, 𝑝, 𝑛 ) isthe subgroup of GL 𝑛 ( C ) consisting of the generalised permutation matrices with entries in 𝜇 𝑚 := ⟨ 𝜁 ⟩ such that the product of all non-zero entries is an ( 𝑚/𝑝 ) -th root of unity. Thegroup 𝐺 ( 𝑚, 𝑝, 𝑛 ) is a normal subgroup of index 𝑝 in 𝐺 ( 𝑚, , 𝑛 ) . For a partition 𝜆 of an integer | 𝜆 | ≤ 𝑛 let S 𝜆 be the corresponding Young subgroup of the symmetric group S | 𝜆 | . We havean obvious embedding S 𝜆 × 𝐺 ( 𝑚, 𝑚, 𝑛 − | 𝜆 | ) ˓ → 𝐺 ( 𝑚, 𝑚, 𝑛 ) . The following lemma can bededuced from [39, 3.11]. Lemma 4.6.
Up to 𝐺 ( 𝑚, , 𝑛 ) -conjugacy the parabolic subgroups of 𝐺 ( 𝑚, 𝑚, 𝑛 ) are the stan-dard parabolic subgroups S 𝜆 × 𝐺 ( 𝑚, 𝑚, 𝑛 − | 𝜆 | ) for partitions 𝜆 of 𝑛 .We note that for the 𝐺 ( 𝑚, 𝑚, 𝑛 ) -conjugacy classes of parabolic subgroups of 𝐺 ( 𝑚, 𝑚, 𝑛 ) some 𝐺 ( 𝑚, , 𝑛 ) -conjugates of the above standard parabolic subgroups have to be taken intoaccount (see [39, 3.11]). For us, however, it is sufficient to know the 𝐺 ( 𝑚, , 𝑛 ) -conjugacyclasses because of Lemma 4.5. By Lemma 4.6 the maximal parabolic subgroups of 𝐺 ( 𝑚, 𝑚, 𝑛 ) are up to 𝐺 ( 𝑚, , 𝑛 ) -conjugacy of the form S 𝑘 × 𝐺 ( 𝑚, 𝑚, 𝑛 − 𝑘 ) for ≤ 𝑘 ≤ 𝑛 . Proof of Theorem 4.2.
Clearly, we can assume that 𝑊 acts irreducibly on h and that 𝑃 is amaximal parabolic subgroup. It is well-known (see [29, Corollary 3.24]) that the centre 𝑍 ( 𝑊 ) of 𝑊 is a cyclic group Z ℓ = ⟨ 𝜎 ⟩ . If 𝜎 ̸ = 1 , then 𝜎 fixes only the origin and so 𝜎 / ∈ 𝑊 ′ for anyproper parabolic subgroup of 𝑊 . Hence, if | 𝑍 ( 𝑊 ) | > , then the claim holds by Lemma 4.3.The classification of irreducible complex reflection groups shows that | 𝑍 ( 𝑊 ) | = 1 implies WYN B ELLAMY AND U LRICH T HIEL that 𝑊 ≃ 𝐺 ( 𝑚, 𝑚, 𝑛 ) for some 𝑚, 𝑛 . By Lemma 4.6 and Lemma 4.5 we can assume that 𝑃 = S 𝑘 × 𝐺 ( 𝑚, 𝑚, 𝑛 − 𝑘 ) , where ≤ 𝑘 ≤ 𝑛 . Let 𝜆 ∈ Irr ( 𝑃 ) .We assume first that 𝑚 > . The module 𝜋 𝜆 is isomorphic to 𝜋 ′ 𝜆 (cid:2) 𝜋 𝜇 for some 𝜋 ′ 𝜆 ∈ Irr ( S 𝑘 ) and 𝜋 𝜇 ∈ Irr ( 𝐺 ( 𝑚, 𝑚, 𝑛 − 𝑘 )) . Note that 𝑃 ⊂ 𝐺 ( 𝑚, 𝑚, 𝑘 ) × 𝐺 ( 𝑚, 𝑚, 𝑛 − 𝑘 ) ⊂ 𝑊 .If 𝑘 > then it suffices to show that Ind 𝐺 ( 𝑚,𝑚,𝑘 ) S 𝑘 𝜋 ′ 𝜆 is not irreducible. That is, we mayassume 𝑘 = 𝑛 . The symmetric group S 𝑛 is a quotient of 𝐺 ( 𝑚, 𝑚, 𝑛 ) , the morphism given bysending an element to the underlying permutation. Then we may consider 𝜋 ′ 𝜆 as an irreducible 𝐺 ( 𝑚, 𝑚, 𝑛 ) -module 𝜋 ′′ 𝜆 . Clearly 𝜋 ′′ 𝜆 | S 𝑛 = 𝜋 ′ 𝜆 . Hence [︁ Ind 𝐺 ( 𝑚,𝑚,𝑛 ) S 𝑛 𝜋 ′ 𝜆 : 𝜋 ′′ 𝜆 ]︁ ≥ . On the otherhand, dim Ind 𝐺 ( 𝑚,𝑚,𝑛 ) S 𝑛 𝜋 ′ 𝜆 = 𝑚 𝑛 − dim 𝜋 ′ 𝜆 . Hence it is not irreducible.In the case 𝑘 = 1 , we have 𝑃 = 𝐺 ( 𝑚, 𝑚, 𝑛 − ⊂ 𝐺 ( 𝑚, 𝑚, 𝑛 ) . Let 𝑄 := 𝐺 ( 𝑚, , 𝑛 − ⊆ 𝐺 ( 𝑚, , 𝑛 ) and note that 𝑃 = 𝑄 ∩ 𝐺 ( 𝑚, 𝑚, 𝑛 ) . If we can show that 𝑏 𝑄 ( 𝐺 ( 𝑚, , 𝑛 )) ≥ 𝑚 + 1 ,then Lemma 4.4 shows that 𝑃 is branching in 𝐺 ( 𝑚, 𝑚, 𝑛 ) . But this follows from the branchingrule ([36, Theorem 10]) which shows that, when viewing 𝜆 as an 𝑚 -multipartition, we have atleast 𝑚 + 1 constituents in Ind 𝐺 ( 𝑚, ,𝑛 ) 𝐺 ( 𝑚, ,𝑛 − 𝜋 𝜆 obtained by adding boxes to 𝜆 .Finally, we need to deal with the case 𝑚 = 1 , i.e. 𝑊 = S 𝑛 . In this case we have 𝑃 = S 𝑘 × S 𝑛 − 𝑘 and it is known that Ind S 𝑛 S 𝑘 × S 𝑛 − 𝑘 𝜋 𝜆 (cid:2) 𝜋 𝜇 = ∑︀ 𝜈 𝑐 𝜈𝜆,𝜇 𝜋 𝜈 , where 𝑐 𝜈𝜆,𝜇 are theLittlewood-Richardson coefficients. We need to show that ∑︀ 𝜈 𝑐 𝜈𝜆,𝜇 > . Presumably, this iswell-known. We will deduce it from the fact that for the Weyl group of Type 𝐵 𝑛 we have(7) Ind 𝐵 𝑛 𝐵 𝑘 × 𝐵 𝑛 − 𝑘 𝜋 ( 𝜆 (1) ,𝜆 (2) ) (cid:2) 𝜋 ( 𝜇 (1) ,𝜇 (2) ) = ∑︁ ( 𝜈 (1) ,𝜈 (2) ) 𝑐 𝜈 (1) 𝜆 (1) ,𝜇 (1) 𝑐 𝜈 (2) 𝜆 (2) ,𝜇 (2) 𝜋 ( 𝜈 (1) ,𝜈 (2) ) . Take 𝜆 (1) = 𝜆, 𝜆 (2) = ∅ , 𝜇 (1) = 𝜇 and 𝜇 (2) = ∅ . Then (7) implies that it suffices to showthat Ind 𝐵 𝑛 𝐵 𝑘 × 𝐵 𝑛 − 𝑘 𝜋 ( 𝜆 (1) ,𝜆 (2) ) (cid:2) 𝜋 ( 𝜇 (1) ,𝜇 (2) ) is not an irreducible 𝐵 𝑛 -module. But 𝐵 𝑛 contains anon-trivial central element that does not belong to either 𝐵 𝑛 − 𝑘 or 𝐵 𝑘 . This implies by Lemma4.3 that the induced module is not irreducible. (cid:4) Proposition 4.7. If 𝐿 is rigid, then 𝐿 ≃ 𝐿 c ( 𝜆 ) is a H c ( 𝑊 ) -module (isomorphic to 𝜆 as a 𝑊 -module), for some 𝜆 ∈ Irr 𝑊 . Proof. If 𝐿 is not a simple H c ( 𝑊 ) -module, then either the set-theoretic support of 𝐿 as a C [ h ] -module is not contained in { } , or the set-theoretic support of 𝐿 as a C [ h * ] -module isnot contained in { } . Without loss of generality, we assume that the set-theoretic support of 𝐿 as a C [ h ] -module is not contained in { } . Thus, there exists some 𝑏 ̸ = 0 in h such that m 𝑏 · 𝐿 ̸ = 0 , where m 𝑏 is the maximal ideal defining 𝑏 in h . The stabiliser 𝑊 𝑏 of 𝑏 is a propersubgroup of 𝑊 . Thus, the Bezrukavnikov–Etingof isomorphism, see [2, Theorem 4.3], impliesthat 𝐿 ≃ Ind 𝑊𝑊 𝑏 𝐿 ′ for some H c ′ ( 𝑊 𝑏 ) -module 𝐿 ′ . By Theorem 4.2, this implies that 𝐿 is notrigid. Thus, 𝐿 is a simple H c ( 𝑊 ) -module. The simple H c ( 𝑊 ) -modules are of the form 𝐿 c ( 𝜆 ) and 𝜆 always appears in the restriction to 𝑊 of 𝐿 c ( 𝜆 ) with non-zero multiplicity. The resultfollows. (cid:4) Remark 4.8.
Proposition 4.7 implies that if 𝐿 is a rigid module then h · 𝐿 = 0 = h * · 𝐿 . Inparticular, every rigid module is of “one- 𝑊 -type”, as recently defined in [14].The following lemma explains our choice of terminology since it is standard in finite-dimensional representation theory to say that a simple module 𝐿 of dimension 𝑑 for a finite-dimensional algebra 𝐴 is rigid if the set of points 𝑀 in the representation scheme Rep 𝑑 ( 𝐴 ) satisfying 𝑀 ≃ 𝐿 is open. Lemma 4.9.
Let 𝐿 be a rigid H c ( 𝑊 ) -module and set 𝑑 := dim 𝐿 . Let Rep 𝑑 ( H c ( 𝑊 )) be thescheme parameterizing all 𝑑 -dimensional representations of H c ( 𝑊 ) . Let 𝑋 be the set of points 𝑀 in Rep 𝑑 ( H c ( 𝑊 )) such that 𝑀 ≃ 𝐿 . Then 𝑋 is a connected component of Rep 𝑑 ( H c ( 𝑊 )) . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Proof.
Let 𝑆 be a reduced, irreducible affine C -variety and ℱ a flat family of H c ( 𝑊 ) -modulesover 𝑆 such that the fiber ℱ 𝑠 is isomorphic to 𝐿 , for some 𝑠 ∈ 𝑆 . Then it suffices to provethat h and h * act identically by zero on ℱ . Since C [ 𝑆 ] is a domain, its radical is zero, and henceit suffices to show that h and h * act as zero on every fiber ℱ 𝑠 of ℱ for 𝑠 ∈ MaxSpec ( C [ 𝑆 ]) .We may consider ℱ as a flat family of C [ h ] (cid:111) 𝑊 -modules instead and prove the claim inthis setting (repeating the argument for C [ h * ] (cid:111) 𝑊 ). Then the claim is a consequence ofTheorem 4.2 together with the (easy) classification of simple C [ h ] (cid:111) 𝑊 -modules. Firstly,since 𝑆 is connected ℱ 𝑠 ≃ 𝐿 as a 𝑊 -module for all 𝑠 . This is well-known and followsfor instance from [18, Corollary 1.4]. Therefore, it suffices to show that if 𝑀 is any simple C [ h ] (cid:111) 𝑊 -module such that h * ⊂ C [ h ] does not act identically zero, then 𝑀 ̸≃ 𝐿 . If h * does not act identically zero then there exists a non-zero character 𝜒 : C [ h ] → C such that 𝑀 𝜒 = { 𝑚 ∈ 𝑀 | 𝑥 · 𝑚 = 𝜒 ( 𝑥 ) 𝑚, ∀ 𝑥 ∈ h * } is non-zero. Let 𝑊 ′ (cid:40) 𝑊 be the stabilizer of 𝜒 ∈ h . Since 𝑀 is simple, 𝑀 𝜒 is a simple 𝑊 ′ -module and 𝑀 ≃ Ind C [ h ] (cid:111) 𝑊 C [ h ] (cid:111) 𝑊 ′ 𝑀 𝜒 . By Theorem4.2, 𝑀 is not irreducible. Hence 𝑀 ̸≃ 𝐿 as required.To deduce the statement of the lemma, take 𝑆 to be any irreducible component (with reducedscheme structure) of Rep 𝑑 ( H c ( 𝑊 )) containing 𝐿 . (cid:4) Notice that Lemma 4.9 shows that the set of all rigid modules in
Rep 𝑑 ( H c ( 𝑊 )) is open. Ingeneral, the connected component 𝑋 has a very non-trivial scheme structure. This can be seenfrom Voigt’s Lemma [18] which implies that(8) dim 𝑋 − dim 𝑋 red = dim Ext H c ( 𝑊 ) ( 𝐿, 𝐿 ) . One can compute, using the projective resolution (2.5) of [16, page 259], that for a rigid module 𝐿 we have Ext ∙ H c ( 𝑊 ) ( 𝐿, 𝐿 ) ≃ ∧ ∙ 𝑉 ⊗ 𝑊 End C ( 𝐿 ) , where 𝑉 = h ⊕ h * . In particular, it is easy to construct examples of rigid modules wherethe right hand side of (8) is strictly positive. Also, the variety Rep 𝑑 ( H c ( 𝑊 )) can have manyconnected components. This can be seen, for instance, by considering the case c = 0 .Via the bijection Irr 𝑊 → Irr H c ( 𝑊 ) given by Proposition 1.3, the element 𝜆 ∈ Irr 𝑊 issaid to be c -rigid if 𝐿 c ( 𝜆 ) is a rigid H c ( 𝑊 ) -module. The following is the main theorem of thissection. Theorem C.
Let 𝑊 be a complex reflection group. If 𝜆 ∈ Irr 𝑊 is c -rigid, then 𝜆 lies in acuspidal Calogero–Moser c -family. Proof.
Let ℱ be the Calogero–Moser c -family of 𝐿 c ( 𝜆 ) and let 𝜒 be the correspondingpoint of X c ( 𝑊 ) . Suppose that ℱ is not cuspidal. Then by Theorem 3.5 there is a para-bolic subgroup 𝑊 ′ of 𝑊 , a cuspidal symplectic leaf 𝜒 ′ of X c ′ ( 𝑊 ′ ) , and an equivalence Φ 𝜒 ′ ,𝜒 : H c ′ ,𝜒 ′ ( 𝑊 ′ ) - mod ∼ −→ H c ,𝜒 ( 𝑊 ) - mod such that Φ 𝜒 ′ ,𝜒 ( 𝑀 ) ≃ Ind 𝑊𝑊 ′ 𝑀 as 𝑊 -modulesfor all 𝑀 ∈ H c ′ ,𝜒 ′ ( 𝑊 ′ ) - mod . In particular, there must exist a 𝑊 ′ -module 𝑀 with Ind 𝑊𝑊 ′ 𝑀 ≃ 𝐿 c ( 𝜆 ) ≃ 𝜆 . But this is not possible by Theorem 4.2. (cid:4) Of course, the major advantage of rigid modules is that they are easily detected.
Lemma 4.10.
Let 𝜆 : 𝐺 → GL 𝑟 ( C ) be an irreducible representation of 𝑊 . Then 𝐿 c ( 𝜆 ) is arigid module for H c ( 𝑊 ) if and only if(9) ∑︁ 𝑠 ∈ Ref ( 𝑊 ) c ( 𝑠 )( 𝑦, 𝛼 𝑠 )( 𝛼 ∨ 𝑠 , 𝑥 ) 𝜆 ( 𝑠 ) = 0 for all 𝑦 ∈ h and 𝑥 ∈ h * . Proof.
The module 𝐿 c ( 𝜆 ) is rigid if and only if it is as a 𝑊 -module isomorphic to 𝜆 . Moreover,by Remark 4.8 both h and h * act trivially on 𝐿 c ( 𝜆 ) . Hence, 𝐿 c ( 𝜆 ) is rigid if and only if the WYN B ELLAMY AND U LRICH T HIEL representation ̂︀ 𝜆 : 𝑇 ( h ⊕ h * ) (cid:111) 𝑊 → Mat 𝑟 ( C ) with h and h * acting trivially and 𝑊 acting by 𝜆 descends to H c ( 𝑊 ) . This is the case if and only if ̂︀ 𝜆 ([ 𝑦, 𝑥 ]) = 0 , and this is equivalent to theasserted equation. (cid:4) Lemma 4.11.
For any 𝑊 we have Ω cusp0 ( 𝑊 ) = Lus cusp0 ( 𝑊 ) , i.e. Conjecture B holds for c = 0 . Proof.
Recall from Lemma 2.3 that Ω ( 𝑊 ) = Lus ( 𝑊 ) = { Irr 𝑊 } . Furthermore, recall fromExample 2.1 that truncated induction is for c = 0 just usual induction, i.e. j 𝑊𝑊 𝐼 = Ind 𝑊𝑊 𝐼 fora parabolic subgroup 𝑊 𝐼 of 𝑊 . Now, if the unique Lusztig family were not cuspidal, thenthe irreducible characters of 𝑊 would all be induced from a proper parabolic subgroup of 𝑊 ,but this is not possible by Theorem 4.2. Hence, the unique Lusztig family is cuspidal. On theother hand, all 𝜆 ∈ Irr 𝑊 are rigid for c = 0 by Lemma 4.10. Hence, each 𝜆 ∈ Irr 𝑊 lies in acuspidal Calogero–Moser family by Theorem C. As there is just one Calogero–Moser family,this one is cuspidal. (cid:4) Remark 4.12.
