CUSPIDAL CALOGERO–MOSER AND LUSZTIG FAMILIES
FOR COXETER GROUPS

arXiv:1505.00486v3 [math.RT] 13 Jun 2016

GWYN BELLAMY AND ULRICH THIEL
A BSTRACT. The goal of this paper is to compute the cuspidal Calogero–Moser families for
all infinite families of finite Coxeter groups, at all parameters. We do this by first computing
the 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 between
Calogero–Moser families and Lusztig families. Our classification shows that, additionally, the
cuspidal Calogero–Moser families equal cuspidal Lusztig families for the infinite families of
Coxeter 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 Cherednik algebras by Etingof and Ginzburg [16] that there is a very close connection between
these algebras and cyclotomic Hecke algebras [11]. This connection is encoded in the Knizhnik–Zamolodchikov functor, introduced in [24], and is a key tool in the representation theory of
rational Cherednik algebras at 𝑡 ̸= 0.
In the quasi-classical limit 𝑡 = 0 the Knizhnik–Zamolodchikov functor no longer exists and
no functorial connection to Hecke algebras is currently known. Astonishingly, as first noticed
by Gordon and Martino [26], it seems that there is still, none the less, a close relationship
between rational Cherednik algebras in 𝑡 = 0 and Hecke algebras, suggesting that there may be
an asymptotic Knizhnik–Zamolodchikov functor in the quasi-classical limit. The aim of this
article is to add weight to this expectation by comparing cuspidal Calogero–Moser families
with 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 invariant
under 𝑊 -conjugation, the rational Cherednik algebra Hc (𝑊 ) at 𝑡 = 0. The spectrum of the
centre of this algebra is an affine Poisson deformation Xc (𝑊 ) of the symplectic singularity
(h × h* )/𝑊 , called the Calogero–Moser space. This theory exists in particular for finite Coxeter
groups 𝑊 . In this case, one can also attach to 𝑊 the Hecke algebra ℋ𝐿 (𝑊 ) depending on a
weight 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 those
coming from rational Cherednik algebras.
Gordon [25] has defined the notion of Calogero–Moser c-families of Irr(𝑊 ), which on the
geometric side correspond to the C* -fixed points of the Calogero–Moser space Xc (𝑊 ). Work
of 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. Algebra
G WYN B ELLAMY, School of Mathematics and Statistics, University Gardens, University of Glasgow, Glasgow,
G12 8QW, UK, gwyn.bellamy@glasgow.ac.uk
U LRICH T HIEL, Universität Stuttgart, Fachbereich Mathematik, Institut für Algebra und Zahlentheorie, Lehrstuhl
für Algebra, Pfaffenwaldring 57, 70569 Stuttgart, Germany, thiel@mathematik.uni-stuttgart.de

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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, 𝐼2 (𝑚) or 𝐻3 , then the Lusztig c-families equal
the 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 indeed
true for all finite Coxeter groups; see also Bonnafé–Rouquier [9]. There is so far no conceptual
explanation for this connection. Bonnafé and Rouquier [9] furthermore constructed analogs
of constructible characters and cells on the Calogero–Moser side, and collected evidence
supporting 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 is
Lusztig’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 for
a proper parabolic subgroup. Cuspidal families play a key role in describing certain unipotent
representations for the corresponding finite groups of Lie type. In [2] the first author also
introduced 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 symplectic
leaf of the Calogero–Moser space. In this article we determine the cuspidal Calogero–Moser
families for the Coxeter groups of type 𝐴, 𝐵, 𝐷 and 𝐼2 (𝑚). Our main result states (see §3):
Theorem A. If 𝑊 is of type 𝐴, 𝐵, 𝐷 or 𝐼2 (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 methods
we 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 cuspidal
Calogero–Moser c-families for all real parameters c.
Because of Theorem A this conjecture remains open only for the six exceptional Coxeter
groups 𝐻3 , 𝐻4 , 𝐹4 , 𝐸6 , 𝐸7 , 𝐸8 .
Rigid representations
The main ingredient for calculating the cuspidal Calogero–Moser families, and hence confirming
Theorem A, is the notion of a rigid module: a Hc (𝑊 )-module is said to be rigid if it is
irreducible as a 𝑊 -module. These have already played a role in the representation theory of
rational Cherednik algebras at 𝑡 ̸= 0, see e.g. [6] or [17], and at 𝑡 = 0 they were studied by the
second 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 Rep𝑑 (Hc (𝑊 )),
the variety parameterizing representations of dimension 𝑑, is open. Therefore, though these
modules often appear in families with respect to the parameter c, the module structure (for
fixed parameter c) on a rigid module cannot be deformed to a continuous family. This is the
first clue that there is a strong connection between rigid representations and zero-dimensional
leaves of Xc (𝑊 ) (and hence to cuspidal Calogero-Moser families).
In this article we classify the rigid modules for all non-exceptional Coxeter groups and all
parameters. The importance of these modules is explained by our second main result which we
prove in §4:
Theorem C. Let 𝑊 be an arbitrary finite complex reflection group. If the simple module 𝐿c (𝜆),
where 𝜆 ∈ Irr(𝑊 ), is a rigid Hc (𝑊 )-module, then the Calogero–Moser c-family to which it
belongs is cuspidal.

G WYN B ELLAMY AND U LRICH T HIEL

3

Rigid modules are easily computed, and using Theorem C this allows us to identify certain
cuspidal families. Remarkably, for the non-exceptional Coxeter groups we can show that
the cuspidal Calogero–Moser families are precisely those containing the rigid modules. The
cuspidal Lusztig families are similarly characterized.
Remark. While this paper was in preparation, the preprint [14] appeared, where rigid modules
also play a key role (though the definition there is slightly different). Based on the analogy
with 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 cuspidal
family in our sense is cuspidal in the sense of [14]. However, it is clear that for most complex
reflection 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 𝐸7 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 that
the Calogero–Moser space Xc (𝑊 ) is stratified by finitely many symplectic leaves. These leaves
are naturally labeled by conjugacy classes of parabolic subgroups (𝑊 ′ ) of 𝑊 . There are two
natural partial orderings on the set of symplectic leaves: a geometric one given in terms of
the closures of leaves, and another, algebraic one given in terms of inclusions of parabolic
subgroups. 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 𝐼2 (𝑚) in terms of the conjugacy classes of parabolic subgroups. We also describe
the two orderings on the set of symplectic leaves in these cases (see Theorem 6.2, Theorem 7.2
and [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 the
groups of type 𝐵 using the fact that 𝐷𝑛 C 𝐵𝑛 . More generally, we consider a complex reflection
group (h, 𝑊 ) and a normal subgroup 𝐾 C 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–Moser
space Xc (𝐾) such that Xc (𝑊 ) = Xc (𝐾)/Γ. This allows us to deduce the Calogero–Moser
families for 𝐾 from the Calogero–Moser families for 𝑊 , generalising results of [4]. Cuspidal
families and rigid representations behave well under this correspondence. We also describe the
symplectic leaves in Xc (𝐾) in terms of those of Xc (𝑊 ).
Acknowledgements. The authors would like to thank Cédric Bonnafé and Meinolf Geck for
many fruitful discussions. We also thank Dan Ciubotaru for informing us about his preprint [14]
and his result that for 𝐸7 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.

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C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

C ONTENTS
§1. Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§2. Lusztig families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§3. Cuspidal Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§4. Calculating cuspidal Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§5. Type 𝐴 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§6. Type 𝐵 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§7. Type 𝐷 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§8. Type 𝐼2 (𝑚). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
6
9
13
21
22
34
35

§1. Calogero–Moser families
We begin by recalling the definition of the main protagonists of this paper—the Calogero–Moser
families for complex reflection groups. They are obtained from the block structure of the
restricted rational Cherednik algebra studied by Gordon [25], which is a finite-dimensional
quotient 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 finite
subgroup of GL(h) for some finite-dimensional complex vector space h such that 𝑊 is generated
by its set Ref(𝑊 ) of reflections, i.e., by those elements 𝑠 ∈ 𝑊 such that Ker(idh −𝑠) is of
codimension one in h. Let (·, ·) : h × h* → C be the natural pairing defined by (𝑦, 𝑥) = 𝑥(𝑦).
For 𝑠 ∈ Ref(𝑊 ) we fix 𝛼𝑠 ∈ h* to be a basis of the one-dimensional space Im(𝑠 − 1)|h* and
𝛼𝑠∨ ∈ h to be a basis of the one-dimensional space Im(𝑠 − 1)|h , normalised so that 𝛼𝑠 (𝛼𝑠∨ ) = 2.
Our discussion will not depend on the choice of 𝛼𝑠 and 𝛼𝑠∨ . 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 H𝑡,c (𝑊 ), as introduced by Etingof and Ginzburg [16], is the
quotient of the skew group algebra of the tensor algebra, 𝑇 (h ⊕ h* ) o 𝑊 , by the ideal generated
by the relations [𝑥, 𝑥′ ] = [𝑦, 𝑦 ′ ] = 0 for all 𝑥, 𝑥′ ∈ h* and 𝑦, 𝑦 ′ ∈ h, and
(1)

[𝑦, 𝑥] = 𝑡(𝑦, 𝑥) −

∑︁

c(𝑠)(𝑦, 𝛼𝑠 )(𝛼𝑠∨ , 𝑥)𝑠 ,

∀ 𝑦 ∈ h, 𝑥 ∈ h* .

𝑠∈Ref(𝑊 )

We concentrate on the case 𝑡 = 0 and set Hc := H0,c . For any 𝛼 ∈ C∖{0}, the algebras H𝛼c (𝑊 )
and Hc (𝑊 ) are naturally isomorphic. Therefore we are free to rescale c by 𝛼 whenever this
is convenient. A fundamental result for rational Cherednik algebras, proved by Etingof and
Ginzburg [16, Theorem 1.3], is that the PBW property holds for all c, i.e., the natural map
(2)
C[h] ⊗C C𝑊 ⊗C C[h* ] → Hc (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) = −1 for 𝑦 ∈ h, and deg(𝑤) = 0 for 𝑤 ∈ 𝑊 . We note that no
such grading exists for general symplectic reflection algebras.
§1B. Calogero–Moser space
The centre Zc (𝑊 ) of Hc (𝑊 ) is an affine domain. We shall denote by Xc (𝑊 ) := Spec(Zc (𝑊 ))
the corresponding affine variety. It is called the (generalized) Calogero–Moser space associated
to 𝑊 at parameter c. These varieties define a flat family of deformations of (h ⊕ h* )/𝑊 over
the affine C-space of dimension | Ref(𝑊 )/𝑊 |. The following was shown for Coxeter groups
in [16, Proposition 4.15], and the general case is due to [25, Proposition 3.6].