Ciubotaru [14] has recently classified the rigid H c ( 𝑊 ) -modules for all Weylgroups and all parameters. We will independently obtain this classification for non-exceptionalCoxeter groups from sections §5 to §8. Ciubotaru furthermore shows for all Weyl groups atequal parameters—except 𝐸 —that the rigid modules always lie in a single Calogero–Moserfamily, and that this family contains the (unique) cuspidal Lusztig family; for 𝐹 and 𝐸 this isin fact an equality. Using Theorem C, this shows that one direction of Conjecture B also holdsfor 𝐹 and 𝐸 for equal parameters. However, a classification of the cuspidal symplectic leavesis still open in all cases not covered by our Theorem A. §4B. Cuspidal reduction II For a conjugacy class ( 𝑊 ′ ) of parabolic subgroups of 𝑊 we denote by PSpec ( 𝑊 ′ ) ( Z c ( 𝑊 )) thesubset of PSpec ( Z c ( 𝑊 )) of Poisson prime ideals p with 𝑊 ℒ p = ( 𝑊 ′ ) . This set might be empty.If 𝑊 ⊆ 𝐺 ⊆ 𝑁 GL ( h ) ( 𝑊 ) is a finite subgroup such that c : Ref ( 𝑊 ) → C is 𝐺 -invariant, then 𝐺 acts on H c ( 𝑊 ) by algebra automorphisms. This induces an action of 𝐺 by Poisson algebraautomorphisms on Z c ( 𝑊 ) and, since 𝑊 acts trivially, this action factors through 𝐺/𝑊 . Ap-plied to a parabolic subgroup 𝑊 ′ of 𝑊 and an arbitrary 𝑊 -invariant function c : Ref ( 𝑊 ) → C this shows that Ξ( 𝑊 ′ ) acts on Z c ′ ( 𝑊 ′ ) . Here, and below, c ′ denotes the restriction of c to Ref ( 𝑊 ) ∩ 𝑊 ′ = Ref ( 𝑊 ′ ) .The following was shown by Losev [31, Theorem 1.3.2]. Theorem 4.13.
Let 𝑊 ′ be a parabolic subgroup of 𝑊 . The group Ξ( 𝑊 ′ ) acts on the set PMax ( Z c ′ ( 𝑊 ′ )) such that there is a bijection PSpec ( 𝑊 ′ ) ( Z c ( 𝑊 )) ←→ PMax ( Z c ′ ( 𝑊 ′ )) / Ξ( 𝑊 ′ ) . Losev considers in [31] a different completion of the rational Cherednik algebra than the oneused in [2] (which is based on a construction by Bezrukavnikov and Etingof). Therefore we willnow show that Theorem 4.13 still holds in the context of Bezrukavnikov–Etingof completions.Fix a parabolic subgroup 𝑊 ′ of 𝑊 and let 𝑁 := 𝑁 𝑊 ( 𝑊 ′ ) . Let 𝑈 be an affine open subsetof h /𝑊 such that 𝑈 ∩ h 𝑊 ′ reg /𝑊 is closed, but non-empty, in 𝑈 . Let 𝑉 denote the preimage of 𝑈 in h . Then 𝑉 is 𝑊 -stable and 𝑉 𝑊 ′ = h 𝑊 ′ reg ∩ 𝑉 is closed in 𝑉 . Let k denote the 𝑊 ′ -modulecomplement to h 𝑊 ′ in h . It is an 𝑁 -module.Let 𝐴 := C [ 𝑈 ] and set 𝑍 := 𝐴 ⊗ C [ h ] 𝑊 Z c ( 𝑊 ) . The prime ideal of 𝐴 defining 𝑈 ∩ h 𝑊 ′ reg /𝑊 is denoted q . Let ̂︀ 𝐴 q be the completion of 𝐴 along q and set ̂︀ X c ( 𝑊 ) := Spec ( ̂︀ 𝐴 q ⊗ 𝐴 𝑍 ) . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Morally speaking, ̂︀ X c ( 𝑊 ) should be thought of as the formal neighbourhood of 𝜋 − ( h 𝑊 ′ reg /𝑊 ) in X c ( 𝑊 ) . However, since 𝑍 is not a finite 𝐴 -module, this is not strictly true.Let 𝐴 ′ = C [ k /𝑊 ′ × 𝑉 𝑊 ′ ] and q ′ the prime ideal defining { } × 𝑉 𝑊 ′ in k /𝑊 ′ × 𝑉 𝑊 ′ . Then ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) := Spec (︁ ̂︀ 𝐴 ′ q ′ ⊗ 𝐴 ′ Z c ′ ( 𝑊 ′ ) ⊗ C [ 𝑇 * 𝑉 𝑊 ′ ] )︁ , where 𝑇 * 𝑉 𝑊 ′ is the cotangent bundle of 𝑉 𝑊 ′ . The group Ξ( 𝑊 ′ ) acts on ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) . Thefollowing is an analogue of the isomorphism Θ in section 3.7 of [7]; a complete proof is givenin [5]. Theorem 4.14.
There is an isomorphism of affine Poisson varieties
Φ : ̂︀ X c ( 𝑊 ) ∼ −→ ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) / Ξ( 𝑊 ′ ) . In order to deduce Theorem 4.13 from Theorem 4.14, we require the following lemma.
Lemma 4.15.
The map p ↦→ ̂︀ 𝐴 q ⊗ 𝐴 p defines a bijection between PSpec ( 𝑊 ′ ) ( X c ( 𝑊 )) and theset of Poisson prime ideals of ̂︀ 𝐴 q ⊗ 𝐴 𝑍 of height dim k . Proof.
First, we must show that ̂︀ 𝐴 q ⊗ 𝐴 p is prime in ̂︀ 𝐴 q ⊗ 𝐴 𝑍 . Let 𝑌 = 𝑈 ∩ h 𝑊 ′ reg /𝑊 and denoteby 𝜋 ( ℒ p ) the closure of 𝜋 ( ℒ p ) ∩ 𝑈 in 𝑈 . Recall that 𝜋 ( ℒ p ) ∩ h 𝑊 ′ reg /𝑊 is dense in h 𝑊 ′ reg /𝑊 .This implies that 𝜋 ( ℒ p ) ∩ 𝑌 is dense in the closed, irreducible set 𝑌 , i.e. 𝜋 ( ℒ ) ∩ 𝑌 = 𝑌 .Similarly, 𝜋 ( ℒ p ) ∩ 𝑌 is dense in 𝜋 ( ℒ p ) . Thus, 𝜋 ( ℒ p ) = 𝑌 . This implies that q ⊂ p ∩ 𝐴 andhence 𝑍 · q ⊂ p . Since ̂︀ 𝐴 q is flat over 𝐴 , we have a short exact sequence → ̂︀ 𝐴 q ⊗ 𝐴 p → ̂︀ 𝐴 q ⊗ 𝐴 𝑍 → ̂︀ 𝐴 q ⊗ 𝐴 ( 𝑍/ p ) → . The order filtration on H c ( 𝑊 ) defines an increasing filtration ℱ 𝑖 𝑍 on 𝑍 such that each pieceis a coherent 𝐴 -module. This restricts to a filtration on p and we have a short exact sequence → ℱ 𝑖 p → ℱ 𝑖 𝑍 → ℱ 𝑖 𝑍/ ℱ 𝑖 p → of coherent 𝐴 -modules. Since tensor products commutewith colimits, ̂︀ 𝐴 q ⊗ 𝐴 ( 𝑍/ p ) = lim 𝑖 →∞ ̂︀ 𝐴 q ⊗ 𝐴 ( ℱ 𝑖 𝑍/ ℱ 𝑖 p ) . But 𝑍 · q ⊂ p implies that ̂︀ 𝐴 q ⊗ 𝐴 ( ℱ 𝑖 𝑍/ ℱ 𝑖 p ) = lim ∞← 𝑚 ( ℱ 𝑖 𝑍/ ℱ 𝑖 p ) q 𝑚 ( ℱ 𝑖 𝑍/ ℱ 𝑖 p ) = ℱ 𝑖 𝑍/ ℱ 𝑖 p . Thus, ̂︀ 𝐴 q ⊗ 𝐴 ( 𝑍/ p ) = 𝑍/ p is a domain and ̂︀ 𝐴 q ⊗ 𝐴 p is prime. It is clearly Poisson; see [2,Lemma 3.5]. Moreover, the fact that ̂︀ 𝐴 q ⊗ 𝐴 ( 𝑍/ p ) = 𝑍/ p shows that ̂︀ 𝐴 q ⊗ 𝐴 p = ̂︀ 𝐴 q ⊗ 𝐴 p if and only if p = p . Lemma 3.3 of loc. cit. says that ht( ̂︀ 𝐴 q ⊗ 𝐴 p ) = ht( p ) , which equals 𝑊 ′ ) . Thus, the map we have written down is injective.On the other hand, if p ′ is a Poisson prime in ̂︀ 𝐴 q ⊗ 𝐴 𝑍 of height dim k , then Lemmata 3.3and 3.5 of loc. cit. say that p := p ′ ∩ 𝑍 is a Poisson prime of height dim k . Therefore, we justneed to show that 𝑌 ∩ 𝜋 ( ℒ p ) is dense in 𝑌 . The prime p belongs to PSpec ( 𝑊 ′′ ) ( X c ( 𝑊 )) forsome parabolic 𝑊 ′′ of 𝑊 of the same rank as 𝑊 ′ . The sets 𝑈 ∩ h 𝑊 ′′ reg /𝑊 and 𝑌 are disjointif 𝑊 ′′ / ∈ ( 𝑊 ′ ) , which implies that the image in ̂︀ 𝐴 q of the ideal defining 𝑈 ∩ h 𝑊 ′′ reg /𝑊 is thewhole of ̂︀ 𝐴 q . Therefore the image of p ∩ 𝐴 in ̂︀ 𝐴 q would also be the whole of ̂︀ 𝐴 q if 𝑊 ′′ / ∈ ( 𝑊 ′ ) .But since ̂︀ 𝐴 q ⊗ 𝐴 p is contained in p ′ , this cannot happen and thus 𝑊 ′′ ∈ ( 𝑊 ′ ) as required. (cid:4) Proof of Theorem 4.13.
Since the symplectic structure on 𝑇 * 𝑉 𝑊 ′ is non-degenerate, the onlyPoisson prime in C [ 𝑇 * 𝑉 𝑊 ′ ] is the zero ideal. Therefore, every Poisson prime in Z c ′ ( 𝑊 ′ ) ⊗ C [ 𝑇 * 𝑉 𝑊 ′ ] has height at most dim k and the Poisson primes of height dim k are in bijectionwith the Poisson maximal ideals of Z c ′ ( 𝑊 ′ ) . Repeating the arguments of Lemma 4.15, there WYN B ELLAMY AND U LRICH T HIEL is a bijection between the Poisson primes in Z c ′ ( 𝑊 ′ ) ⊗ C [ 𝑇 * 𝑉 𝑊 ′ ] of height dim k and thePoisson primes of height dim k in C [︁̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) ]︁ .By Lemma 4.15 and Theorem 4.14, the set PSpec ( 𝑊 ′ ) ( X c ( 𝑊 )) is in bijection with the sym-plectic leaves in ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) / Ξ( 𝑊 ′ ) of dimension dim h − dim k ) . Since Ξ( 𝑊 ′ ) acts freely on 𝑉 𝑊 ′ it also acts freely on ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) . Therefore the symplectic leaves in ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) / Ξ( 𝑊 ′ ) of dimension dim h − dim k ) are in bijection with the Ξ( 𝑊 ′ ) -orbits of symplectic leaves in ̂︀ X c ′ ( 𝑊 ′ , 𝑉 ) of dimension dim h − dim k ) . But, as explained above, this is the same as the Ξ( 𝑊 ′ ) -orbits of Poisson maximal ideals Z c ′ ( 𝑊 ′ ) . (cid:4) §4C. Clifford theory Throughout this section we fix an irreducible complex reflection group ( h , 𝑊 ) . Moreover, weassume that there exists a normal subgroup 𝐾 (cid:67) 𝑊 such that 𝐾 acts, via inclusion in 𝑊 , on h as a complex reflection group (though h need not be irreducible as a 𝐾 -module). Since 𝐾 isnormal in 𝑊 , the group 𝑊 acts on Ref ( 𝐾 ) by conjugation. Let us fix a 𝑊 -equivariant function c : Ref ( 𝐾 ) → C . We extend this to a 𝑊 -equivariant function c : Ref ( 𝑊 ) → C by setting c ( 𝑠 ) = 0 for all 𝑠 ∈ Ref ( 𝑊 ) (cid:114) Ref ( 𝐾 ) . A 𝐾 -equivariant function on Ref ( 𝐾 ) is not always 𝑊 -equivariant. For our choice of parameter c , the inclusion 𝐾 ˓ → 𝑊 extends to an algebraembedding H c ( 𝐾 ) ˓ → H c ( 𝑊 ) , which is the identity on h and h * . Let Γ =
𝑊/𝐾 . As explainedin [4, Section 4.1], the group 𝑊 acts on H c ( 𝐾 ) by conjugation. Thus, it acts on Z c ( 𝐾 ) . Thisaction factors through Γ .We will require the following lemma. Lemma 4.16.
Under the graded 𝑊 -module identification H c ( 𝑊 ) = ⨁︀ 𝑤 ∈ 𝑊 C [ h ] ⊗ C [ h * ] ⊗ 𝑤 ,every non-zero element 𝑧 = ∑︀ 𝑤 ∈ 𝑊 𝑧 𝑤 · 𝑤 ∈ Z c ( 𝑊 ) satisfies 𝑧 ̸ = 0 . Proof.
A reformulation of the PBW property is that, under the filtration ℱ 𝑖 H c ( 𝑊 ) putting h and h * in degree one, 𝑊 in degree zero and ℱ − := 0 , the associated graded gr ℱ H c ( 𝑊 ) equals C [ h ⊕ h * ] (cid:111) 𝑊 . An easy induction on 𝑘 shows that ℱ 𝑘 H c ( 𝑊 ) = ( C [ h ] ⊗ C [ h * ]) ≤ 𝑘 ⊗ C 𝑊 as a 𝑊 -module, where ( C [ h ] ⊗ C [ h * ]) ≤ 𝑘 is the sum of all graded pieces of degree at most 𝑘 . Thenthe short exact sequences → ℱ 𝑘 − → ℱ 𝑘 → ℱ 𝑘 / ℱ 𝑘 − → can be identified, as short exactsequences of 𝑊 -modules, with → ( C [ h ] ⊗ C [ h * ]) ≤ 𝑘 − ⊗ C 𝑊 → ( C [ h ] ⊗ C [ h * ]) ≤ 𝑘 ⊗ C 𝑊 → ( C [ h ] ⊗ C [ h * ]) 𝑘 ⊗ C 𝑊 → . The image of Z c ( 𝑊 ) under gr ℱ equals C [ h × h * ] 𝑊 . Therefore, 𝑧 = ∑︀ 𝑤 ∈ 𝑊 𝑧 𝑤 · 𝑤 ∈ ℱ 𝑘 (cid:114) ℱ 𝑘 − then its (non-zero!) image in ( C [ h ] ⊗ C [ h * ]) 𝑘 ⊗ C 𝑊 belongs to ( C [ h ] ⊗ C [ h * ]) 𝑊𝑘 . In particular, 𝑧 ̸ = 0 . (cid:4) Proposition 4.17.
The centre Z c ( 𝑊 ) of H c ( 𝑊 ) equals the subalgebra Z c ( 𝐾 ) Γ of Z c ( 𝐾 ) .Moreover, the embedding Z c ( 𝑊 ) ˓ → Z c ( 𝐾 ) is as Poisson algebras. Proof.
Clearly, Z c ( 𝑊 ) ∩ Z c ( 𝐾 ) ⊆ Z c ( 𝐾 ) 𝑊 . Therefore, we just need to show that Z c ( 𝑊 ) ⊂ Z c ( 𝐾 ) . Fix coset representatives 𝑤 , . . . , 𝑤 ℓ of 𝐾 in 𝑊 . Then H c ( 𝑊 ) = ⨁︀ ℓ𝑖 =1 H c ( 𝐾 ) 𝑤 𝑖 as a left H c ( 𝐾 ) -module. Let 𝑧 = ∑︀ ℓ𝑖 =1 𝑧 𝑖 𝑤 𝑖 denote an element in Z c ( 𝑊 ) with 𝑧 𝑖 ∈ H c ( 𝐾 ) forall 𝑖 . We wish to show that 𝑧 𝑖 = 0 for 𝑖 ̸ = 1 . Let 𝑓 ∈ H c ( 𝐾 ) . Then [ 𝑓, 𝑧 ] = ℓ ∑︁ 𝑖 =1 ([ 𝑓, 𝑧 𝑖 ] + 𝑧 𝑖 ( 𝑓 − 𝑤 𝑖 ( 𝑓 ))) 𝑤 𝑖 . Since [ 𝑓, 𝑧 𝑖 ] + 𝑧 𝑖 ( 𝑓 − 𝑤 𝑖 ( 𝑓 )) ∈ H c ( 𝐾 ) for all 𝑖 , we must have [ 𝑓, 𝑧 𝑖 ] + 𝑧 𝑖 ( 𝑓 − 𝑤 𝑖 ( 𝑓 )) = 0 . Inparticular, this implies that 𝑧 ∈ Z c ( 𝐾 ) ∩ Z c ( 𝑊 ) . Without loss of generality, 𝑧 = 0 . But nowit follows from Lemma 4.16 that 𝑧 = 0 . Thus, Z c ( 𝑊 ) = Z c ( 𝐾 ) 𝑊 . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
It is clear that the embedding is as Poisson algebras; one can see this directly from theconstruction or simply by noting that the bracket is Γ -invariant and hence restricts to Z c ( 𝐾 ) Γ . (cid:4) Thus, geometrically we have a Poisson morphism 𝜂 : X c ( 𝐾 ) → X c ( 𝑊 ) identifying X c ( 𝑊 ) with X c ( 𝐾 ) / Γ . It is a finite, surjective map which is generically a Γ -covering. This fits into acommutative diagram X c ( 𝐾 ) X c ( 𝑊 ) h /𝐾 × h * /𝐾 h /𝑊 × h * /𝑊 𝜂 ϒ c ,𝐾 ϒ c ,𝑊 Lemma 4.18. If ℒ is a leaf of X c ( 𝐾 ) , then 𝜂 ( ℒ ) is a finite union of leaves of X c ( 𝑊 ) . Proof.