G WYN B ELLAMY AND U LRICH T HIEL

5

Proposition 1.1 (Etingof–Ginzburg, Gordon). The subspace 𝐷(𝑊 ) := C[h]𝑊 ⊗C C[h* ]𝑊 of
Hc (𝑊 ) is a central subalgebra and Zc (𝑊 ) is a free 𝐷(𝑊 )-module of rank |𝑊 |.
The inclusions C[h]𝑊 ˓→ 𝑍c (𝑊 ) and C[h* ]𝑊 ˓→ 𝑍c (𝑊 ) define finite surjective morphisms
𝜋c : Xc (𝑊 )  h/𝑊 and 𝜛c : Xc (𝑊 )  h* /𝑊 .
We write
ϒc := 𝜋c × 𝜛c : Xc (𝑊 )  h/𝑊 × h* /𝑊
for the product morphism. It is a finite, and hence closed, surjective morphism. Note that
both Zc (𝑊 ) and 𝐷(𝑊 ) are graded subalgebras of Hc (𝑊 ). This implies that Xc (𝑊 ) and
h/𝑊 × h* /𝑊 carry a C* -action making ϒc a C* -equivariant morphism.
§1C. Restricted rational Cherednik algebras
The inclusion of algebras 𝐷(𝑊 ) ˓→ Zc (𝑊 ) allows us to define the restricted rational Cherednik algebra Hc (𝑊 ) as the quotient
Hc (𝑊 )
,
Hc (𝑊 ) =
𝐷(𝑊 )+ · 𝐻c (𝑊 )
where 𝐷(𝑊 )+ denotes the ideal in 𝐷(𝑊 ) of elements with zero constant term. This algebra
was originally introduced, and extensively studied, by Gordon [25]. The PBW theorem implies
that
(3)
Hc (𝑊 ) ≃ C[h]co𝑊 ⊗C C𝑊 ⊗C C[h* ]co𝑊
as C-vector spaces. Here,
C[h]co𝑊 = C[h]/⟨C[h]𝑊
+⟩
*
co𝑊
is the coinvariant algebra of 𝑊 and C[h ]
is defined analogously. Since 𝑊 is a reflection
group, the coinvariant algebra C[h]co𝑊 is of dimension |𝑊 | and is isomorphic to the regular
representation as a 𝑊 -module. Thus, dim Hc (𝑊 ) = |𝑊 |3 . The restricted rational Cherednik
algebra is a quotient of Hc (𝑊 ) by an ideal generated by homogeneous elements and so it is
also a graded algebra. This combined with the triangular decomposition (3) of Hc (𝑊 ) implies
that the representation theory of Hc (𝑊 ) has a rich combinatorial structure. The following is
due to Gordon [25], based on an abstract framework by Holmes and Nakano [27]. First of all,
note that the skew-group algebra C[h* ]co𝑊 o 𝑊 is a graded subalgebra of Hc (𝑊 ).
Definition 1.2. The baby Verma module of Hc (𝑊 ) associated to a 𝑊 -module 𝜆 is
Δc (𝜆) := Hc (𝑊 ) ⊗C[h* ]co𝑊 o𝑊 𝜆 ,
where C[h* ]co𝑊
acts on 𝜆 as zero.
+
The baby Verma module Δc (𝜆) is naturally a graded Hc (𝑊 )-module, where 1 ⊗ 𝜆 sits
in degree zero. By studying quotients of baby Verma modules, it is possible to completely
classify the simple Hc (𝑊 )-modules. We denote by Irr 𝑊 the set of simple 𝑊 -modules (up to
isomorphism). Similarly, we understand Irr Hc (𝑊 ).
Proposition 1.3 (Gordon). Let 𝜆, 𝜇 ∈ Irr 𝑊 .
(1) The baby Verma module Δc (𝜆) has a simple head. We denote it by 𝐿c (𝜆).
(2) 𝐿c (𝜆) is isomorphic to 𝐿c (𝜇) if and only if 𝜆 ≃ 𝜇.
(3) The map Irr 𝑊 → Irr Hc (𝑊 ), 𝜆 → 𝐿c (𝜆), is a bijection.
The bijection in the proposition allows us to transform representation theoretic information
about Hc (𝑊 ) into combinatorial c-dependent data about 𝑊 . The Calogero–Moser families are
the primary example of this process.

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§1D. Calogero–Moser families
Since the algebra Hc (𝑊 ) is finite-dimensional, it has a block decomposition Hc (𝑊 ) =
𝑖=1 𝐵𝑖 , with each 𝐵𝑖 an indecomposable algebra. If 𝑏𝑖 is the identity element of 𝐵𝑖 then
the identity element 1 of Hc (𝑊 ) is the sum 1 = 𝑏1 + . . . + 𝑏𝑘 of the 𝑏𝑖 . For each simple
Hc (𝑊 )-module 𝐿, there exists a unique 𝑖 such that 𝑏𝑖 · 𝐿 ̸= 0. In this case we say that 𝐿
belongs to the block 𝐵𝑖 . By Proposition 1.3, we can (and will) identify Irr Hc (𝑊 ) with Irr 𝑊 .
Let Ωc (𝑊 ) be the set of equivalence classes of Irr 𝑊 under the equivalence relation 𝜆 ∼ 𝜇 if
and only if 𝐿c (𝜆) and 𝐿c (𝜇) belong to the same block. These equivalence classes are called
the Calogero–Moser c-families of 𝑊 .
These families have an important geometric interpretation. The image of the natural map
Zc (𝑊 )/𝐷(𝑊 )+ · Zc (𝑊 ) → Hc (𝑊 )
is clearly contained in the centre of Hc (𝑊 ). In general it is not equal to the centre of Hc (𝑊 ).
However, it is a consequence of a theorem by Müller, see [13, Corollary 2.7], that the primitive
central idempotents of Hc (𝑊 ), the block idempotents 𝑏𝑖 above, are precisely the images
of the primitive idempotents of Zc (𝑊 )/𝐷(𝑊 )+ · Zc (𝑊 ). This shows that the natural map
Irr 𝑊 → ϒ−1
c (0), 𝜆 ↦→ Supp 𝐿c (𝜆) = 𝜒𝐿c (𝜆) , factors through the Calogero–Moser partition.
−1
Here, ϒc (0) is considered as the set theoretic fibre over the origin 0 of h/𝑊 × h* /𝑊 . In
other words, we have a natural bijection between Ωc (𝑊 ) and ϒ−1
c (0). Now, recall that ϒc
is C* -equivariant. The only C* -fixed point of h/𝑊 × h* /𝑊 is the origin 0 and therefore
C*
ϒ−1
c (0) = Xc (𝑊 ) . Hence, we can identify the Calogero–Moser families Ωc (𝑊 ) with the
C* -fixed closed points of the Calogero–Moser space Xc (𝑊 ).
The next theorem follows from the fact that the Azumaya locus of Hc (𝑊 ) is equal to the
smooth locus of Zc (𝑊 ), which in turn follows from results by Etingof–Ginzburg [16, Theorem
1.7] and Brown (see [25, Lemma 7.2]).
⨁︀𝑘