Since the stratification of X c ( 𝑊 ) by symplectic leaves is finite, it suffices to show that 𝜂 ( ℒ ) is a union of leaves, i.e. invariant under Hamiltonian flows. After a suitable localization, wemay assume that ℒ is closed in X c ( 𝐾 ) . Then 𝜂 ( ℒ ) is closed. It is invariant under Hamiltonianflows if and only if the semi-prime ideal 𝐼 ( 𝜂 ( ℒ )) is Poisson. But 𝐼 ( 𝜂 ( ℒ )) = 𝐼 ( ℒ ) ∩ Z c ( 𝐾 ) Γ .Since 𝐼 ( ℒ ) is Poisson and the bracket is invariant under Γ , if 𝑧 ∈ 𝐼 ( 𝜂 ( ℒ )) ∩ Z c ( 𝐾 ) Γ and ℎ ∈ Z c ( 𝐾 ) Γ , then { 𝑧, ℎ } ∈ 𝐼 ( 𝜂 ( ℒ )) ∩ Z c ( 𝐾 ) Γ , as required. (cid:4) Note that, in general, the preimage of a leaf of X c ( 𝑊 ) is not a leaf. Let X c ( 𝐾 ) sing be thesingular locus of X c ( 𝐾 ) , let X c ( 𝐾 ) sm be the smooth locus and let X c ( 𝐾 ) free be the locuswhere Γ acts freely. The following is the geometric counterpart of [4, Lemma 4.12]. Proposition 4.19.
The preimage 𝜂 − ( X c ( 𝑊 ) sm ) equals X c ( 𝐾 ) sm ∩ X c ( 𝐾 ) free . Proof.
Since Γ preserves the Poisson structure on X c ( 𝐾 ) , for each 𝑝 ∈ X c ( 𝐾 ) sm , the group Γ 𝑝 acts symplectically on the tangent space 𝑇 𝑝 X c ( 𝐾 ) sm . Thus, ( 𝑇 𝑝 X c ( 𝐾 ) sm ) / Γ 𝑝 is smoothif and only if Γ 𝑝 = 1 . Using the fact that one can linearize the action of a finite group in theformal neighborhood of any fixed point, this implies that the smooth locus of X c ( 𝐾 ) sm / Γ equals ( X c ( 𝐾 ) sm ∩ X c ( 𝐾 ) free ) / Γ . Hence 𝜂 − ( X c ( 𝑊 ) sm ) ∩ X c ( 𝐾 ) sm = X c ( 𝐾 ) sm ∩ X c ( 𝐾 ) free . On the other hand, X c ( 𝐾 ) sing is a union of symplectic leaves ℒ with dim ℒ < dim X c ( 𝐾 ) .Therefore Lemma 4.18 implies that 𝜂 ( X c ( 𝐾 ) sing ) ⊂ X c ( 𝑊 ) sing . (cid:4) The following was stated in [4] in the case Γ is a cyclic group. We give a simple geometricproof. Theorem 4.20.
Let c : Ref ( 𝐾 ) → C be 𝑊 -equivariant.(a) The group Γ acts on Ω c ( 𝐾 ) such that 𝜎 ℱ = { 𝜎 𝜆 | 𝜆 ∈ ℱ } for 𝜎 ∈ Γ and ℱ ∈ Ω c ( 𝐾 ) .(b) There is a natural bijection between Ω c ( 𝑊 ) and Ω c ( 𝐾 ) / Γ given by Ω c ( 𝑊 ) ∋ ℱ ←→ { 𝜆 ∈ Irr ( 𝐾 ) | 𝜆 ⊂ Res 𝑊𝐾 𝜇 for some 𝜇 ∈ ℱ } ∈ Ω c ( 𝐾 ) / Γ . Proof.
Recall the notation from §1C. We will use the notation and results from [4, §3, §4].Let Z c ( 𝑊 ) denote the quotient of Z c ( 𝑊 ) by the ideal generated by 𝐷 ( 𝑊 ) + , Z c ( 𝐾 ) thequotient of Z c ( 𝐾 ) by the ideal generated by 𝐷 ( 𝐾 ) + and ̃︀ Z c ( 𝐾 ) the quotient of Z c ( 𝐾 ) bythe ideal generated by 𝐷 ( 𝑊 ) + . We also let ̃︀ H c ( 𝐾 ) denote the quotient of H c ( 𝐾 ) by the idealgenerated by 𝐷 ( 𝑊 ) + . The Satake isomorphism [16, Theorem 3.1] implies that the natural map Z c ( 𝑊 ) → ̃︀ Z c ( 𝐾 ) is an embedding. The group Γ acts on ̃︀ Z c ( 𝐾 ) and Proposition 4.17 nowimplies that Z c ( 𝑊 ) = ̃︀ Z c ( 𝐾 ) Γ . Thus, Z c ( 𝑊 ) = ̃︀ Z c ( 𝐾 ) Γ ˓ → ̃︀ Z c ( 𝐾 ) (cid:16) Z c ( 𝐾 ) . WYN B ELLAMY AND U LRICH T HIEL The kernel of the surjection ̃︀ Z c ( 𝐾 ) (cid:16) Z c ( 𝐾 ) is nilpotent. Therefore it identifies the primitiveidempotents in both algebras.Let { 𝑑 𝑖 } 𝑖 ∈ Ω c ( 𝑊 ) , { 𝑏 ′ 𝑗 } 𝑗 ∈ Ω c ( 𝐾 ) , { 𝑏 𝑗 } 𝑗 ∈ Ω c ( 𝐾 ) , denote the primitive idempotents in Z c ( 𝑊 ) , resp. ̃︀ Z c ( 𝐾 ) and Z c ( 𝐾 ) . Then Γ acts on { 𝑏 ′ 𝑗 } 𝑗 ∈ Ω c ( 𝐾 ) and the rule 𝑏 ′ 𝑗 ↦→ ∑︁ 𝜎 ∈ Γ / Stab Γ ( 𝑏 ′ 𝑗 ) 𝜎 𝑏 ′ 𝑗 defines a bijection { 𝑑 𝑖 } 𝑖 ∈ Ω c ( 𝑊 ) 1:1 ←→ { 𝑏 ′ 𝑗 } / Γ . There is a natural surjective map ̃︀ H c ( 𝐾 ) (cid:16) H c ( 𝐾 ) and the kernel of this map is generatedby certain central nilpotent elements in ̃︀ H c ( 𝐾 ) . In particular, the kernel is contained in theradical of ̃︀ H c ( 𝐾 ) and so the map induces a bijection between the simple modules. We can thusconsider any simple H c ( 𝐾 ) -module 𝐿 c ( 𝜆 ) as a simple H c ( 𝐾 ) -module, and to be precise wedenote this as ̃︀ 𝐿 c ( 𝜆 ) .Now, 𝑏 ′ 𝑖 · ˜ 𝐿 c ( 𝜆 ) ̸ = 0 if and only if ( 𝜎 𝑏 ′ 𝑖 ) · ( 𝜎 ˜ 𝐿 c ( 𝜆 )) ̸ = 0 . The statements of the theorem thenfollow from the Clifford theoretic fact, compare [4, Proposition 4.7], that Res 𝐴 𝑊 𝐴 𝐾 𝐿 c ( 𝜆 ) = ⨁︁ 𝜎 ∈ Γ / Stab Γ ( 𝜇 ) 𝜎 ˜ 𝐿 c ( 𝜇 ) , for some (any) simple summand 𝜇 of Res 𝑊𝐾 𝜆 , where 𝐴 𝑊 := H c ( 𝑊 ) / Rad H c ( 𝑊 ) and 𝐴 𝐾 = ̃︀ H c ( 𝐾 ) / Rad ̃︀ H c ( 𝐾 ) are the maximal semisimple quotients of H c ( 𝑊 ) and ̃︀ H c ( 𝐾 ) , respectively. (cid:4) Remark 4.21.
Geometrically, Theorem 4.20 is simply saying that ϒ − c ,𝐾 (0) = 𝜂 − (ϒ − c ,𝑊 (0)) is a union of Γ -orbits.Let Ω c ( 𝑊 ) rigid denote the set of Calogero–Moser c -families containing a rigid module. Proposition 4.22.
Let c : Ref ( 𝐾 ) → C be 𝑊 -equivariant.(a) The set Ω c ( 𝐾 ) cusp is Γ -stable and the bijection of Theorem 4.20(a) restricts to anembedding Ω c ( 𝐾 ) cusp / Γ ˓ → Ω c ( 𝑊 ) cusp .(b) The set Ω c ( 𝐾 ) rigid is Γ -stable and the bijection of Theorem 4.20(b) restricts to abijection Ω c ( 𝑊 ) rigid 1:1 ←→ Ω c ( 𝐾 ) rigid / Γ . Proof.
Part (a) follows from Lemma 4.18 which implies that the image of a zero-dimensionalleaf is a zero-dimensional leaf. If 𝐿 c ( 𝜆 ) is a rigid H c ( 𝑊 ) -module and 𝜆 ′ an irreduciblesummand of Res 𝑊𝐾 𝜆 , then 𝐿 c ( 𝜆 ′ ) is a rigid H c ( 𝐾 ) -module. Conversely, if 𝐿 c ( 𝜇 ) is a rigid H c ( 𝐾 ) -module and 𝜇 ′ an irreducible summand of Ind 𝑊𝐾 𝜇 , then 𝐿 c ( 𝜇 ′ ) is a rigid H c ( 𝑊 ) -module. This implies part (b). (cid:4) Remark 4.23.
The embedding of Proposition 4.22 (1) is not generally a bijection since thepreimage of a zero-dimensional leaf under 𝜂 is not always a union of zero-dimensional leaves. §5. Type 𝐴 Let 𝑊 be the Weyl group of type 𝐴 𝑛 . This is simply the symmetric group S 𝑛 +1 . It has an 𝑛 -dimensional irreducible reflection representation. There is just one conjugacy class of reflectionsso that our parameter c for rational Cherednik algebras is just a complex number. By Lemma4.11 we know that Conjecture B holds for c = 0 , so we can assume that c > . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Etingof and Ginzburg [16, Proposition 16.4] have shown that the Calogero–Moser space X c ( 𝑊 ) is smooth. Theorem 1.4 now implies that the Calogero–Moser c -families are singletonsand Lemma 3.7 shows that none of the Calogero–Moser c -families is cuspidal.Lusztig [33, Lemma 22.5] on the other hand has shown that for integral c > we have Con c ( 𝑊 ) = Irr ( 𝑊 ) . Using Lemma 2.5 we conclude that Con c ( 𝑊 ) = Irr ( 𝑊 ) for arbitraryreal c > . It then follows that the Lusztig c -families are singletons and using Lemma 2.6 wefurthermore see that no Lusztig c -family is cuspidal.Comparing both results proves Theorem 2.4 and Theorem A for 𝑊 of type 𝐴 . §6. Type 𝐵 Weyl groups of type 𝐵 are much more difficult to handle than those of type 𝐴 , in particularas we now have to deal with a two-dimensional parameter space. We have split the discussioninto several parts, some just dealing with the Calogero–Moser families, some just dealing withthe Lusztig families. At the very end we combine these results to obtain the proof of Theorem A.§6A. The group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22§6B. Reflections and parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22§6C. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23§6D. The rational Cherednik algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23§6E. Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23§6F. Symplectic leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24§6G. Parabolic subgroups attached to symplectic leaves . . . . . . . . . . . . . . . . . . . . . . . . 25§6H. Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25§6I. Simple H c ( 𝑊 ) -modules in the degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . 25§6J. Lusztig families in the non-degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26§6K. Lusztig families in the degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28§6L. Calogero–Moser families vs. Lusztig families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28§6M. Cuspidal Lusztig families in the non-degenerate case . . . . . . . . . . . . . . . . . . . . . . 31§6N. Cuspidal Lusztig families in the degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . 31§6O. Rigid modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33§6P. Cuspidal Lusztig families vs. cuspidal Calogero–Moser families . . . . . . . . . . . . 33 §6A. The group Let 𝑊 be the Weyl group of type 𝐵 𝑛 . This group is isomorphic to the group 𝐺 (2 , , 𝑛 ) of generalized permutation matrices in GL 𝑛 ( C ) with entries in 𝜇 := { , − } ⊆ C , and thisdefines at the same time an irreducible reflection representation of 𝐵 𝑛 . Note that 𝑊 = 𝜇 𝑛 (cid:111) S 𝑛 ,where S 𝑛 acts on 𝜇 𝑛 by coordinate permutation. For each ≤ 𝑖 ≤ 𝑛 we have a naturalembedding 𝜀 𝑖 of 𝜇 into 𝑊 , sending 𝑢 ∈ 𝜇 to the diagonal matrix (1 , . . . , 𝑢, . . . , with 𝑢 in the 𝑖 -th place. For ≤ 𝑖 < 𝑗 ≤ 𝑛 let 𝑠 𝑖𝑗 be the transposition ( 𝑖, 𝑗 ) ∈ S 𝑛 . For 𝑢 ∈ 𝜇 set 𝑠 𝑖𝑗,𝑢 := 𝑠 𝑖𝑗 𝜀 𝑖 ( 𝑢 ) − 𝜀 𝑗 ( 𝑢 ) . Note that 𝑠 𝑖𝑗, = 𝑠 𝑖𝑗 . The group 𝑊 is generated by 𝜀 ( − and thetranspositions 𝑠 𝑖𝑗 . §6B. Reflections and parabolic subgroups Let ( 𝑦 , . . . , 𝑦 𝑛 ) be the standard basis of h := C 𝑛 with dual basis ( 𝑥 , . . . , 𝑥 𝑛 ) . For any ≤ 𝑗 ≤ 𝑛 the element 𝜀 𝑗 ( − is a reflection with coroot 𝛼 ∨ 𝑗 := 𝑦 𝑗 and root 𝛼 𝑗 := 2 𝑥 𝑗 . Also,for any 𝑢 ∈ 𝜇 and ≤ 𝑖 < 𝑗 ≤ 𝑛 the element 𝑠 𝑖𝑗,𝑢 is a reflection with coroot 𝛼 ∨ 𝑖𝑗,𝑢 := 𝑢𝑦 𝑖 − 𝑦 𝑗 and root 𝛼 𝑖𝑗,𝑢 := 𝑢 − 𝑥 𝑖 − 𝑥 𝑗 = 𝑢𝑥 𝑖 − 𝑥 𝑗 . These elements are precisely the reflections in 𝑊 . WYN B ELLAMY AND U LRICH T HIEL We can now easily compute that(10) ( 𝑦 𝑘 , 𝛼 𝑗 )( 𝛼 ∨ 𝑗 , 𝑥 𝑙 ) = {︂ if 𝑘 = 𝑗 = 𝑙 elseand(11) ( 𝑦 𝑘 , 𝛼 𝑖𝑗,𝑢 )( 𝛼 ∨ 𝑖𝑗,𝑢 , 𝑥 𝑙 ) = ⎧⎨⎩ if 𝑘, 𝑙 ∈ { 𝑖, 𝑗 } with 𝑘 = 𝑙 − 𝑢 if 𝑘, 𝑙 ∈ { 𝑖, 𝑗 } with 𝑘 ̸ = 𝑙 else.The conjugacy classes of reflections in 𝑊 are 𝒮 := { 𝑠 𝑖𝑗,𝑢 | 𝑢 ∈ 𝜇 , ≤ 𝑖 < 𝑗 ≤ 𝑛 } and 𝒮 := { 𝜀 𝑗 ( − | ≤ 𝑗 ≤ 𝑛 } . We have |𝒮 | = 𝑛 − 𝑛 and 𝒮 = 𝑛 . The parabolic subgroups of 𝑊 are, up to conjugacy, ofthe form S 𝜆 × 𝐵 𝑛 −| 𝜆 | for partitions 𝜆 of integers ≤ 𝑛 . §6C. Representations Since 𝑊 = 𝜇 𝑛 (cid:111) S 𝑛 = 𝜇 ≀ S 𝑛 , the irreducible representations of 𝑊 are labeled by bipartitions 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) of 𝑛 . Let 𝜋 𝜆 denote the representation labeled by 𝜆 . The trivial representationof 𝑊 is 𝜋 ( 𝑛, ∅ ) . The representation 𝛾 := 𝜋 ( ∅ ,𝑛 ) is a linear character of 𝑊 with 𝛾 ( 𝑠 ) = 1 for all 𝑠 ∈ 𝒮 and 𝛾 ( 𝑠 ) = − for 𝑠 ∈ 𝒮 . We denote by 𝛾𝜋 𝜆 the 𝛾 -twist of 𝜋 𝜆 .The symmetric group S 𝑛 is a quotient of 𝐵 𝑛 by sending 𝜀 𝑗 ( − to . We can thus consider (ir-reducible) S 𝑛 -modules 𝜋 𝜆 for partitions 𝜆 of 𝑛 as (irreducible) 𝑊 -modules. If 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) is a bipartition of 𝑛 and 𝑟 := | 𝜆 (0) | , then 𝜋 𝜆 (0) (cid:2) 𝛾𝜋 𝜆 (1) is an irreducible ( 𝐵 𝑟 × 𝐵 𝑛 − 𝑟 ) -subrepresentation of 𝜋 𝜆 with(12) 𝜋 𝜆 = Ind 𝐵 𝑛 𝐵 𝑟 × 𝐵 𝑛 − 𝑟 𝜋 𝜆 (0) (cid:2) 𝛾𝜋 𝜆 (1) . §6D. The rational Cherednik algebra Fix a 𝑊 -equivariant function c : Ref ( 𝑊 ) → C and define 𝑐 := c ( 𝒮 ) and 𝜅 := c ( 𝒮 ) . In terms of the Coxeter diagram of type 𝐵 𝑛 the weight function c is determined as follows: . . . 𝑐 𝜅 𝜅 𝜅 𝜅 Using equations (10) and (11) we see that the defining relation (1) for H c ( 𝑊 ) becomes(13) [ 𝑦 𝑖 , 𝑥 𝑖 ] = − 𝑐 𝜀 𝑖 ( − − 𝜅 ∑︁ 𝑢 ∈ 𝜇 𝑛 ∑︁ 𝑗 =1 𝑗 ̸ = 𝑖 𝑠 𝑖𝑗,𝑢 and(14) [ 𝑦 𝑖 , 𝑥 𝑗 ] = 𝜅 ∑︁ 𝑢 ∈ 𝜇 𝑢𝑠 𝑖𝑗,𝑢 . for 𝑖 ̸ = 𝑗 . These are the same relations and parameters as in [35].Recall from Lemma 4.11 that Conjecture B holds for c = 0 .We assume from now on that c ̸ = 0 , i.e. 𝑐 ̸ = 0 or 𝜅 ̸ = 0 . §6E. Isomorphisms Recall that for any 𝛼 ∈ C * , the algebras H c ( 𝑊 ) and H 𝛼 c ( 𝑊 ) are isomorphic. Given abipartition 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) , we define 𝜆 𝜏 to be ( 𝜆 (1) , 𝜆 (0) ) . The following proposition followsfrom [9, 4.6B]. USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Proposition 6.1.
The linear character 𝛾 of 𝑊 defined in §6C extends to an isomorphism 𝜏 : H ( 𝑐 ,𝜅 ) ( 𝐵 𝑛 ) ∼ −→ H ( − 𝑐 ,𝜅 ) ( 𝐵 𝑛 ) with 𝜏 ( 𝑥 ) = 𝑥 , 𝜏 ( 𝑦 ) = 𝑦 and 𝜏 ( 𝑤 ) = 𝛾 ( 𝑤 ) 𝑤 for all 𝑥 ∈ h * , 𝑦 ∈ h and 𝑤 ∈ 𝑊 . Moreover,(a) 𝜏 − 𝐿 ( 𝑐 ,𝜅 ) ( 𝜆 ) ≃ 𝐿 ( − 𝑐 ,𝜅 ) ( 𝜆 𝜏 ) .(b) 𝜆 and 𝜇 belong to the same Calogero–Moser ( 𝑐 , 𝜅 ) -family if and only if 𝜆 𝜏 and 𝜇 𝜏 belong to the same Calogero–Moser ( − 𝑐 , 𝜅 ) -family.(c) 𝜆 is cuspidal, resp. rigid, for H ( 𝑐 ,𝜅 ) ( 𝑊 ) if and only if 𝜆 𝜏 is cuspidal, resp. rigid, for H ( − 𝑐 ,𝜅 ) ( 𝑊 ) .In the case 𝜅 = 0 the defining relations (13) and (14) of H c ( 𝑊 ) show that we have analgebra isomorphism H c ( 𝑊 ) ≃ H 𝑐 ( Z ) ⊗ 𝑛 (cid:111) S 𝑛 , where S 𝑛 naturally acts on the 𝑛 -fold tensorproduct of the rational Cherednik algebra at 𝑐 for the cyclic group of order . From this weget an isomorphism of Poisson varieties X c ( 𝑊 ) ≃ 𝑆 𝑛 ( X 𝑐 ( Z )) , where 𝑆 𝑛 denotes the 𝑛 -thsymmetric power. Since 𝑐 ̸ = 0 , the Calogero–Moser space X 𝑐 ( Z ) is a smooth symplecticsurface by [16, 16.2]. §6F. Symplectic leaves It was shown by Etingof and Ginzburg [16, 16.2] that the Calogero–Moser space of type 𝐵 is smooth for generic parameters. In this case the Calogero–Moser families are singletonsby Theorem 1.4 and none of them is cuspidal by Lemma 3.7. Using the relation betweenCalogero–Moser spaces and representation varieties of deformed preprojective algebras, Mar-tino has determined in his Ph.D thesis [34, Section 5] for precisely which parameters theCalogero–Moser space is smooth and gave a parametrization of the symplectic leaves. Tosimplify notations we set [ 𝑎, 𝑏 ] := { 𝑎, . . . , 𝑏 } and denote by ± [ 𝑎, 𝑏 ] the set [ − 𝑏, − 𝑎 ] ∪ [ 𝑎, 𝑏 ] forintegers 𝑎 ≤ 𝑏 . Note that ± [0 , 𝑏 ] = [ − 𝑏, 𝑏 ] . Theorem 6.2 (Martino) . Let c = ( 𝜅, 𝑐 ) .(a) X c ( 𝑊 ) is singular if and only if 𝜅 = 0 or 𝑐 = 𝑚𝜅 for some 𝑚 ∈ ± [0 , 𝑛 − .(b) If 𝜅 = 0 , then the symplectic leaves of X c ( 𝑊 ) are parameterised by the set 𝒫 ( 𝑛 ) ofpartitions of 𝑛 . For 𝜆 ∈ 𝒫 ( 𝑛 ) , the corresponding leaf ℒ 𝜆 has dimension ℓ ( 𝜆 ) , where ℓ ( 𝜆 ) is the length of 𝜆 .(c) If 𝑐 = 𝑚𝜅 , with 𝜅 ̸ = 0 , then there is a bijection 𝑘 ↦→ ℒ 𝑘 , { symplectic leaves ℒ of X c ( 𝑊 ) } ←→ { 𝑘 ∈ N ≥ | 𝑘 ( 𝑘 + 𝑚 ) ≤ 𝑛 } . Moreover, dim ℒ 𝑘 = 2( 𝑛 − 𝑘 ( 𝑘 + 𝑚 )) .We say that c is singular if X c ( 𝑊 ) is singular. Moreover, we call singular parameters with 𝜅 ̸ = 0 non-degenerate and those with 𝜅 = 0 degenerate . By the formulas for the dimensionsof the symplectic leaves we can immediately deduce when zero-dimensional leaves (and thuscuspidal Calogero–Moser families) exist. Corollary 6.3.
The space X c ( 𝑊 ) has a zero-dimensional symplectic leaf if and only if 𝑐 = 𝑚𝜅 for some 𝑚 ∈ ± [0 , 𝑛 − such that 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) for some 𝑘 > . In this case there is aunique zero-dimensional leaf and thus a unique cuspidal Calogero–Moser family.If our parameter c is as in Corollary 6.3 we say that it is cuspidal . Remark 6.4.
For a given 𝑛 and ± 𝑚 ∈ [0 , 𝑛 − there is at most one 𝑘 ≥ with 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) . In [34] the parameters are named ( 𝑐 𝛾 , 𝑐 ) instead of ( 𝑐 , 𝜅 ) . WYN B ELLAMY AND U LRICH T HIEL §6G. Parabolic subgroups attached to symplectic leaves We would like to parameterise the symplectic leaves of X c ( 𝐵 𝑛 ) by conjugacy classes ofparabolic subgroups and work out the geometric ordering. Lemma 6.5. If 𝑐 = 𝑚𝜅 for some 𝑚 ∈ ± [0 , 𝑛 − , then the leaf ℒ 𝑘 is labeled by the conjugacyclass of the parabolic 𝐵 𝑘 ( 𝑘 + 𝑚 ) and ℒ 𝑘 ≺ ℒ 𝑘 ′ ⇐⇒ ( 𝐵 𝑘 ( 𝑘 + 𝑚 ) ) ≤ ( 𝐵 𝑘 ′ ( 𝑘 ′ + 𝑚 ) ) ⇐⇒ 𝑘 ≥ 𝑘 ′ . Proof. If ℒ 𝑘 is labeled by the parabolic 𝑊 ′ then X c ′ ( 𝑊 ′ ) contains at least one zero-dimensionalleaf and 𝑊 ′ must have rank 𝑛 − dim ℒ 𝑘 = 𝑘 ( 𝑘 + 𝑚 ) . Since 𝜅 ̸ = 0 , the parabolic must beof the form 𝐵 𝑚 for some 𝑚 . Hence 𝑊 ′ = 𝐵 𝑘 ( 𝑘 + 𝑚 ) . It is a consequence of the proof of [34,Proposition 5.7] that ℒ 𝑘 ≺ ℒ 𝑘 ′ if and only if 𝑘 ≥ 𝑘 ′ . (cid:4) In the degenerate case 𝜅 = 0 , recall from §6E that there is an isomorphism of Poissonvarieties X c ( 𝐵 𝑛 ) ≃ 𝑆 𝑛 ( X 𝑐 ( Z )) . Then ℒ 𝜆 = 𝑆 𝜆 ( X 𝑐 ( Z )) , where 𝑆 𝜆 ( 𝑋 ) is the image in 𝑆 𝑛 ( 𝑋 ) of the set {︁∑︀ ℓ ( 𝜆 ) 𝑖 =1 𝜆 𝑖 · 𝑥 𝑖 | 𝑥 𝑖 ̸ = 𝑥 𝑗 ∈ 𝑋 }︁ . This implies that ℒ 𝜆 is labeled by the classof the parabolic subgroup S 𝜆 = S 𝜆 × · · · × S 𝜆 ℓ ( 𝜆 ) and ℒ 𝜆 ≺ ℒ 𝜇 ⇔ ( S 𝜆 ) ≤ ( S 𝜇 ) . Moreover, in this case, if ϒ − c (0) = { 𝑝, 𝑞 } for H 𝑐 ( Z ) , where 𝑝 = Supp 𝐿 𝑐 (1 Z ) and 𝑞 = Supp 𝐿 𝑐 ( sgn Z ) , then in X c ( 𝐵 𝑛 ) we have(15) ϒ − c (0) = { 𝑛 · 𝑝 + 𝑛 · 𝑞 | 𝑛 + 𝑛 = 𝑛, 𝑛 𝑖 ≥ } . The point 𝑛 · 𝑝 + 𝑛 · 𝑞 belongs to the leaf ℒ ( 𝑛 ,𝑛 ) . §6H. Calogero–Moser families The Calogero–Moser families in type 𝐵 𝑛 have been first described by Gordon and Martino [26]using the notion of 𝐽 -hearts, and later by Martino [35] using the notion of residues . We recallthe description given in [35] now.Let 𝜆 = ( 𝜆 , 𝜆 , . . . ) be a partition. We think of 𝜆 as a stack of boxes, left justified, with thebottom row containing 𝜆 boxes, the next row containing 𝜆 boxes and so forth. The content ct( (cid:3) ) of a box (cid:3) = ( 𝑖, 𝑗 ) ∈ 𝜆 is defined to be 𝑗 − 𝑖 . We consider the group ring Z [ C ] of theadditive group C and write 𝑥 𝛼 for the element corresponding to 𝛼 ∈ C . The residue of 𝜆 is theelement Res 𝜆 ( 𝑥 ) := ∑︁ (cid:3) ∈ 𝜆 𝑥 ct( (cid:3) ) ∈ Z [ Z ] ⊆ Z [ C ] . Just as in [10, §3A], we define for a triple m = ( 𝑚 , 𝑚 , 𝑚 ′ ) of complex numbers (the charge ),and a bipartition 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) , the charged residue as Res m 𝜆 ( 𝑥 ) := 𝑥 𝑚 Res 𝜆 (0) ( 𝑥 𝑚 ′ ) + 𝑥 𝑚 Res 𝜆 (1) ( 𝑥 𝑚 ′ ) ∈ Z [ C ] . The following theorem is [35, Theorem 5.5]. The additional parameters ( ℎ, 𝐻 , 𝐻 ) used in loc. cit. are given by ℎ = − 𝜅 , 𝐻 = − 𝑐 , and 𝐻 = 𝑐 . Theorem 6.6 (Martino) . Two bipartitions 𝜆 and 𝜇 lie in the same Calogero–Moser c -family ifand only if Res ^ c 𝜆 ( 𝑥 ) = Res ^ c 𝜇 ( 𝑥 ) with respect to the charge ^ c := (0 , 𝑐 , − 𝜅 ) . §6I. Simple H c ( 𝑊 ) -modules in the degenerate case In the degenerate case 𝜅 = 0 it is possible to determine the structure of the simple H c ( 𝑊 ) -modules 𝐿 c ( 𝜆 ) as 𝑊 -modules. Lemma 6.7. If 𝜅 = 0 , then 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) ∈ 𝒫 ( 𝑛 ) and 𝜇 = ( 𝜇 (0) , 𝜇 (1) ) ∈ 𝒫 ( 𝑛 ) lie in thesame Calogero–Moser c -family if and only if | 𝜆 (1) | = | 𝜇 (1) | . In particular, there are 𝑛 + 1 USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Calogero–Moser families ℱ deg0; 𝑛 , . . . , ℱ deg 𝑛 ; 𝑛 with ℱ deg 𝑖 ; 𝑛 = { 𝜆 ∈ 𝒫 ( 𝑛 ) | | 𝜆 (0) | = 𝑖 } . Proof.
In the case 𝜅 = 0 we have Res ^ c 𝜆 ( 𝑥 ) = ∑︁ (cid:3) ∈ 𝜆 (0) ct (cid:3) + 𝑥 𝑐 ∑︁ (cid:3) ∈ 𝜆 (1) ct (cid:3) = | 𝜆 (0) | + 𝑥 𝑐 | 𝜆 (1) | = 𝑛 − | 𝜆 (1) | + 𝑥 𝑐 | 𝜆 (1) | . The claim follows directly from Theorem 6.6. (cid:4)
Proposition 6.8.
Assume that 𝜅 = 0 . Then the family ℱ deg 𝑖 ; 𝑛 is labeled by the class of theparabolic S 𝑖 × S 𝑛 − 𝑖 ⊂ 𝐵 𝑛 and we have a bijection Irr ( S 𝑖 × S 𝑛 − 𝑖 ) ∼ −→ ℱ deg 𝑖 ; 𝑛 , sending thepair of partitions ( 𝜆 (0) , 𝜆 (1) ) to itself (thought of as a bipartition) such that 𝐿 c ( 𝜆 (0) , 𝜆 (1) ) ≃ Ind 𝐵 𝑛 S 𝑖 × S 𝑛 − 𝑖 𝜋 𝜆 (0) (cid:2) 𝜋 𝜆 (1) as 𝑊 -modules. Proof.
Since H c ( 𝐵 𝑛 ) ≃ ( H 𝑐 ( Z ) ⊗ ) (cid:111) S 𝑛 in this case (see §6E), we have 𝐿 c ( 𝜆 (0) , 𝜆 (1) ) ≃ Ind 𝐵 𝑛 ( H 𝑐 ( Z ) ⊗ ) (cid:111) ( S 𝑖 × S 𝑛 − 𝑖 ) 𝐿 𝑐 (1 Z ) ⊗ 𝐿 𝑐 ( sgn Z ) ⊗ ( 𝜋 𝜆 (0) (cid:2) 𝜋 𝜆 (1) ) , Since 𝑐 ̸ = 0 , both 𝐿 𝑐 (1 Z ) and 𝐿 𝑐 ( sgn Z ) are isomorphic to the regular representation as Z -modules. Recall that we have described ϒ − c (0) in (15). If 𝑖 = | 𝜆 (0) | , so that 𝑛 − 𝑖 = | 𝜆 (1) | ,then the support of 𝐿 c ( 𝜆 (0) , 𝜆 (1) ) is 𝑖 · 𝑝 + ( 𝑛 − 𝑖 ) · 𝑞 , which lies on the the leaf labeled by theparabolic S 𝑖 × S 𝑛 − 𝑖 . The result follows. (cid:4) §6J. Lusztig families in the non-degenerate case For the description of the Lusztig families in the non-degenerate case we first argue that we canrestrict to the so-called integral case where 𝑐 is an integral multiple of 𝜅 . Proposition 6.9.
Suppose that 𝜅 > . If there is no 𝑚 ∈ N with 𝑐 = 𝑚𝜅 , then Con c 𝑊 = Irr 𝑊 and so the Lusztig c -families are singletons. Proof.