Theorem 1.4 (Etingof–Ginzburg, Brown). A C* -fixed closed point of Xc (𝑊 ) is smooth if and
only if the corresponding Calogero–Moser family is a singleton, i.e. it consists only of one
irreducible character of 𝑊 .
Example 1.5. Consider the special case c = 0. In this case X0 (𝑊 ) = (h ⊕ h* )/𝑊 . The
quotient morphism h ⊕ h* → (h ⊕ h* )/𝑊 is C* -equivariant and finite, hence X0 (𝑊 ) has only
one C* -fixed closed point, namely the origin. In particular, there is only one Calogero–Moser
family.

§2. Lusztig families
In this section we give a short summary of the other protagonist of this paper—Lusztig’s
families. 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 weight
function 𝐿 on (𝑊, 𝑆), i.e., a function 𝐿 : 𝑊 → R satisfying 𝐿(𝑤𝑤′ ) = 𝐿(𝑤) + 𝐿(𝑤′ ) for
all 𝑤, 𝑤′ ∈ 𝑊 with ℓ(𝑤𝑤′ ) = ℓ(𝑤) + ℓ(𝑤′ ), where ℓ is the length function of (𝑊, 𝑆). Let
𝐴 := Z𝑊 [R] be the group ring of the additive group R over the subring Z𝑊 of C generated by
the values of the irreducible complex characters of 𝑊 . This is an integral domain and we denote
by 𝑞 𝛼 the element of 𝐴 corresponding to 𝛼 ∈ R. Note that 𝑞 𝛼 𝑞 𝛽 = 𝑞 𝛼+𝛽 . Set 𝑞𝑤 := 𝑞 𝐿(𝑤) for
𝑤 ∈ 𝑊 . Let ℋ := ℋ𝐿 (𝑊, 𝑆) be the Hecke algebra of (𝑊, 𝑆) over 𝐴 with respect to 𝐿. This is
the free 𝐴-algebra with basis {𝑇𝑤 | 𝑤 ∈ 𝑊 } whose multiplication is uniquely determined by

G WYN B ELLAMY AND U LRICH T HIEL

7

the relations
{︂

(4)

𝑇𝑠 𝑇𝑤 =

𝑇𝑠𝑤
if ℓ(𝑠𝑤) > ℓ(𝑤)
𝑇𝑠𝑤 + (𝑞𝑠 − 𝑞𝑠−1 )𝑇𝑤 if ℓ(𝑠𝑤) < ℓ(𝑤)

for all 𝑠 ∈ 𝑆 and 𝑤 ∈ 𝑊 . It is a standard fact that the scalar extension ℋ𝐾 of ℋ to the fraction
field 𝐾 of 𝐴 is split semisimple. It is then a consequence of Tits’s deformation theorem that
there is a natural bijection between Irr 𝑊 and Irr ℋ𝐾 . We write 𝐸𝑞𝜆 for the simple ℋ𝐾 -module
corresponding to the simple 𝑊 -module 𝜆 under this bijection. It is also well-known that ℋ is a
symmetric 𝐴-algebra. This implies that the scalar extension ℋ𝐾 is symmetric and so by the
theory in [23, §7] there is a Schur element s𝜆 ∈ 𝐴 attached to every simple module 𝐸𝑞𝜆 . There
is a unique element a𝜆 ∈ R≥0 satisfying 𝑞 2a𝜆 s𝜆 ∈ Z𝑊 [R≥0 ] and 𝑞 2a𝜆 s𝜆 ≡ 𝑓𝜆 mod Z𝑊 [R>0 ]
for some 𝑓𝜆 > 0. This is called Lusztig’s a-invariant of 𝜆. The Schur elements and a-invariants
are known for all Coxeter groups and all weight functions. Note that despite the notation the
Schur elements s𝜆 and the a-invariants a𝜆 depend on 𝐿.
§2B. Truncated induction
Recall that if 𝐼 ⊆ 𝑆 is any subset, then (𝑊𝐼 , 𝐼) is naturally a Coxeter system, where 𝑊𝐼 is the
group generated by 𝐼. This is called a (standard) parabolic subgroup of (𝑊, 𝑆). The restriction
𝐿𝐼 of our weight function 𝐿 to 𝑊𝐼 is a weight function on (𝑊𝐼 , 𝐼). For any simple module 𝜇
of 𝑊𝐼 , Lusztig defined the truncated induction (or j-induction) as
∑︁
(5)
j𝑊
⟨Ind 𝑊
𝑊𝐼 𝜇 :=
𝑊𝐼 𝜇, 𝜆⟩𝜆 ,
𝜆∈Irr 𝑊
a𝜆 = a𝜇