By Lemma 2.5, we may assume that 𝜅 = 1 . The statement of the proposition hasbeen shown by Lusztig [33, Proposition 22.25] when 𝑐 is rational. We reduce the generalcase to the rational case. Let 𝜆 be an irreducible representation of a parabolic subgroup 𝑊 ′ of 𝑊 . The explicit formula given for the Schur element s 𝜆 , see [23, Theorem 10.5.2] and[33, Lemma 22.12], shows that there exist finitely many integers 𝑟 𝜆 , 𝑟 𝜆 , . . . , 𝑠 𝜆 , 𝑠 𝜆 , . . . , with ( 𝑟 𝑖 , 𝑠 𝑖 ) ̸ = ( 𝑟 𝑗 , 𝑠 𝑗 ) for 𝑖 ̸ = 𝑗 , and rational numbers 𝑓 𝜆 , 𝑓 𝜆 , . . . such that s 𝜆 = ∑︁ 𝑖 𝑓 𝜆𝑖 𝑞 ( 𝑟 𝜆𝑖 𝜅 + 𝑠 𝜆𝑖 𝑐 ) . Recall that 𝜅 = 1 and 𝑐 / ∈ Q . We claim that a 𝜆 = min { 𝑟 𝜆𝑖 𝜅 + 𝑠 𝜆𝑖 𝑐 | 𝑖 = 1 , . . . } . Note thatthis is not the case in general since there might be some cancellation between the 𝑓 𝜆𝑖 when 𝑟 𝜆𝑖 𝜅 + 𝑠 𝜆𝑖 𝑐 = 𝑟 𝜆𝑗 𝜅 + 𝑠 𝜆𝑗 𝑐 for some 𝑖 ̸ = 𝑗 . However, in our case the fact that 𝜅 = 1 and 𝑐 isirrational implies that 𝑟 𝜆𝑖 𝜅 + 𝑠 𝜆𝑖 𝑐 = 𝑟 𝜆𝑗 𝜅 + 𝑠 𝜆𝑗 𝑐 if and only if 𝑟 𝜆𝑖 = 𝑟 𝜆𝑗 and 𝑠 𝜆𝑖 = 𝑠 𝜆𝑗 , i.e. 𝑖 = 𝑗 .The claim follows.The definition of j -induction and constructible representations makes it clear that if we aregiven two parameters c and c ′ such that(16) a 𝜆 = a 𝜇 ⇔ a ′ 𝜆 = a ′ 𝜇 for all irreducible representations 𝜆 and 𝜇 of all parabolic subgroups of 𝑊 , then Con c 𝑊 = Con c ′ 𝑊 . Since there are only finitely many 𝑟 𝜆𝑖 and 𝑠 𝜆𝑗 as 𝜆 ranges over all irreducible repre-sentations of all parabolic subgroups of 𝑊 , one can easily choose a rational number 𝑐 ′ > WYN B ELLAMY AND U LRICH T HIEL with | 𝑐 − 𝑐 ′ | very small and 𝑐 ′ not an integer such that 𝑟 𝜆𝑖 + 𝑠 𝜆𝑖 𝑐 < 𝑟 𝜇𝑗 + 𝑠 𝜇𝑗 𝑐 ⇔ 𝑟 𝜆𝑖 + 𝑠 𝜆𝑖 𝑐 ′ < 𝑟 𝜇𝑗 + 𝑠 𝜇𝑗 𝑐 ′ for all 𝜆 , 𝜇 and 𝑖, 𝑗 . In particular, for c ′ = ( 𝑐 ′ , equation (16) holds. Moreover, since 𝑐 ′ isrational, every constructible representation in Con c ′ 𝑊 is irreducible by [33, Proposition 22.25].Hence Con c 𝑊 = Irr 𝑊 , too. (cid:4) We can thus restrict to the case 𝑐 = 𝑚𝜅 for some 𝑚 ∈ N , which by Lemma 2.5 is the sameas c = ( 𝑚, . The Lusztig families in this case have been described by Lusztig [33, §22] usingthe notion of symbols. We review the notion of symbols for general integral parameters.We assume that 𝜅 > and that c = ( 𝑐 , 𝜅 ) ≥ is integral .We can uniquely write 𝑐 = 𝑚𝜅 + 𝑟 for some 𝑚, 𝑟 ∈ N ≥ with 𝑟 < 𝜅 . Fix an arbitraryinteger 𝑁 > . A symbol for 𝐵 𝑛 with respect to 𝑁 at parameter c is a list of the form(17) 𝑆 = (︂ 𝛽 𝛽 · · · · · · · · · 𝛽 𝑁 + 𝑚 − 𝛽 𝑁 + 𝑚 𝛾 𝛾 · · · 𝛾 𝑁 )︂ , where ≤ 𝛽 < · · · < 𝛽 𝑁 + 𝑚 are congruent to 𝑟 modulo 𝜅 and ≤ 𝛾 < · · · < 𝛾 𝑁 aredivisible by 𝜅 , such that(18) ∑︁ 𝑖 𝛽 𝑖 + ∑︁ 𝑗 𝛾 𝑗 = 𝑛𝜅 + 𝜅𝑁 + 𝑁 ( 𝑐 − 𝜅 ) + 𝜅 (︃ 𝑚 )︃ + 𝑟𝑚 . Let Sy 𝑁 c ; 𝑛 denote the set of all such symbols. We have an embedding Sy 𝑁 c ; 𝑛 ˓ → Sy 𝑁 +1 c ; 𝑛 sendinga symbol 𝑆 as above to the symbol(19) 𝑆 [1] = (︂ 𝑟 𝛽 + 𝜅 𝛽 + 𝜅 · · · · · · 𝛽 𝑁 + 𝑚 + 𝜅 𝛾 + 𝜅 𝛾 + 𝜅 · · · 𝛾 𝑁 + 𝜅 )︂ . For 𝑖 ∈ N we denote by 𝑆 [ 𝑖 ] the 𝑖 -fold composition of the above map applied to 𝑆 and call thisthe shift of 𝑆 by 𝑖 . Let Sy c ; 𝑛 be the direct limit of the Sy 𝑁 c ; 𝑛 with respect to the above maps. Wesay that 𝑁 is large enough for a bipartition 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) of 𝑛 if 𝜆 (0) 𝑁 + 𝑚 +1 = 0 = 𝜆 (1) 𝑁 +1 . Wethen define the corresponding symbol Sy 𝑁 c ; 𝑛 ( 𝜆 ) = (︀ 𝛽𝛾 )︀ ∈ Sy 𝑁 c ; 𝑛 via(20) 𝛽 𝑖 := 𝜅 (︁ 𝜆 (0) 𝑁 + 𝑚 − 𝑖 +1 + 𝑖 − )︁ + 𝑟 for 𝑖 ∈ [1 , 𝑁 + 𝑚 ] 𝛾 𝑗 := 𝜅 (︁ 𝜆 (1) 𝑁 − 𝑗 +1 + 𝑗 − )︁ for 𝑗 ∈ [1 , 𝑁 ] . If 𝑁 is large enough for all bipartitions of 𝑛 , e.g., 𝑁 ≥ 𝑛 , the map 𝜆 ↦→ Sy 𝑁 c ; 𝑛 ( 𝜆 ) defines abijection between the set 𝒫 ( 𝑛 ) of bipartitions of 𝑛 and Sy 𝑁 c ; 𝑛 . For a symbol 𝑆 we then denoteby 𝜋 𝑆 the representation of 𝑊 labeled by the bipartition corresponding to 𝑆 . The content ct ( 𝑆 ) of a symbol 𝑆 ∈ Sy 𝑁 c ; 𝑛 is the multiset of its entries, i.e., the list of entries with repetitions butignoring positions. We can, and will, equally well write the content as a polynomial ∑︀ 𝑖 ≥ 𝑛 𝑖 𝑥 𝑖 ,where 𝑛 𝑖 denotes the multiplicity of the entry 𝑖 in 𝑆 . It is clear from the definition of a symbolthat it has at least 𝑁 + 𝑚 distinct entries and the multiplicity of an entry in a symbol is atmost . Example 6.10.
Let 𝜆 = (︁ , )︁ and ( 𝑐 , 𝜅 ) = (1 , . Then Sy , ( 𝜆 ) = (︂ )︂ ∈ Sy , . This symbol is in fact the shift of (︂ )︂ ∈ Sy , by . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Theorem 6.11 (Lusztig) . Let c = ( 𝑐 , 𝜅 ) ≥ with 𝜅 > and 𝑐 = 𝑚𝜅 for some 𝑚 ∈ N . Thentwo bipartitions 𝜆 and 𝜇 lie in the same Lusztig c -family if and only if ct ( Sy 𝑁 ( 𝑚, 𝑛 ( 𝜆 )) = ct ( Sy 𝑁 ( 𝑚, 𝑛 ( 𝜇 )) for 𝑁 sufficiently large. Proof.
Because of Lemma 2.5 we can assume that c = ( 𝑚, . Then the description of the c -constructible characters in [33, Proposition 22.24] along with [33, Lemma 22.22] proves theclaim. (cid:4) §6K. Lusztig families in the degenerate case The description of the Lusztig families in the degenerate case is given in [21, Example 7.13]and follows from the general theory in [22, §2.4.3].
Lemma 6.12. If 𝜅 = 0 , then 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) ∈ 𝒫 ( 𝑛 ) and 𝜇 = ( 𝜇 (0) , 𝜇 (1) ) ∈ 𝒫 ( 𝑛 ) lie inthe same Lusztig c -family if and only if | 𝜆 (1) | = | 𝜇 (1) | . In particular, there are precisely 𝑛 + 1 Lusztig families ℱ deg0; 𝑛 , . . . , ℱ deg 𝑛 ; 𝑛 with ℱ deg 𝑖 ; 𝑛 = { 𝜆 ∈ 𝒫 ( 𝑛 ) | | 𝜆 (0) | = 𝑖 } . §6L. Calogero–Moser families vs. Lusztig families We can now prove Theorem 2.4 for type 𝐵 𝑛 . Corollary 6.13.
For type 𝐵 𝑛 and any parameter c ≥ the Lusztig c -families are equal to theCalogero–Moser c -families. Proof. If 𝜅 = 0 , the claim follows from Lemma 6.7 and Lemma 6.12. Now, assume that 𝜅 ̸ = 0 .If 𝑐 ̸ = 𝑚𝜅 for all 𝑚 ∈ N ≥ , then we know from Theorem 6.2(a) and Proposition 6.9 thatboth the Calogero–Moser c -families and the Lusztig c -families are singletons. So, supposethat 𝑐 = 𝑚𝜅 for some 𝑚 ∈ N ≥ . Because of Lemma 2.5 and Lemma 3.8 we can assume that 𝜅 = 1 . It follows from [10, Proposition 3.4] that ct ( Sy 𝑁 c ; 𝑛 ( 𝜆 )) = ct ( Sy 𝑁 c ; 𝑛 ( 𝜇 )) for 𝑁 largeenough if and only if Res ^ c 𝜆 ( 𝑥 ) = Res ^ c 𝜇 ( 𝑥 ) . By Theorem 6.6 and Theorem 6.11 this shows that Ω c ( 𝑊 ) = Lus c ( 𝑊 ) . (cid:4) §6M. Cuspidal Lusztig families in the non-degenerate case In the non-singular case, Proposition 6.9 and Lemma 2.6 immediately imply the followingresult.
Lemma 6.14. If 𝜅 > and there is no 𝑚 ∈ N with 𝑐 = 𝑚𝜅 , then there are no cuspidal Lusztig c -families.Lemma 2.5 implies that we can restrict to the following situation.We assume that 𝜅 = 1 and that 𝑐 = 𝑚𝜅 = 𝑚 for some 𝑚 ∈ N ≥ .At equal parameters, i.e. 𝑚 = 1 , the cuspidal families are described by Lusztig in [32,Section 8.1]. It seems difficult to find an explicit description of the cuspidal families for unequalparameters. Therefore we derive the classification here in Theorem 6.21 using the results of[33].We choose 𝑁 sufficiently large for all bipartitions of 𝑛 (see §6J). For a Lusztig c -family ℱ we denote by Sy 𝑁 c ; 𝑛 ( ℱ ) the set of symbols Sy 𝑁 c ; 𝑛 ( 𝜆 ) with 𝜆 ∈ ℱ . Using the combinatorics ofsymbols, we can explicitly determine the size of ℱ . WYN B ELLAMY AND U LRICH T HIEL Lemma 6.15.
Let
ℱ ∈
Lus c ( 𝑊 ) . Let ∑︀ 𝑖 ≥ 𝑛 𝑖 𝑥 𝑖 be the content of one (any) 𝑆 ∈ Sy 𝑁 c ; 𝑛 ( ℱ ) .Set 𝑘 ℱ := 𝑁 − |{ 𝑖 | 𝑛 𝑖 = 2 }| . Then 𝑘 ℱ ≥ , 𝑘 ℱ ( 𝑘 ℱ + 𝑚 ) ≤ 𝑛 , and |ℱ | = (︀ 𝑘 ℱ + 𝑚𝑘 ℱ )︀ . Proof.
Let 𝑆 ∈ Sy 𝑁 c ; 𝑛 ( ℱ ) and set 𝑘 := 𝑘 ℱ . The multiplicity of an entry in 𝑆 is at most equalto and 𝑆 has at least 𝑁 + 𝑚 distinct entries. Since 𝑆 has exactly 𝑁 + 𝑚 entries withmultiplicity, this immediately shows that 𝑘 ≥ . Let 𝐸 be the set (not multiset) of entries of 𝑆 .By definition of 𝑘 we have | 𝐸 | = 𝑁 + 𝑘 + 𝑚 . Any 𝑁 -element subset of 𝐸 containing the set { 𝑖 | 𝑛 𝑖 = 2 } defines a unique symbol in Sy 𝑁 c ; 𝑛 ( ℱ ) , and in this way all symbols of Sy 𝑁 c ; 𝑛 ( ℱ ) areobtained. The number of such sets is equal to (︀ 𝑁 + 𝑘 + 𝑚 − ( 𝑁 − 𝑘 ) 𝑁 − ( 𝑁 − 𝑘 ) )︀ = (︀ 𝑘 + 𝑚𝑘 )︀ .It remains to show that 𝑘 ( 𝑘 + 𝑚 ) ≤ 𝑛 . Since 𝑆 ∈ Sy 𝑁 c ; 𝑛 , equation (18) says that(21) ∑︁ 𝑖 𝛽 𝑖 + ∑︁ 𝑗 𝛾 𝑗 − 𝑁 − 𝑁 ( 𝑚 − − (︃ 𝑚 )︃ = 𝑛 , where the 𝛽 𝑖 and 𝛾 𝑗 are the entries of 𝑆 . Hence, it suffices to show that the left hand sideis at least as big as 𝑘 ( 𝑘 + 𝑚 ) . Recall that 𝑁 − 𝑘 is equal to the number of pairs ( 𝑖, 𝑗 ) suchthat 𝛽 𝑖 = 𝛾 𝑗 . Since 𝛽 𝑖 , 𝛾 𝑗 ≥ , the expression on the left is minimal if 𝛽 𝑖 = 𝛾 𝑖 = 𝑖 − for 𝑖 = 1 , . . . , 𝑁 − 𝑘 and the remaining 𝑘 + 𝑚 entries are in { 𝑁 − 𝑘, . . . , 𝑁 + 𝑘 + 𝑚 − } . Thenthe left hand side of equation (21) becomes 𝑁 − 𝑘 ∑︁ 𝑖 =1 𝑖 −
1) + 𝑁 + 𝑘 + 𝑚 − ∑︁ 𝑖 = 𝑁 − 𝑘 𝑖 − 𝑁 − 𝑁 ( 𝑚 − − (︃ 𝑚 )︃ = 𝑘 ( 𝑘 + 𝑚 ) . (cid:4) Definition 6.16.
A symbol 𝑆 ∈ Sy 𝑁 c ; 𝑛 with content ∑︀ 𝑖 ≥ 𝑛 𝑖 𝑥 𝑖 is called cuspidal if 𝑛 𝑖 ≥ 𝑛 𝑖 +1 for all 𝑖 = 0 , , . . . If 𝑆 is a cuspidal symbol then 𝑆 [1] is also cuspidal.Suppose that 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) for some 𝑘 ∈ N > . Then we have the box partition ( 𝑘 𝑘 + 𝑚 ) of 𝑛 . If 𝜆 is any partition such that ℓ ( 𝜆 ) ≤ 𝑘 + 𝑚 and 𝜆 ≤ 𝑘 , then 𝜆 ⊆ ( 𝑘 𝑘 + 𝑚 ) . Adding zeros,we may assume that ℓ ( 𝜆 ) = 𝑘 + 𝑚 . Define the partition 𝜆 † by 𝜆 † 𝑖 := { 𝑗 ∈ [1 , 𝑘 + 𝑚 ] | 𝑘 − 𝜆 𝑘 + 𝑚 +1 − 𝑗 ≥ 𝑖 } = 𝑘 + 𝑚 + 1 − min { 𝑗 ∈ [1 , 𝑘 + 𝑚 ] | 𝑘 − 𝑖 ≥ 𝜆 𝑗 } . This is simply the transpose of the reverse of the complement 𝑘 − 𝜆 of 𝜆 in the box ( 𝑘 𝑘 + 𝑚 ) .Since | 𝜆 | + | 𝜆 † | = 𝑘 ( 𝑘 + 𝑚 ) = 𝑛 , we get in this way a bipartition ( 𝜆, 𝜆 † ) of 𝑛 . Let ℱ cusp 𝑘,𝑚 := { ( 𝜆, 𝜆 † ) | ℓ ( 𝜆 ) ≤ 𝑘 + 𝑚, 𝜆 ≤ 𝑘 } . Example 6.17. If 𝑛 = 6 and 𝑚 = 1 , we can write 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) with 𝑘 = 2 and get ℱ cusp2 , = {︃(︁ , )︁ , (︁ ∅ , )︁ , (︃ , ∅ )︃ , (︁ , )︁ , (︃ , )︃ , (︁ , )︁ , (︁ , )︁ , (︃ , )︃ , (︃ , )︃ , (︁ , )︁}︃ . If 𝑛 = 3 and 𝑚 = 2 , we can write 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) with 𝑘 = 1 and get ℱ cusp1 , = {︃ ( ∅ , ) , ( , ) , (︁ , )︁ , (︃ , ∅ )︃}︃ . Lemma 6.18.
Suppose that 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) . The content of the symbol Sy 𝑘 c ; 𝑛 ( 𝜆, 𝜆 † ) is equalto ∑︀ 𝑘 + 𝑚 − 𝑖 =0 𝑥 𝑖 for any ( 𝜆, 𝜆 † ) ∈ ℱ cusp 𝑘,𝑚 . In particular, Sy 𝑘 c ; 𝑛 ( 𝜆, 𝜆 † ) is cuspidal and ℱ cusp 𝑘,𝑚 is aLusztig c -family with |ℱ cusp 𝑘,𝑚 | = (︀ 𝑘 + 𝑚𝑘 )︀ . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Proof.