where ⟨Ind 𝑊
𝑊𝐼 𝜇, 𝜆⟩ denotes the multiplicity of 𝜆 in the induction of 𝜇 from 𝑊𝐼 to 𝑊 . Keep in
mind that the a-invariant a𝜇 is computed using the restriction 𝐿𝐼 of 𝐿 to 𝑊𝐼 . It is shown in [19,
Lemma 3.5] that for any 𝜇 ∈ Irr 𝑊 ′ there is a 𝜆 ∈ Irr 𝑊 with a𝜆 = a𝜇 so that the above sum
is never empty. This operation extends to a morphism j𝑊
𝑊𝐼 : K0 (𝑊𝐼 -mod) → K0 (𝑊 -mod) of
𝑊
𝑊
𝐼
Grothendieck groups. It is transitive in the sense that j𝑊𝐼 ∘ j𝑊
𝑊𝐽 = j𝑊𝐽 for 𝐽 ⊆ 𝐼 ⊆ 𝑆.
§2C. Constructible characters and families
Using truncated induction, Lusztig inductively defined the set Con𝐿 (𝑊 ) of 𝐿-constructible representations of 𝑊 as follows: if 𝑊 is trivial, then Con𝐿 (𝑊 ) consists of the unit representation,
𝑊
and otherwise Con𝐿 (𝑊 ) consists of the 𝑊 -modules of the form j𝑊
𝑊𝐼 𝐸 and (j𝑊𝐼 𝐸) ⊗ sgn𝑊
for all proper subsets 𝐼 ( 𝑆 and all 𝐸 ∈ Con𝐿𝐼 (𝑊𝐼 ). Here sgn𝑊 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 ∈ Con𝐿 (𝑊 ) such that ⟨𝐸, 𝜆⟩ =
̸ 0.
The constructible graph is the graph 𝒞𝐿 (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 . The
connected 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 Lus𝐿 (𝑊 ). Lusztig’s families are known for all
finite Coxeter groups (see [33, §22] and also §5 to §8).
Example 2.1. Consider the special case 𝐿 = 0. The map 𝑇𝑤 ↦→ 𝑤 extends to an algebra isomorphism from ℋ0 (𝑊, 𝑆) to the group algebra 𝐴𝑊 which is compatible with the symmetrising
|𝑊 |
traces. Hence, s𝜆 = dim
𝜆 ∈ Z𝑊 [R>0 ] by [23, 7.2.5] and so a𝜆 = 0. This in turn immediately
𝑊
𝑊
shows that j𝑊𝐼 𝜇 = Ind 𝑊𝐼 𝜇 for any parabolic subgroup 𝑊𝐼 of 𝑊 and 𝜇 ∈ Irr 𝑊𝐼 . We then see
that there is only one constructible representation, namely the regular representation of 𝑊 . In
particular, the constructible graph is connected and there is only one Lusztig family.

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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 of
dimension 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 Cherednik
algebra Hc (𝑊 ) and have the notion of Calogero–Moser c-families Ωc (𝑊 ) of Irr 𝑊 . It follows
from Matsumoto’s lemma (see [22, 1.1.5]) that a weight function 𝐿 on 𝑊 is already uniquely
determined by the values on the 𝑊 -conjugacy classes of 𝑆, and that conversely every collection
of elements 𝑐𝑠 ∈ R for 𝑠 ∈ 𝑆 with 𝑐𝑠 = 𝑐𝑡 whenever 𝑐𝑠 and 𝑐𝑡 are conjugate defines
a 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 . We
will thus use both notions interchangeably.
Whenever we write c ≥ 0, resp. c > 0, we mean that c takes values in R≥0 , resp. R>0 .
Similarly, we write 𝐿 ≥ 0, resp. 𝐿 > 0, if 𝐿(𝑠) ≥ 0, resp. 𝐿(𝑠) > 0, for all 𝑠 ∈ 𝑆.
We can twist by linear characters of 𝑊 in order to ensure that we are always in the situation
c ≥ 0. Namely, let 𝛿 : 𝑊 → R× be a linear character. Clearly 𝛿 is uniquely defined by
its values on 𝑆, where it is ±1. Conversely, for any assignment of ±1 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 𝑇𝑤 ↦→ 𝛿(𝑤)𝑇𝑤 defines an algebra isomorphism ℋ𝐿 (𝑊, 𝑆) −→ ℋ𝛿𝐿 (𝑊, 𝑆). Given a
representation 𝜆 of 𝑊 , 𝛿 𝜆 denotes the twist of 𝜆 by 𝛿. It is immediate from the definition of
Lusztig families that 𝜆 and 𝜇 belong to the same 𝐿-family if and only if 𝛿 𝜆 and 𝛿 𝜇 belong
to 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 𝛿c-family. Moreover, a family ℱ is c-cuspidal (see below) if and only if 𝛿 ℱ
is 𝛿c-cuspidal. Therefore, to prove Theorem A, it suffices to make the following assumption, as
in [22]:
We assume that 𝐿 ≥ 0.
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 Ωc (𝑊 ) =
Lusc (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 functions
taking values in Q>0 . Moreover, both in [26] and [9] it was conjectured that Ωc (𝑊 ) coincides
with the partition of Irr 𝑊 into Kazhdan–Lusztig families. Assuming Lusztig’s conjectures P1
to 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 way
by Bonnafé [8] for parameters c > 0) 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 Ω0 (𝑊 ) = Lus0 (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 holds
in many cases.

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9

Theorem 2.4. If 𝑊 is of type 𝐴, 𝐵, 𝐷, 𝐼2 (𝑚), or 𝐻3 , then Ωc (𝑊 ) = Lusc (𝑊 ) for any
c : Ref(𝑊 ) → R.
Except for type 𝐻3 , which follows from [40], the proof of this theorem is also obtained here
from §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 ∈ Lus𝐿 (𝑊 )
is j-induced from a parabolic subgroup 𝑊𝐼 of 𝑊 in the sense that there is a Lusztig family
′
′
ℱ ′ ∈ Lus𝐿𝐼 (𝑊𝐼 ) such that j𝑊
𝑊𝐼 induces a bijection between ℱ and ℱ or between ℱ and
ℱ ⊗ sgn𝑊 . Lusztig called a family cuspidal if it is not j-induced from a proper parabolic
subgroup of 𝑊 . Let Luscusp
𝐿 (𝑊 ) ⊆ Lus𝐿 (𝑊 ) be the set of cuspidal Lusztig families. These
families 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>0 we have Con𝛼𝐿 (𝑊 ) = Con𝐿 (𝑊 ), Lus𝛼𝐿 (𝑊 ) = Lus𝐿 (𝑊 ) and
cusp
Luscusp
𝛼𝐿 (𝑊 ) = Lus𝐿 (𝑊 ).
Proof. As in [22, 1.1.9] one can introduce a universal Hecke algebra ℋ over Z𝑊 [R𝑛 ], where
𝑛 is the number of 𝑊 -conjugacy classes in 𝑆. The Hecke algebra ℋ𝐿 for a particular weight
function 𝐿 : 𝑆 → R is then obtained by specialisation of ℋ. The algebra ℋ admits Schur
elements s𝜆 ∈ Z𝑊 [R𝑛 ] and it follows from the theory in [23, §7] that s𝜆 specialises to the
Schur element s𝜆 of ℋ𝐿 . From this one can deduce that the a-invariant a𝜆 of ℋ𝛼𝐿 is obtained
from the one of ℋ𝐿 by multiplication by 𝛼. This immediately proves the claim.

The key fact (6) implies:
Lemma 2.6. If ℱ = {𝜆} is a Lusztig family such that 𝜆 ∈ Conc (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, the
first author has introduced in [2] the notion of cuspidal Calogero–Moser families. These are also
minimal with respect to a certain condition, but this time they have a geometric interpretation
via the Poisson structure on Calogero–Moser spaces. Despite their name, the two notions of
cuspidality have, a priori, nothing in common. None the less, we will show that they coincide
for all infinite families of Coxeter groups. In this paragraph we will review the foliation of
Calogero–Moser spaces into symplectic leaves and the notion of cuspidal Calogero–Moser
families.
§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 Poisson
bracket 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 Xc (𝑊 ) is a flat Poisson deformation of (h⊕h* )/𝑊 . The Poisson
structure on Xc (𝑊 ) comes from the commutation in the rational Cherednik algebra at 𝑡 ̸= 0 as
follows. Let t be an indeterminate. Clearly, Hc (𝑊 ) = Ht,c (𝑊 )/tHt,c (𝑊 ) and therefore we
can lift elements 𝑧1 , 𝑧2 ∈ Zc (𝑊 ) to elements 𝑧̂︀1 , 𝑧̂︀2 ∈ Ht,c (𝑊 ). Now, define
{𝑧1 , 𝑧2 } := [𝑧̂︀1 , 𝑧̂︀2 ]t=0 ,

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C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

where [𝑧̂︀1 , 𝑧̂︀2 ] is the commutator of 𝑧̂︀1 and 𝑧̂︀2 in Ht,c (𝑊 ) and [𝑧̂︀1 , 𝑧̂︀2 ]t=0 is the projection of
this commutator to Hc (𝑊 ) = Ht,c (𝑊 )/tHt,c (𝑊 ). This is indeed an element in Zc (𝑊 ) and
defines 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 an
ideal 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 of
any prime ideal is a Poisson prime ideal. We denote by PSpec(𝐴) the set of all Poisson prime
ideals of 𝐴 and by PMax(𝐴) the set of all Poisson maximal ideals.
§3B. Symplectic leaves
The (analytification of the) smooth part (Xc (𝑊 ))sm of Xc (𝑊 ) is a Poisson manifold and admits
a foliation into symplectic leaves; that is, a stratification into smooth connected strata such
that the rank of the bracket is maximal along strata. The strata are the symplectic leaves of
the manifold (see [43]). By continuing this process on the complement Xc (𝑊 ) ∖ (Xc )sm we
end up with a decomposition of Xc (𝑊 ) into symplectic leaves. Brown and Gordon [12] have
shown that the leaves obtained in this way are in fact algebraic, i.e., locally closed in the Zariski
topology and finite in number. The leaf of a closed point m of Xc (𝑊 ) consists of all closed
points n ∈ Xc (𝑊 ) such that m and n have the same Poisson core. Furthermore, it is shown
in 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(m𝜒 )) of the Poisson core of its defining
maximal ideal. This shows in particular that the closure of each symplectic leaf is an irreducible
affine Poisson variety.
Lemma 3.1. The set of symplectic leaves of Xc (𝑊 ) is naturally in bijection with the set
PSpec(Zc (𝑊 )) of Poisson prime ideals of Zc (𝑊 ).
Proof. Let ℒ be a symplectic leaf. As we noted above, the closure ℒ is an irreducible affine
variety and therefore the defining ideal pℒ = I(ℒ) is a prime ideal. Moreover, as ℒ = V(𝒫(m𝜒 ))
for any closed point 𝜒 of ℒ, it follows that p𝐿 = 𝒫(m𝜒 ) is a Poisson prime ideal. The map
ℒ ↦→ pℒ is injective since if pℒ = pℒ′ , then ℒ = V(pℒ ) = V(pℒ′ ) = ℒ′ , and this implies
ℒ = ℒ′ as the symplectic leaves form a stratification of Xc (𝑊 ). Now, let p be an arbitrary
Poisson prime ideal of Zc (𝑊 ). Then ℒp := {m ∈ Max(Zc (𝑊 )) | 𝒫(m) = p} is a symplectic
leaf by the description of symplectic leaves due to Brown and Gordon. By construction pℒp = p
and therefore the map ℒ ↦→ pℒ is also surjective.