First, note that 𝑁 = 𝑘 is large enough for any ( 𝜆, 𝜆 † ) ∈ ℱ cusp 𝑘,𝑚 . The symbol Sy 𝑘 c ; 𝑛 ( 𝜆, 𝜆 † ) = (︀ 𝛽𝛾 )︀ is then given by 𝛽 𝑖 := 𝜆 𝑘 + 𝑚 − 𝑖 +1 + 𝑖 − for 𝑖 ∈ [1 , 𝑘 + 𝑚 ] 𝛾 𝑗 := 𝜆 † 𝑘 − 𝑗 +1 + 𝑗 − for 𝑗 ∈ [1 , 𝑘 ] . Our assertion about the content of this symbol is equivalent to showing that the symbol containsthe entry , all entries are bounded above by 𝑘 + 𝑚 − , and that 𝛽 𝑖 ̸ = 𝛾 𝑗 for all 𝑖, 𝑗 . First, wehave 𝛽 = 𝜆 𝑘 + 𝑚 and 𝛾 = 𝜆 † 𝑘 = 𝑘 + 𝑚 + 1 − min { 𝑗 ∈ [ 𝑘 + 𝑚 | ≥ 𝜆 𝑗 } = 𝑘 + 𝑚 − ℓ ( 𝜆 ) . We immediately see that either 𝛽 = 0 or 𝛾 = 0 . On the other hand, we have 𝛽 𝑘 + 𝑚 = 𝜆 + 𝑘 + 𝑚 − and 𝛾 𝑘 = 𝜆 † + 𝑘 − 𝑘 + 𝑚 − min { 𝑗 ∈ [1 , 𝑘 + 𝑚 ] | 𝑘 > 𝜆 𝑗 } . This shows that the entries of the symbol are at most equal to 𝑘 + 𝑚 − . Showing that 𝛽 𝑖 ̸ = 𝛾 𝑗 for all 𝑖 ∈ [1 , 𝑘 + 𝑚 ] and 𝑗 ∈ [1 , 𝑚 ] is equivalent to showing that 𝛽 𝑘 + 𝑚 − 𝑖 +1 ̸ = 𝛾 𝑘 − 𝑗 +1 for all 𝑖 ∈ [1 , 𝑘 + 𝑚 ] and 𝑗 ∈ [1 , 𝑚 ] . Now, 𝛽 𝑘 + 𝑚 − 𝑖 +1 = 𝛾 𝑘 − 𝑗 +1 ⇔ 𝜆 𝑖 + 𝑚 − 𝑖 = 𝜆 † 𝑗 + 𝑘 − 𝑗 ⇔ 𝜆 𝑖 − 𝑖 = 𝑘 + 1 − min { 𝑙 ∈ [1 , 𝑘 + 𝑚 ] | 𝑘 − 𝑗 ≥ 𝜆 𝑙 } − 𝑗 . (22)Suppose that 𝑘 − 𝑗 ≥ 𝜆 𝑖 . Then min { 𝑙 | 𝑘 − 𝑗 ≥ 𝜆 𝑙 } ≤ 𝑖 and we get from equation (22) theestimate 𝜆 𝑖 − 𝑖 ≥ 𝑘 + 1 − 𝑖 − 𝑗 . This implies 𝜆 𝑖 > 𝑘 − 𝑗 , contradicting the assumption. On theother hand, if 𝑘 − 𝑗 ≤ 𝜆 𝑖 , we similarly deduce the estimate 𝜆 𝑖 < 𝑘 − 𝑗 , again a contradiction.Hence, 𝛽 𝑖 ̸ = 𝛾 𝑗 for all 𝑖, 𝑗 .The number of elements in ℱ cusp 𝑘,𝑚 equals the number of sub-partitions 𝜆 of ( 𝑘 𝑘 + 𝑚 ) , and thisnumber is equal to (︀ 𝑘 + 𝑚𝑘 )︀ . We have just seen that ℱ cusp 𝑘,𝑚 is contained in a single Lusztig family ℱ . Since the multiplicity of each entry in the content we have just computed is equal to , itfollows from Lemma 6.15 that |ℱ | = (︀ 𝑘 + 𝑚𝑘 )︀ . Hence, ℱ cusp 𝑘,𝑚 = ℱ is a Lusztig family. (cid:4) The symbols in Lemma 6.18 are in fact the minimal representatives of cuspidal symbols.
Lemma 6.19.
Suppose that 𝑆 ∈ Sy 𝑁 c ; 𝑛 is a cuspidal symbol. Then 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) for some 𝑘 and 𝑆 ∈ ℱ cusp 𝑘,𝑚 . Proof.
Recall that 𝑆 cuspidal means that 𝑛 𝑖 ≥ 𝑛 𝑖 +1 for 𝑖 = 0 , , . . . . Since there is at least one 𝑖 such that 𝑛 𝑖 ̸ = 0 , we have 𝑛 ̸ = 0 . As in Lemma 6.15, let 𝑘 := 𝑁 − |{ 𝑖 | 𝑛 𝑖 = 2 }| . Because 𝑆 is cuspidal, the symbol 𝑆 ′ := 𝑆 [ − ( 𝑁 − 𝑘 )] ∈ Sy 𝑘 c ; 𝑛 is well-defined. By definition of shift,the content of this symbol is equal to ∑︀ 𝑘 + 𝑚 − 𝑖 =0 𝑥 𝑖 . Equation (18) for 𝑆 ′ says that 𝑛 = 𝑘 + 𝑚 − ∑︁ 𝑖 =0 𝑖 − 𝑘 − 𝑘 ( 𝑚 − − (︃ 𝑚 )︃ = 𝑘 ( 𝑘 + 𝑚 ) . Hence, 𝑆 ∈ ℱ cusp 𝑘,𝑚 by Lemma 6.18 and Theorem 6.11. (cid:4) For a symbol 𝑆 ∈ Sy 𝑁 c ; 𝑛 as in (17) we define the symbol 𝑆 ∈ Sy 𝑁 ′ c ; 𝑛 for certain 𝑁 ′ as follows.Choose 𝑡 ≥ max { 𝛽 𝑁 + 𝑚 , 𝛾 𝑁 } . Note that 𝑡 ≥ 𝑚 since 𝑆 has at least 𝑁 + 𝑚 distinct entries.Now, the first row of 𝑆 is the set { , , . . . , 𝑡 } (cid:114) { 𝑡 − 𝛾 , . . . , 𝑡 − 𝛾 𝑁 } and the second row is { , , . . . , 𝑡 } (cid:114) { 𝑡 − 𝛽 , . . . , 𝑡 − 𝛽 𝑁 + 𝑚 } . By [33, 22.8], the symbol 𝑆 belongs to Sy 𝑡 +1 − 𝑁 − 𝑚 c ; 𝑛 and by [33, Lemma 22.18] we have 𝜋 𝑆 ⊗ sgn 𝑊 = 𝜋 𝑆 . Example 6.20.
Let 𝜆 = (︁ , )︁ and ( 𝑐 , 𝜅 ) = (1 , . Recall from Example 6.10 that Sy , ( 𝜆 ) = (︂ )︂ ∈ Sy , . WYN B ELLAMY AND U LRICH T HIEL Choosing 𝑡 = 5 we get Sy , ( 𝜆 ) = (︂ )︂ = Sy , (︁ , )︁ . Indeed, (︁ , )︁ ⊗ ⎛⎝ ∅ , ⎞⎠⏟ ⏞ = sgn 𝑊 = (︁ , )︁ . Theorem 6.21.
There exists a cuspidal Lusztig family if and only if there is a 𝑘 > such that 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) . In this case there is a unique cuspidal family, and it is equal to ℱ cusp 𝑘,𝑚 . Proof.
Because of the transitivity of j -induction, Lusztig families of 𝐵 𝑛 induced from someparabolic subgroup are also induced from some maximal parabolic subgroup. These subgroupsare all of the form 𝐵 𝑙 × S 𝑛 − 𝑙 for some ≤ 𝑙 ≤ 𝑛 − . The restriction of the parameter c to S 𝑛 − 𝑙 is equal to 𝜅 > and so the Lusztig families of S 𝑛 − 𝑙 are singletons by §5. A Lusztigfamily of 𝐵 𝑙 × S 𝑛 − 𝑙 is thus of the form { 𝜋 𝑆 (cid:2) 𝜋 𝜆 | 𝜋 𝑆 ∈ ℱ } for some Lusztig family ℱ of 𝐵 𝑙 and some fixed 𝜆 ∈ 𝒫 ( 𝑛 − 𝑙 ) . Since any given number can only appear at most twicein 𝑆 , either the set of 𝑛 − 𝑙 largest entries in 𝑆 is well-defined or there is a choice of twopossible "largest 𝑛 − 𝑙 -entries". Notice that this depends only on the content of 𝑆 . By adding to the 𝑛 − 𝑙 largest entries, we get either a new symbol 𝑆 ′ or two new symbols 𝑆 𝐼 and 𝑆 𝐼𝐼 . Then, as explained in [33, Section 22.15], j 𝐵 𝑛 𝐵 𝑙 × S 𝑛 − 𝑙 𝜋 𝑆 (cid:2) sgn 𝑊 equals 𝜋 𝑆 ′ or 𝜋 𝑆 𝐼 ⊕ 𝜋 𝑆 𝐼𝐼 .In the latter case, the j -induction of 𝜋 𝑆 (cid:2) 𝜋 𝜆 is not irreducible and so j -induction does notinduce a Lusztig family of 𝐵 𝑛 . We thus assume we are in the former case. Let ∑︀ 𝑖 ≥ 𝑛 𝑖 𝑥 𝑖 bethe content of 𝑆 ′ . Here 𝑛 𝑖 ∈ { , , } and 𝑛 𝑖 = 0 for 𝑖 ≫ . Assume that there exists some 𝑖 such that 𝑛 𝑖 > 𝑛 𝑖 − . There are two possibilities, either 𝑛 𝑖 − = 0 or ( 𝑛 𝑖 − , 𝑛 𝑖 ) = (1 , .Consider first the former. We let 𝑙 be defined such that the 𝑛 − 𝑙 largest numbers in 𝑆 are { 𝑛 𝑖 · 𝑖, 𝑛 𝑖 +1 · ( 𝑖 + 1) , . . . } . Here, 𝑛 𝑖 · 𝑖 means that 𝑖 occurs with multiplicity 𝑛 𝑖 . Since 𝑛 𝑖 − = 0 ,we can remove from each of the 𝑛 − 𝑙 largest numbers and still have a well-defined symbol 𝑆 ′′ .Moreover, j 𝐵 𝑛 𝐵 𝑙 × S 𝑛 − 𝑙 𝜋 𝑆 ′′ ⊗ sgn 𝑊 = 𝜋 𝑆 ′ . This applies to all symbols in the family to which 𝑆 belongs. Hence this family is not cuspidal. The other case is where ( 𝑛 𝑖 − , 𝑛 𝑖 ) = (1 , . In thiscase it suffices to show that 𝜋 𝑆 = 𝜋 𝑆 ⊗ sgn 𝑊 is not cuspidal. If the content of 𝑆 is ∑︀ 𝑖 ≥ 𝑛 𝑖 𝑥 𝑖 ,then the content of 𝑆 equals ∑︀ 𝑖 ≥ (2 − 𝑛 𝑡 − 𝑖 ) 𝑥 𝑖 for some 𝑡 ≫ . Thus, there exists some 𝑗 suchthat ( 𝑛 𝑗 − , 𝑛 𝑗 ) equals (0 , in the content of 𝑆 . By our previous argument, the family to which 𝑆 belongs is induced from some parabolic subgroup.Above, we assumed that there exists some 𝑖 such that 𝑛 𝑖 > 𝑛 𝑖 − . When 𝑛 𝑖 ≤ 𝑛 𝑖 − for all 𝑖 ,Lemma 6.19 implies that 𝜋 𝑆 belongs to the family ℱ cusp 𝑘,𝑚 for some 𝑘 with 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) . If ℱ ′ is a family in Irr ( 𝐵 𝑙 × S 𝑛 − 𝑙 ) for some 𝑙 < 𝑛 , then Lemma 6.15 implies that |ℱ ′ | < |ℱ | . Hence ℱ cannot be induced and must be cuspidal. (cid:4) §6N. Cuspidal Lusztig families in the degenerate case In this section we consider the case 𝜅 = 0 . Recall from Lemma 6.12 that the Lusztig familiesare labeled ℱ deg 𝑖 ; 𝑛 for 𝑖 = 0 , . . . , 𝑛 . Lemma 6.22.
For any 𝑖 , tensoring with the sign character yields a bijection ℱ deg 𝑖 ; 𝑛 ∼ −→ ℱ deg 𝑛 − 𝑖 ; 𝑛 . Proof. If 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) ∈ 𝒫 ( 𝑛 ) , then 𝜋 𝜆 ⊗ sgn 𝑊 = 𝜋 (( 𝜆 (1) ) * , ( 𝜆 (0) ) * ) by [23, Theorem5.5.6(c)]. Hence, if 𝜋 𝜆 ∈ ℱ deg 𝑖 ; 𝑛 , then 𝜋 𝜆 ⊗ sgn 𝑊 ∈ ℱ deg 𝑛 − 𝑖 ; 𝑛 . As sgn 𝑊 is an automorphism on Irr 𝑊 , the claim follows. (cid:4) USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Proposition 6.23. If 𝜅 = 0 but 𝑐 ̸ = 0 , there are no cuspidal Lusztig c -families. Proof.
We claim that j 𝐵 𝑛 𝐵 𝑖 × S 𝑛 − 𝑖 induces a bijection between the Lusztig family ℱ deg 𝑖 ; 𝑖 × Irr S 𝑛 − 𝑖 of the parabolic subgroup 𝐵 𝑖 × S 𝑛 − 𝑖 of 𝐵 𝑛 and ℱ deg 𝑖 ; 𝑛 for ≤ 𝑖 < 𝑛 . This shows that all ℱ deg 𝑖 ; 𝑛 with 𝑖 < 𝑛 are induced, and since ℱ deg0; 𝑛 ⊗ sgn 𝑊 = ℱ deg 𝑛 ; 𝑛 by Lemma 6.22, this proves the claim.So, assume that ≤ 𝑖 < 𝑛 . In [21, Example 7.13] (see also [22, §2.4.3]) it is shown that the a -invariant in the degenerate case of 𝜇 ∈ 𝒫 ( 𝑖 ) is a 𝜇 = 𝑐 | 𝜇 (1) | . Moreover, in the degeneratecase, the restriction of the parameter to S 𝑛 − 𝑖 is zero so that the a -invariants 𝑎 𝜈 are zero for all 𝜈 ∈ 𝒫 ( 𝑛 − 𝑖 ) by Example 2.1. Hence, j 𝐵 𝑛 𝐵 𝑖 × S 𝑛 − 𝑖 𝜋 𝜇 (cid:2) 𝜋 𝜈 = ∑︁ 𝜆 ∈𝒫 ( 𝑛 ) | 𝜆 (1) | = | 𝜇 (1) | ⟨ Ind 𝐵 𝑛 𝐵 𝑖 × S 𝑛 − 𝑖 𝜋 𝜇 (cid:2) 𝜋 𝜈 , 𝜋 𝜆 ⟩ 𝜋 𝜆 for 𝜇 ∈ 𝒫 ( 𝑖 ) and 𝜈 ∈ 𝒫 ( 𝑛 − 𝑖 ) . We will show that if ( 𝜇 , 𝜈 ) ∈ ℱ deg 𝑖 ; 𝑖 × Irr S 𝑛 − 𝑖 , then(23) j 𝐵 𝑛 𝐵 𝑖 × S 𝑛 − 𝑖 𝜋 𝜇 (cid:2) 𝜋 𝜈 = 𝜋 ( 𝜈,𝜇 (1) ) . From this equation, the claim follows immediately. Since S 𝑛 − 𝑖 is a subgroup of 𝐵 𝑛 − 𝑖 , we getthe following relation using the branching rules: Ind 𝐵 𝑛 𝐵 𝑖 × S 𝑛 − 𝑖 𝜋 𝜇 (cid:2) 𝜋 𝜈 = Ind 𝐵 𝑛 𝐵 𝑖 × 𝐵 𝑛 − 𝑖 Ind 𝐵 𝑖 × 𝐵 𝑛 − 𝑖 𝐵 𝑖 × S 𝑛𝑖 𝜋 𝜇 (cid:2) 𝜋 𝜈 = Ind 𝐵 𝑛 𝐵 𝑖 × 𝐵 𝑛 − 𝑖 (︁ 𝜋 𝜇 (cid:2) Ind 𝐵 𝑛 − 𝑖 S 𝑛 − 𝑖 𝜋 𝜈 )︁ = Ind 𝐵 𝑛 𝐵 𝑖 × 𝐵 𝑛 − 𝑖 ⎛⎝ 𝜋 𝜇 (cid:2) ∑︁ 𝛼 ∈𝒫 ( 𝑛 − 𝑖 ) 𝑐 𝜈 𝛼 𝜋 𝛼 ⎞⎠ = ∑︁ 𝜆 ∈𝒫 ( 𝑛 ) ∑︁ 𝛼 ∈𝒫 ( 𝑛 − 𝑖 ) 𝑐 𝜆 (0) 𝜇 (0) ,𝛼 (0) 𝑐 𝜆 (1) 𝜇 (1) ,𝛼 (1) 𝑐 𝜈 𝛼 𝜋 𝜆 . Note that the sum runs only over those 𝛼 with 𝛼 (0) , 𝛼 (1) ⊆ 𝜈 and | 𝜈 | = | 𝛼 (0) | + | 𝛼 (1) | , andsimilarly only over those 𝜆 which satisfy(24) 𝜇 ( 𝑗 ) , 𝛼 ( 𝑗 ) ⊆ 𝜆 ( 𝑗 ) and(25) | 𝜆 ( 𝑗 ) | = | 𝜇 ( 𝑗 ) | + | 𝛼 ( 𝑗 ) | for 𝑗 ∈ { , } . To show (23) we need to show that among those 𝜆 ∈ 𝒫 ( 𝑛 ) occurring in thissum such that | 𝜆 (1) | = | 𝜇 (1) | we have 𝑐 𝜆 (0) 𝜇 (0) ,𝛼 (0) 𝑐 𝜆 (1) 𝜇 (1) ,𝛼 (1) 𝑐 𝜈 𝛼 = {︃ if 𝜆 = ( 𝜈, 𝜇 (1) ) , 𝜇 = ( ∅ , 𝜇 (1) ) , 𝛼 = ( 𝜈, ∅ )0 else.So, suppose that 𝑐 𝜆 (0) 𝜇 (0) ,𝛼 (0) 𝑐 𝜆 (1) 𝜇 (1) ,𝛼 (1) 𝑐 𝜈 𝛼 ̸ = 0 . By (24) we have 𝜇 (1) ⊆ 𝜆 (1) , which implies that 𝜇 (1) 𝑘 ≤ 𝜆 (1) 𝑘 for all 𝑘 . Hence, as | 𝜆 (1) | = | 𝜇 (1) | by assumption, we must have 𝜇 (1) = 𝜆 (1) . Incombination with (25) this shows that 𝛼 (1) = ∅ .By definition of the Littlewood–Richardson coefficients, the coefficient 𝑐 𝜈 𝛼 = 𝑐 𝜈𝛼 (0) , ∅ is equalto the coefficient of the symmetric polynomial 𝑠 𝜈 in the product 𝑠 𝛼 (0) · 𝑠 ∅ = 𝑠 𝛼 (0) · 𝑠 𝛼 (0) .Hence, 𝑐 𝜈 𝛼 = {︃ if 𝜈 = 𝛼 (0) elseand therefore we must have 𝜈 = 𝛼 (0) . With the same argumentation we see that 𝑐 𝜆 (1) 𝜇 (1) ,𝛼 (1) = 𝑐 𝜆 (1) 𝜆 (1) , ∅ = 1 . WYN B ELLAMY AND U LRICH T HIEL Since 𝜇 ∈ ℱ deg 𝑖 ; 𝑖 , we have 𝜇 (0) = ∅ . Hence, 𝑐 𝜆 (0) 𝜇 (0) ,𝛼 (0) = 𝑐 𝜆 (0) ∅ ,𝜈 = {︃ if 𝜆 (0) = 𝜈 elseso that 𝜆 (0) = 𝜈 . This proves the claim. (cid:4) §6O. Rigid modules We will now show that rigid modules exist precisely in the cuspidal cases and describe themexplicitly. In this section, we consider again an arbitrary complex parameter c . Theorem 6.24.