We immediately obtain the following.
Corollary 3.2. The set of zero-dimensional symplectic leaves of Xc (𝑊 ) is naturally in bijection
with the set PMax(Zc (𝑊 )) of Poisson maximal ideals of Zc (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 parabolic
subgroups 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 𝑊h′ of a subspace h′ of h. By
a theorem of Steinberg [38, Theorem 1.5] the pair (h′ , 𝑊h′ ) is itself a complex reflection group.
Moreover, 𝑊h′ is the stabiliser 𝑊𝑏 of a generic point 𝑏 of h′ . Hence, parabolic subgroups of 𝑊
are in fact the stabilisers of points of h.

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11

Define the rank of a complex reflection group 𝑊 to be the dimension of a faithful reflection
representation 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 = h𝑊 ⊕ (h*𝑊 )⊥ is a decomposition of h as a 𝑊 ′ -module and (h*𝑊 )⊥ is a faithful
′
reflection representation of 𝑊 ′ of minimal rank. Hence, the rank of 𝑊 ′ is dim(h*𝑊 )⊥ . We will
always 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 ′ ) is
a 𝑊 ′ -equivariant function and we understand the rational Cherednik algebra Hc′ (𝑊 ′ ) to be
defined with respect to this reflection representation of 𝑊 ′ .
The group 𝑊 acts on its set of parabolic subgroups by conjugation. Given a parabolic
subgroup 𝑊 ′ the corresponding conjugacy class will be denoted (𝑊 ′ ). We also require the
partial ordering on conjugacy classes of parabolic subgroups of 𝑊 defined by (𝑊1 ) ≥ (𝑊2 ) if
and only if 𝑊1 is conjugate to a subgroup of 𝑊2 . The ordering is chosen in this way so that it
agrees with a geometric ordering to be introduced in the next paragraph.
′
𝑊′
Finally, for a given parabolic subgroup 𝑊 ′ of 𝑊 , we denote by h𝑊
reg the subset of h
consisting of those points whose stabiliser in 𝑊 is equal to 𝑊 ′ . This is a locally closed subset
of h. We denote by Ξ(𝑊 ′ ) the quotient 𝑁𝑊 (𝑊 ′ )/𝑊 ′ , where 𝑁𝑊 (𝑊 ′ ) is the normaliser of
′
𝑊 ′ in 𝑊 . The group Ξ(𝑊 ′ ) acts freely on h𝑊
reg .
Remark 3.3. Suppose that (𝑊, 𝑆) is a Coxeter group. In §2D we already used the standard
parabolic subgroups 𝑊𝐼 of 𝑊 for subsets 𝐼 ⊆ 𝑆. Let h be the (complexified) geometric
representation of 𝑊 so that (h, 𝑊 ) is a complex reflection group. Then 𝑊𝐼 is a parabolic
subgroup 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 standard
parabolic subgroups 𝑊𝐼 .
§3D. Parabolic subgroup attached to a symplectic leaf
′

′

𝑊
If 𝑊 ′ is a parabolic subgroup of 𝑊 then h𝑊
reg /𝑊 denotes the image of hreg in h/𝑊 . The
symplectic leaves of Xc (𝑊 ) are natural labeled by conjugacy classes of parabolic subgroups.

Theorem 3.4. The following holds:
(a) For any symplectic leaf ℒ ⊆ Xc (𝑊 ) there exists a unique conjugacy class 𝑊ℒ := (𝑊 ′ )
′
of parabolic subgroups of 𝑊 such that 𝜋c (ℒ) ∩ h𝑊
reg /𝑊 is dense in 𝜋c (ℒ).
(b) If ℒ, ℒ′ ⊆ 𝑋c (𝑊 ) are symplectic leaves with ℒ ⊆ ℒ′ , then 𝑊ℒ ≤ 𝑊ℒ′ .
Proof. The bijection of Lemma 3.1 is denoted p ↦→ ℒp . The proof of [2, Proposition 4.8]
shows that, for each Poisson prime p, there is a unique conjugacy class (𝑊 ′ ) with 2 dim h =
2 rk(𝑊 ′ ) + dim ℒp such that
′

′

𝑊
dim 𝜋c (ℒp ) ∩ h𝑊
reg /𝑊 = dim hreg /𝑊.
′

′

𝑊
Since 𝜋c (ℒ) is irreducible and dim 𝜋c (ℒ) = dim h𝑊
reg /𝑊 , this implies that 𝜋c (ℒp ) ∩ hreg /𝑊
is dense in 𝜋c (ℒ).