There is a rigid H c ( 𝑊 ) -module if and only if c is cuspidal, i.e., 𝑐 = 𝑚𝜅 forsome 𝑚 ∈ ± [0 , 𝑛 − with 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) for some 𝑘 > . In this case there are exactly tworigid H c ( 𝑊 ) -modules, namely 𝐿 c ( 𝜆 ) with 𝜆 = (( 𝑘 𝑘 + 𝑚 ) , ∅ ) , or 𝜆 = ( ∅ , ( 𝑘 + 𝑚 ) 𝑘 ) . Proof.
First of all, by Theorem C and Corollary 6.3 there can only exist a rigid H c ( 𝑊 ) -moduleif c is cuspidal. So, assume that 𝑐 = 𝑚𝜅 for some 𝑚 ∈ ± [0 , 𝑛 − with 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) forsome 𝑘 > . By Proposition 6.1, we can assume that 𝑚 ∈ [0 , 𝑛 − . Let 𝜆 = ( 𝜆 (0) , 𝜆 (1) ) be abipartition of 𝑛 . By Lemma 4.10 and equations (13) and (14) the representation 𝜋 𝜆 is c -rigid ifand only if(26) 𝑚𝜀 𝑖 ( − · 𝑣 + 𝑛 ∑︁ 𝑗 =1 𝑗 ̸ = 𝑖 ( 𝑠 𝑖𝑗 + 𝑠 𝑖𝑗, − ) · 𝑣 = 0 , ∀ 𝑣 ∈ 𝜋 𝜆 , 𝑖 = 1 , . . . , 𝑛 , and(27) ( 𝑠 𝑖𝑗 − 𝑠 𝑖𝑗, − ) · 𝑣 = 0 , ∀ 𝑣 ∈ 𝜋 𝜆 , 𝑖 ̸ = 𝑗 . Let 𝑟 := | 𝜆 (0) | . Take 𝑣 to be a non-zero vector in the irreducible ( 𝐵 𝑟 × 𝐵 𝑛 − 𝑟 ) -subrepresentation 𝜋 𝜆 (0) ⊗ 𝛾𝜋 𝜆 (1) inducing 𝜋 𝜆 ; see equation (12). Suppose that 𝑟 / ∈ { , 𝑛 } . Then we can find 𝑖 < 𝑗 with 𝑖 ≤ 𝑟 and 𝑟 < 𝑗 . Due to this choice, we have 𝜀 𝑖 ( − · 𝑣 = 𝑣 and 𝜀 𝑗 ( − · 𝑣 = − 𝑣 as wetwist by 𝛾 in the second component. Hence, 𝑠 𝑖𝑗, − · 𝑣 = 𝑠 𝑖𝑗 𝜀 𝑖 ( − 𝜀 𝑗 ( − · 𝑣 = − 𝑠 𝑖𝑗 · 𝑣 . Equation (27) thus says that 𝑠 𝑖𝑗 · 𝑣 = 0 and therefore already 𝑣 = 0 . This is a contradiction, sowe must have 𝑟 ∈ { , 𝑛 } . Assume that 𝑟 = 𝑛 . Then 𝜋 𝜆 = 𝜋 𝜆 (0) . Now, equation (27) says that 𝑛 ∑︁ 𝑗 =1 𝑗 ̸ = 𝑖 𝑠 𝑖𝑗 · 𝑣 = − 𝑚𝑣 , ∀ 𝑣 ∈ 𝜋 𝜆 (0) , 𝑖 = 1 , . . . 𝑛 . In other words, ∑︀ 𝑗 ̸ = 𝑖 𝑠 𝑖𝑗 acts by a scalar on 𝜋 𝜆 (0) . This holds in particular for 𝑖 = 1 , andnow a standard result (see [17, Lemma 2.4]) implies that 𝜆 (0) = ( 𝑙 𝑏 ) is a rectangle for somepositive integers 𝑙, 𝑏 with 𝑙𝑏 = 𝑛 . The 𝑛 -th Jucys–Murphy element 𝑧 𝑛 = ∑︀ 𝑗<𝑛 𝑠 𝑗𝑛 acts on 𝜋 ( 𝑙 𝑏 ) by multiplication by 𝑙 − 𝑏 since every standard tableaux on ( 𝑎 𝑏 ) must have 𝑛 in the topcorner. Thus, 𝑙𝑏 = 𝑛 and 𝑙 + 𝑚 = 𝑏 , so 𝑛 = 𝑙 ( 𝑙 + 𝑚 ) . Because of Remark 6.4 we musthave 𝑙 = 𝑘 , proving the claim. If 𝑟 = 0 , then 𝜋 𝜆 = 𝛾𝜋 𝜆 (1) and the same argument shows that 𝜆 (1) = (( 𝑘 + 𝑚 ) 𝑘 ) . (cid:4) Remark 6.25.
The proof of Theorem 6.24 can be adapted to all the groups 𝐺 ( ℓ, , 𝑛 ) = Z ℓ ≀ S 𝑛 . §6P. Cuspidal Lusztig families vs. cuspidal Calogero–Moser families Combing all of the above results, we arrive at the proof of Theorem A for type 𝐵 𝑛 . Theorem 6.26.
Assume that c ≥ . Then USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS (a) A family (Lusztig = Calogero–Moser) is cuspidal in the sense of Lusztig if and only ifit is cuspidal as a Calogero–Moser family.(b) If 𝜅 ̸ = 0 and 𝑛 = 𝑘 ( 𝑘 + 𝑚 ) for some 𝑘, 𝑚 ∈ N , then both rigid modules lie in the(unique) cuspidal family ℱ cusp 𝑘,𝑚 . §7. Type 𝐷 The group 𝐷 𝑛 is a normal subgroup of 𝐵 𝑛 of index two. By setting c = (0 , 𝜅 ) , we get anembedding H c ( 𝐷 𝑛 ) ˓ → H c ( 𝐵 𝑛 ) . Thus, we are in the situation of section §4C. We assume that 𝜅 ̸ = 0 . Recall that the irreducible representations of 𝐷 𝑛 are essentially given by unorderedbipartitions of 𝑛 . More precisely, if 𝜆 and 𝜇 are a pair of partitions such that 𝜆 ̸ = 𝜇 and | 𝜆 | + | 𝜇 | = 𝑛 then the set { 𝜆, 𝜇 } labels a simple 𝐷 𝑛 -module. If 𝜆 = 𝜇 , then there are twonon-isomorphic simple modules { 𝜆 } and { 𝜆 } labeled by 𝜆 . These modules are defined by 𝜋 ( 𝜆,𝜇 ) | 𝐷 𝑛 = 𝜋 { 𝜆,𝜇 } for 𝜆 ̸ = 𝜇 , and 𝜋 ( 𝜆,𝜆 ) | 𝐷 𝑛 = 𝜋 { 𝜆 } ⊕ 𝜋 { 𝜆 } . Lemma 7.1.
If there exists 𝑘 such that 𝑛 = 𝑘 then there is a unique rigid H c ( 𝐷 𝑛 ) -module,which is 𝐿 c ( { ( 𝑘 𝑘 ) , ∅} ) . Otherwise, there are no rigid modules. Proof.
By Theorem 6.24, if 𝑛 = 𝑘 for some 𝑘 , then the modules 𝐿 c (( 𝑘 𝑘 ) , ∅ ) and 𝐿 c ( ∅ , ( 𝑘 𝑘 )) are the two rigid modules for H c ( 𝐵 𝑛 ) and if there exists no 𝑘 such that 𝑛 = 𝑘 , then thereexists no rigid modules. Therefore, Proposition 4.22 implies that, in this latter case, there existno rigid modules for H c ( 𝐷 𝑛 ) . Moreover, in the case 𝑛 = 𝑘 , the rigid H c ( 𝐷 𝑛 ) -modules areprecisely the modules of the form 𝐿 c ( 𝜆 ) , where 𝜋 𝜆 is an irreducible 𝐷 𝑛 -submodule of 𝜋 (( 𝑘 𝑘 ) , ∅ ) or 𝜋 ( ∅ , ( 𝑘 𝑘 )) . But both of these 𝐵 𝑛 -modules restrict to the irreducible 𝐷 𝑛 -module 𝜋 { ( 𝑘 𝑘 ) , ∅} . (cid:4) Theorem 7.2.
Assume that 𝜅 ̸ = 0 . The symplectic leaves of X c ( 𝐷 𝑛 ) are in bijection with theset { 𝑘 ≥ | 𝑘 ≤ 𝑛 } , such that(a) dim ℒ 𝑘 = 2( 𝑛 − 𝑘 ) ,(b) the leaf ℒ 𝑘 is labeled by the conjugacy class of the parabolic subgroup 𝐷 𝑘 ,(c) ℒ 𝑘 ≺ ℒ 𝑘 ′ if and only if ( 𝐷 𝑘 ) ≤ ( 𝐷 ( 𝑘 ′ ) ) , if and only if 𝑘 ≥ 𝑘 ′ . Proof.
By Lemma 7.1 and Theorem C, there exists at least one zero-dimensional leaf in X c ( 𝐷 𝑛 ) when 𝑛 = 𝑘 . But we know that there is exactly one zero-dimensional leaf in X c ( 𝐵 𝑛 ) when 𝑛 = 𝑘 . Thus, Lemma 4.18 implies that X c ( 𝐷 𝑛 ) contains exactly one zero-dimensional leafwhen 𝑛 = 𝑘 . Since X c ( 𝐵 𝑛 ) contains no zero-dimensional leaves when 𝑛 ̸ = 𝑘 , Lemma 4.18implies that X c ( 𝐷 𝑛 ) also contains no zero-dimensional leaves in this case.Now, we apply Theorem 4.13. By Lemma 4.6, the proper parabolic subgroups of 𝐷 𝑛 are allconjugate to a subgroup of the form 𝐷 𝑚 × S 𝜆 , where ≤ 𝑚 < 𝑛 and 𝜆 is a partition of 𝑛 − 𝑚 .We denote by c ′ the restriction of c to 𝐷 𝑚 × S 𝜆 . Let us consider when X c ′ ( 𝐷 𝑚 × S 𝜆 ) = X c ′ ( 𝐷 𝑚 ) × X c ( S 𝜆 ) admits a zero dimensional leaf. Since 𝜅 ̸ = 0 , there is a zero-dimensionalleaf in X c ′ ( S 𝜆 ) if and only if S 𝜆 = { } , i.e. if 𝜆 = (1 𝑛 − 𝑚 ) . In this case X c ( S 𝜆 ) is a point.Moreover, X c ′ ( 𝐷 𝑚 ) has a zero-dimensional leaf if and only if there exists a 𝑘 such that 𝑚 = 𝑘 .Thus, either 𝑚 = 𝑘 and 𝜆 = (1 𝑛 − 𝑚 ) , in which case there is exactly one zero-dimensional leafin X c ′ ( 𝐷 𝑚 × S 𝜆 ) , or there are no zero-dimensional leaves in X c ′ ( 𝐷 𝑚 × S 𝜆 ) . Hence Theorem4.13 implies the statements of the theorem. (cid:4) By [4, Corollary 6.10], the Calogero–Moser families for H c ( 𝐷 𝑛 ) , with c ̸ = 0 , are de-scribed as follows. If 𝜆 is a partition of 𝑛/ , then the two representations { 𝜆 } and { 𝜆 } eachform a singleton family. Otherwise, { 𝜆, 𝜇 } and { 𝜆 ′ , 𝜇 ′ } are in the same family if and only if Res { 𝜆,𝜇 } ( 𝑥 ) = Res { 𝜆 ′ ,𝜇 ′ } ( 𝑥 ) , where Res { 𝜆,𝜇 } ( 𝑥 ) := Res 𝜆 ( 𝑥 ) + Res 𝜇 ( 𝑥 ) . Theorem 7.3.
Assume that c ≥ . WYN B ELLAMY AND U LRICH T HIEL (a) The Lusztig c -families for 𝐷 𝑛 equal the Calogero–Moser c -families.(b) The cuspidal Lusztig c -families equal the cuspidal Calogero–Moser c -families. Proof.
By Lemma 2.5, Lemma 3.8 and Lemma 4.11 we may assume that ( 𝑐 , 𝜅 ) = (0 , .The first part of the theorem follows from Corollary 6.13, [33, Section 22.26], and the abovedescription of the Calogero–Moser families. As shown in [32, Section 8.1], there is a uniquecuspidal Lusztig family when 𝑛 = 𝑘 and none otherwise. In the case 𝑛 = 𝑘 , it is the uniquefamily containing the symbol(28) (︂ , , . . . , 𝑘 − , , . . . , 𝑘 − )︂ . The content of this symbol is ∑︀ 𝑘 − 𝑖 =0 𝑥 𝑖 , which is the same as the content of the symbol 𝑆 = (︂ 𝑘, 𝑘 + 1 , . . . , 𝑘 − , , . . . , 𝑘 − )︂ . This is the symbol of (( 𝑘 𝑘 ) , ∅ ) in Sy 𝑘 (1 , 𝑛 , which implies that the cuspidal Lusztig familycorresponding to the content of the symbol (28) is the same as the Calogero–Moser family con-taining { ( 𝑘 𝑘 ) , ∅} . By Lemma 7.1 and Theorem 7.2, this is the unique cuspidal Calogero–Moserfamily. (cid:4) §8. Type 𝐼 ( 𝑚 ) In this section we treat the case of dihedral groups. We show that almost all representationsof the restricted rational Cherednik algebra are rigid. From this we easily obtain the proof ofTheorem A. We note that the results here together with [42, Appendix B] give a completedescription of the representation theory of restricted rational Cherednik algebras for dihedralgroups at all parameters. §8A. The group
Throughout, we assume that 𝑚 ≥ and choose a primitive 𝑚 -th root of unity 𝜁 ∈ C . Let 𝑊 be the Coxeter group of type 𝐼 ( 𝑚 ) . This is the dihedral group of order 𝑚 . It has two naturalpresentations, namely the Coxeter presentation ⟨ 𝑠, 𝑡 | 𝑠 = 𝑡 = ( 𝑠𝑡 ) 𝑚 = 1 ⟩ and the geometricpresentation ⟨ 𝑠, 𝑟 | 𝑟 𝑚 = 1 , 𝑠 = 1 , 𝑠 − 𝑟𝑠 = 𝑟 − ⟩ with a generating rotation 𝑟 := 𝑠𝑡 for thesymmetries of a regular 𝑚 -gon. §8B. Representations The representation theory of 𝑊 depends on the parity of 𝑚 . In the following we use the samenotation for the representations as in [23, 5.3.4], which essentially is also the same as in [22].If 𝑚 is odd, the conjugacy classes of 𝑊 are { } , { 𝑟 ± } , { 𝑟 ± } , . . . , { 𝑟 ± ( 𝑚 − / } , { 𝑟 𝑙 𝑠 | ≤ 𝑙 ≤ 𝑚 − } , and so the total number of conjugacy classes is ( 𝑚 + 3) / . There are two irreducible one-dimensional representations: the trivial one 𝑊 and the sign representation 𝜀 : 𝑊 → C with 𝜀 ( 𝑠 ) = − , 𝜀 ( 𝑡 ) = − , 𝜀 ( 𝑟 ) = 1 . The remaining ( 𝑚 + 3) / − 𝑚 − / irreducible representations 𝜙 𝑖 , ≤ 𝑖 ≤ ( 𝑚 − / ,are all two-dimensional and are given by 𝜙 𝑖 ( 𝑠 ) = (︂ )︂ , 𝜙 𝑖 ( 𝑡 ) := (︂ 𝜁 − 𝑖 𝜁 𝑖 )︂ , 𝜙 𝑖 ( 𝑟 ) = (︂ 𝜁 𝑖 𝜁 − 𝑖 )︂ . We denote the character of 𝜙 𝑖 by 𝜒 𝑖 . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS If 𝑚 is even, then the conjugacy classes of 𝑊 are { } , { 𝑟 ± } , { 𝑟 ± } , . . . , { 𝑟 ± 𝑚/ } , { 𝑟 𝑘 𝑠 | ≤ 𝑘 ≤ ( 𝑚/ − } , { 𝑟 𝑘 +1 𝑠 | ≤ 𝑘 ≤ ( 𝑚/ − } , and so the total number of conjugacy classes is ( 𝑚 + 6) / . There are four irreducible one-dimensional representations: the trivial one 𝑊 , the sign representation 𝜀 , and two furtherrepresentations 𝜀 , 𝜀 with 𝜀 ( 𝑠 ) = − , 𝜀 ( 𝑡 ) = − , 𝜀 ( 𝑟 ) = 1 ,𝜀 ( 𝑠 ) = 1 , 𝜀 ( 𝑡 ) = − , 𝜀 ( 𝑟 ) = − ,𝜀 ( 𝑠 ) = − , 𝜀 ( 𝑡 ) = 1 , 𝜀 ( 𝑟 ) = − . The remaining ( 𝑚 + 6) / − 𝑚 − / irreducible representations 𝜙 𝑖 , ≤ 𝑖 ≤ ( 𝑚 − / ,are all two-dimensional and are defined as in case 𝑚 is odd. Again, we denote the character of 𝜙 𝑖 by 𝜒 𝑖 . §8C. Reflections and parameters The two-dimensional faithful irreducible representation 𝜙 of 𝑊 is a reflection representationin which precisely the elements 𝑠 𝑙 := 𝑟 𝑙 𝑠 for ≤ 𝑙 ≤ 𝑚 − act as reflections. We will alwaysfix this representation as the reflection representation of 𝑊 . Let ( 𝑦 , 𝑦 ) be the standard basisof h := C and let ( 𝑥 , 𝑥 ) be the dual basis. We can easily verify that roots and coroots for thereflections 𝑠 𝑙 are given by 𝛼 𝑠 𝑙 = 𝑥 − 𝜁 − 𝑙 𝑥 and 𝛼 ∨ 𝑠 𝑙 = 𝑦 − 𝜁𝑦 . With this we see that the Cherednik coefficients ( 𝑦 𝑖 , 𝑥 𝑗 ) 𝑠 𝑙 = − ( 𝑦 𝑖 , 𝛼 𝑠 𝑙 )( 𝛼 ∨ 𝑠 𝑙 , 𝑥 𝑗 ) are: ( 𝑦 , 𝑥 ) 𝑠 𝑙 = − , ( 𝑦 , 𝑥 ) 𝑠 𝑙 = 𝜁 − 𝑙 , ( 𝑦 , 𝑥 ) 𝑠 𝑙 = 𝜁 𝑙 , ( 𝑦 , 𝑥 ) 𝑠 𝑙 = − . If 𝑚 is odd, there is just one conjugacy class of reflections in 𝑊 , namely the one of 𝑠 which is { 𝑠 𝑙 | ≤ 𝑙 ≤ 𝑚 − } . If 𝑚 is even, there are two conjugacy classes of reflections in 𝑊 , namelythe one of 𝑠 which is { 𝑠 𝑙 | ≤ 𝑙 ≤ 𝑚 − } and the one of 𝑡 which is { 𝑠 𝑙 +1 | ≤ 𝑙 ≤ 𝑚 − } .Note that 𝜙 𝑖 ( 𝑠 𝑙 ) = (︃ 𝜁 𝑖𝑙 𝜁 − 𝑖𝑙 )︃ . If c : Ref ( 𝑊 ) → C is a function which is constant on conjugacy classes, then, as in [22,1.3.7], we define 𝑏 := c ( 𝑠 ) , 𝑎 := c ( 𝑡 ) . We fix such a function from now on and assume that c ̸ = 0 . Note that if 𝑚 is odd, we have 𝑎 = 𝑏 . §8D. Summary We start with a tabular summary of the description of (cuspidal) Calogero–Moser familiesand rigid representations. To simplify notation we denote by ℱ the set of two-dimensionalirreducible characters of 𝑊 . To allow a presentation which is independent of the parity of 𝑚 we set ℛ := {︂ { 𝜙 𝑖 | < 𝑖 ≤ ( 𝑚 − / } = ℱ ∖ { 𝜙 } if 𝑚 is odd { 𝜙 𝑖 | < 𝑖 < ( 𝑚 − / } = ℱ ∖ { 𝜙 , 𝜙 ( 𝑚 − / } if 𝑚 is even.We make the convention that we ignore 𝜀 and 𝜀 whenever 𝑚 is odd. Theorem 8.1.
The (cuspidal) Calogero–Moser families and rigid representations of H c ( 𝑊 ) areas listed in Table 2.In the next three sections we will prove this theorem. WYN B ELLAMY AND U LRICH T HIEL Parameters CM families rigidrepresentations cuspidalCM families 𝑎, 𝑏 ̸ = 0 and 𝑎 ̸ = ± 𝑏 { } , { 𝜀 } , { 𝜀 } , { 𝜀 } , ℱ ℛ ℱ 𝑎 = 0 and 𝑏 ̸ = 0 { , 𝜀 } , { 𝜀, 𝜀 } , ℱ ℛ ℱ 𝑎 ̸ = 0 and 𝑏 = 0 { , 𝜀 } , { 𝜀, 𝜀 } , ℱ ℛ ℱ 𝑎 = 𝑏 ̸ = 0 { } , { 𝜀 } , { 𝜀 , 𝜀 } ∪ ℱ 𝜀 , 𝜀 , 𝜙 |ℱ| , ℛ { 𝜀 , 𝜀 } ∪ ℱ 𝑎 = − 𝑏 ̸ = 0 { 𝜀 } , { 𝜀 } , { , 𝜀 } ∪ ℱ , 𝜀, 𝜙 , ℛ { , 𝜀 } ∪ ℱ Table 2.
The (cuspidal) Calogero–Moser families and rigid representations for dihedral groups. §8E. Calogero–Moser families
We recall from [40] the notion of
Euler c -families . These are defined by the action of the (central)Euler element of H c ( 𝑊 ) on the simple modules and are coarser than the Calogero–Moser c -families. In [40, Corollary 1] a simple character theoretic formula for determining thesefamilies is given: two irreducible characters 𝜆 and 𝜇 of 𝑊 lie in the same Euler c -family if andonly if ∑︁ 𝑥 ∈ Ref ( 𝑊 ) c ( 𝑥 ) (︂ 𝜆 ( 𝑥 ) 𝜆 (1) − 𝜇 ( 𝑥 ) 𝜇 (1) )︂ = 0 . This formula is in our case equivalent to 𝑎 (︂ 𝜆 ( 𝑠 ) 𝜆 (1) − 𝜇 ( 𝑠 ) 𝜇 (1) )︂ + 𝑏 (︂ 𝜆 ( 𝑡 ) 𝜆 (1) − 𝜇 ( 𝑡 ) 𝜇 (1) )︂ = 0 . From this it is easy to deduce that the Euler families are as in Table 2. In [3] the first author hasproven that, for any c , the Euler c -families are in fact already Calogero–Moser c -families when 𝑊 is a dihedral group. §8F. Rigid representations We split the proof of the description of rigid representations into two parts, depending on theparity of 𝑚 . Proposition 8.2.
Assume that 𝑚 is odd. The following holds:(a) The representations , 𝜀 , and 𝜙 are not rigid.(b) The representation 𝜙 𝑖 is a rigid representation for all < 𝑖 ≤ ( 𝑚 − / . Proof.
The representation 𝜙 𝑖 for ≤ 𝑖 ≤ ( 𝑚 − / is rigid if and only if 𝑎 𝑚 − ∑︁ 𝑙 =0 ( 𝑦 𝑘 , 𝑥 𝑗 ) 𝑠 𝑙 𝜙 𝑖 ( 𝑠 𝑙 ) for all 𝑘, 𝑗 ∈ { , } . As 𝑎 ̸ = 0 , this is equivalent to(29) 𝑚 − ∑︁ 𝑙 =0 ( 𝑦 𝑘 , 𝑥 𝑗 ) 𝑠 𝑙 𝜙 𝑖 ( 𝑠 𝑙 ) = 0 for all 𝑘, 𝑗 ∈ { , } . Note that 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 𝑞 ) 𝑙 = {︂ 𝑚 if 𝑞 ∈ 𝑚 Z else.Using the Cherednik coefficients computed in §8C, equation (29) is for 𝑘 = 1 = 𝑗 and for 𝑘 = 2 = 𝑗 equivalent to 𝑚 − ∑︁ 𝑙 =0 (︃ 𝜁 𝑖𝑙 𝜁 − 𝑖𝑙 )︃ = 0 ⇐⇒ 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 𝑖 ) 𝑙 = 0 and 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 − 𝑖 ) 𝑙 = 0 USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS and due to the aforementioned, this condition is satisfied if and only if 𝑖 / ∈ 𝑚 Z . Since ≤ 𝑖 ≤ ( 𝑚 − / , this is always satisfied. For 𝑘 = 1 and 𝑗 = 2 equation (29) is equivalent to 𝑚 − ∑︁ 𝑙 =0 𝜁 − 𝑙 (︃ 𝜁 𝑖𝑙 𝜁 − 𝑖𝑙 )︃ = 0 ⇐⇒ 𝑚 − ∑︁ 𝑙 =0 𝜁 𝑖𝑙 − 𝑙 = 0 and 𝑚 − ∑︁ 𝑙 =0 𝜁 − 𝑖𝑙 − 𝑙 = 0 ⇐⇒ 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 𝑖 − ) 𝑙 = 0 and 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 − 𝑖 − ) 𝑙 = 0 ⇐⇒ 𝑖 − / ∈ 𝑚 Z and 𝑖 + 1 / ∈ 𝑚 Z . Due to ≤ 𝑖 ≤ ( 𝑚 − / the condition 𝑖 + 1 / ∈ 𝑚 Z is always satisfied. Hence, (29) holds for 𝑘 = 1 and 𝑗 = 2 if and only if 𝑖 ̸ = 1 . Finally, for 𝑘 = 2 and 𝑗 = 1 equation (29) is equivalentto 𝑚 − ∑︁ 𝑙 =0 𝜁 𝑙 (︃ 𝜁 𝑖𝑙 𝜁 − 𝑖𝑙 )︃ = 0 ⇐⇒ 𝑚 − ∑︁ 𝑙 =0 𝜁 𝑖𝑙 + 𝑙 = 0 and 𝑚 − ∑︁ 𝑙 =0 𝜁 − 𝑖𝑙 + 𝑙 = 0 ⇐⇒ 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 𝑖 +1 ) 𝑙 = 0 and 𝑚 − ∑︁ 𝑙 =0 ( 𝜁 − 𝑖 +1 ) 𝑙 = 0 ⇐⇒ 𝑖 + 1 / ∈ 𝑚 Z and 𝑖 − / ∈ 𝑚 Z . Again, the condition 𝑖 + 1 / ∈ Z is always satisfied and 𝑖 − / ∈ 𝑚 Z holds if and only if 𝑖 ̸ = 1 .This proves the claim. (cid:4) Proposition 8.3.
Assume that 𝑚 is even. The following holds:(a) For any < 𝑖 < ( 𝑚 − / the representation 𝜙 𝑖 is rigid.(b) The representation 𝜙 is rigid if and only if 𝑎 = − 𝑏 .(c) The representation 𝜙 ( 𝑚 − / is rigid if and only if 𝑎 = 𝑏 .(d) The representations and 𝜀 are rigid if and only if 𝑎 = − 𝑏 .(e) The representations 𝜀 and 𝜀 are rigid if and only if 𝑎 = 𝑏 . Proof.
This follows from a similar direct computation as in the proof of Proposition 8.2. Weomit the details here. (cid:4) §8G. Cuspidal Calogero–Moser families
For 𝑚 ≥ the first author has shown in [2, §5.5] that independent of the parameter c thereis exactly one cuspidal Calogero–Moser family. It thus remains to identify this family. Since 𝑚 ≥ we have ℛ ̸ = ∅ , and as ℛ is always contained in the Calogero–Moser family whichin Table 2 is claimed to be cuspidal, it follows from Theorem C that this family is indeed theunique cuspidal one. §8H. Lusztig families From now on we assume that c ≥ . The Lusztig families of 𝑊 are listed in Table 3 which istaken from [22, 1.7.3]. Parameters Lusztig families 𝑏 = 𝑎 > { 𝑊 } , { 𝜀 } , { 𝜀 , 𝜀 } ∪ ℱ 𝑏 > 𝑎 > or 𝑎 > 𝑏 > { 𝑊 } , { 𝜀 } , { 𝜀 } , { 𝜀 } , ℱ 𝑏 > 𝑎 = 0 { 𝑊 , 𝜀 } , { 𝜀, 𝜀 } , ℱ 𝑎 > 𝑏 = 0 { 𝑊 , 𝜀 } , { 𝜀, 𝜀 } , ℱ Table 3.
Lusztig families
WYN B ELLAMY AND U LRICH T HIEL Comparison with the Calogero–Moser families immediately yields the proof of Theorem 2.4for dihedral groups:
Corollary 8.4.
For any c ≥ the Lusztig c -families are equal to the Calogero–Moser c -families. §8I. Cuspidal Lusztig families In order to determine which of the Lusztig families are cuspidal we explicitly compute the j -induction. The group 𝑊 has two non-trivial proper parabolic subgroups: the group 𝑃 := ⟨ 𝑠 ⟩ and the group 𝑃 := ⟨ 𝑡 ⟩ , which are both Coxeter groups of type A . Let 𝜓 𝑖 be the non-trivialirreducible character of 𝑃 𝑖 and note that this is the sign representation of this Coxeter group. Itis not hard to compute that Ind 𝑊𝑃 𝑃 = 1 𝑊 + 𝜀 + ∑︁ 𝑗 𝜒 𝑗 , Ind 𝑊𝑃 𝜓 = 𝜀 + 𝜀 + ∑︁ 𝑗 𝜒 𝑗 + 𝛿 𝑊 , Ind 𝑊𝑃 𝑃 = 1 𝑊 + 𝜀 + ∑︁ 𝑗 𝜒 𝑗 , Ind 𝑊𝑃 𝜓 = 𝜀 + 𝜀 + ∑︁ 𝑗 𝜒 + 𝛿 𝑊 , where 𝛿 := {︂ if 𝑚 is even if 𝑚 is oddand the sums are taken over all two-dimensional characters.Lusztig’s a -functions a 𝜒 of the irreducible characters 𝜒 of 𝑊 with respect to c are listed inTable 4 which is taken from [22, 1.3.7], where the last row follows by symmetry. Using [22,Parameters 𝜙 𝑖 𝑊 𝜀 𝜀 𝜀 𝑏 = 𝑎 > 𝑎 𝑚𝑎 𝑎 𝑎𝑏 > 𝑎 ≥ 𝑏 𝑚 ( 𝑎 + 𝑏 ) 𝑎 𝑚 ( 𝑏 − 𝑎 ) + 𝑎𝑎 > 𝑏 ≥ 𝑎 𝑚 ( 𝑎 + 𝑏 ) 𝑚 ( 𝑎 − 𝑏 ) + 𝑏 𝑏 Table 4.
The a -function a -functions for the irreducible characters of the parabolic subgroups withrespect to the restriction of c to these groups are as in Table 5. From these tables we can deduce 𝜒 𝑃 𝜓 𝑃 𝜓 a 𝜒 𝑏 𝑎 Table 5.
The a -function for the parabolic subgroups 𝑃 𝑖 . that Lusztig’s j -induction is as in Table 6.Parameters j 𝑊𝑃 𝑃 j 𝑊𝑃 𝜓 j 𝑊𝑃 𝑃 j 𝑊𝑃 𝜓 𝑏 = 𝑎 > 𝑊 𝜀 + ∑︀ 𝑗 𝜒 𝑗 𝑊 𝜀 + ∑︀ 𝑗 𝜒 𝑗 𝑏 > 𝑎 > 𝑊 ∑︀ 𝑗 𝜒 𝑗 𝑊 𝜀 𝑏 > 𝑎 = 0 1 𝑊 + 𝜀 ∑︀ 𝑗 𝜒 𝑗 𝑊 𝜀 𝑎 > 𝑏 > 𝑊 𝜀 𝑊 ∑︀ 𝑗 𝜒 𝑗 𝑎 > 𝑏 = 0 1 𝑊 𝜀 𝑊 + 𝜀 ∑︀ 𝑗 𝜒 𝑗 Table 6. j -induction. Using the table of j -inductions we can now easily determine the cuspidal Lusztig families. Lemma 8.5.
Let c ≥ . There is a unique cuspidal Lusztig family. This family is equal to { 𝜀 , 𝜀 } ∪ ℱ if 𝑏 = 𝑎 , and otherwise it is equal to ℱ . USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
Proof.
The Lusztig families of the parabolic subgroup 𝑃 𝑖 are { 𝑃 𝑖 } and { 𝜓 𝑖 } if 𝑏 ̸ = 0 , respec-tively 𝑎 ̸ = 0 , and they are { 𝑃 𝑖 , 𝜓 𝑖 } if 𝑏 = 0 , respectively 𝑎 = 0 . The claim follows now easilyfrom the definition of cuspidality using the table of j -inductions. (cid:4) Comparison with the cuspidal Calogero–Moser families completes the proof of Theorem A.
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