§3E. Cuspidal reduction I
We recall the main results from [2]. For a closed point 𝜒 of Xc (𝑊 ) with defining maximal ideal
m𝜒 of Zc (𝑊 ) we set
Hc,𝜒 (𝑊 ) := Hc (𝑊 )/m𝜒 · Hc (𝑊 ) .
This is a finite-dimensional C-algebra. We call it cuspidal if 𝜒 is a zero-dimensional symplectic
leaf of Xc (𝑊 ).

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C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

Theorem 3.5. Let ℒ be a symplectic leaf of Xc (𝑊 ) of dimension 2𝑙 and 𝜒 a point on ℒ. Then
there exists a parabolic subgroup 𝑊 ′ of 𝑊 of rank dim h − 𝑙 and a cuspidal algebra 𝐻c′ ,𝜒′ (𝑊 ′ )
such that
Hc,𝜒 (𝑊 ) ≃ Mat|𝑊/𝑊 ′ | (Hc′ ,𝜒′ (𝑊 ′ )).
∼

Moreover, there exists a functor Φ𝜒′ ,𝜒 : Hc′ ,𝜒′ (𝑊 ′ )-mod −→ Hc,𝜒 (𝑊 )-mod defining an
equivalence of categories such that
Φ𝜒′ ,𝜒 (𝑀 ) ≃ Ind 𝑊
∀ 𝑀 ∈ Hc′ ,𝜒′ (𝑊 ′ )-mod
𝑊′ 𝑀
as 𝑊 -modules.
Since there are only finitely many zero dimensional leaves in Xc (𝑊 ) the above result shows
that to describe the 𝑊 -module structure of all the simple modules for a particular rational
Cherednik algebra one only needs to describe the 𝑊 ′ -module structure of the cuspidal simple
modules 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 Ωc (𝑊 ) of Calogero–Moser
−1
families and the points in ϒ−1
c (0). If mℱ denotes the point of ϒc (0) corresponding to the
family ℱ, then mℱ lies on a unique symplectic leaf ℒℱ of Xc (𝑊 ). Using Theorem 3.4 we can
attach a unique conjugacy class 𝑊ℱ := 𝑊ℒℱ of parabolic subgroups of 𝑊 to ℱ. We define a
partial ordering ⪯ on the Calogero–Moser families Ωc (𝑊 ) by
ℱ ⪯ ℱ ′ ⇐⇒ ℒ(ℱ) ⊆ ℒ(ℱ ′ ).
Proposition 3.6. The following holds for any ℱ, ℱ ′ ∈ Ωc (𝑊 ):
(a) ℱ ⪯ ℱ ′ and ℱ ′ ⪯ ℱ if and only if ℱ = ℱ ′ .
(b) ℱ ⪯ ℱ ′ implies that 𝑊ℱ ≤ 𝑊ℱ ′ .
Proof. Part (a) follows from directly from the definition of ⪯ and part (b) is a consequence of
Theorem 3.4(b).

We say that a Calogero–Moser family ℱ is cuspidal if ℒℱ is a zero-dimensional leaf. By
Ωcusp
(𝑊 ) we denote the set of cuspidal Calogero–Moser c-families. It follows from Theorem
c
3.4 that PMax(Zc (𝑊 )) ⊆ ϒ−1
c (0). Hence, the set of zero-dimensional symplectic leaves of
cusp
Xc (𝑊 ) is in bijection with Ωc (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 corresponding
point mℱ of Xc (𝑊 ) is smooth. Therefore it is contained in the unique open leaf of Xc (𝑊 ).
Since dim Xc (𝑊 ) > 0, the open leaf is not zero-dimensional and hence the family is not
cuspidal.

The following well-known lemma is analogous to Lemma 2.5.
≃

Lemma 3.8. For any 𝛼 ∈ C× there is a canonical algebra isomorphism Hc (𝑊 ) −→ H𝛼c (𝑊 ),
≃
which induces an algebra isomorphism Hc (𝑊 ) −→ H𝛼c (𝑊 ) and a Poisson isomorphism
≃
Xc (𝑊 ) −→ X𝛼c (𝑊 ). Moreover, Ωc (𝑊 ) = Ω𝛼c (𝑊 ) and Ωcusp
(𝑊 ) = Ωcusp
c
𝛼c (𝑊 ).
We can now state the main theorem of this paper
Theorem A. If 𝑊 is of type 𝐴, 𝐵, 𝐷 or 𝐼2 (𝑚), then for any parameter c ≥ 0 the cuspidal
Lusztig c-families of 𝑊 equal the cuspidal Calogero–Moser c-families of 𝑊 .

G WYN B ELLAMY AND U LRICH T HIEL

13

Proof. The Weyl groups of type 𝐴 are dealt with in §5. For type 𝐵, see Corollary 6.26, and
type 𝐷 is dealt with in Theorem 7.3. Finally, for the dihedral groups 𝐼2 (𝑚), see §8H.

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 using
several theoretical methods we develop in the next section. We will deduce in Lemma 4.11 that
Conjecture B holds for the special case c = 0 for any 𝑊 . Note that because of Lemma 2.5 and
Lemma 3.8 it is sufficient to prove the conjecture only up to multiplication of the parameter by
positive real numbers.

§4. Calculating cuspidal Calogero–Moser families
To determine the cuspidal Calogero–Moser families we develop several theoretical methods—both of representation theoretic and geometric nature. On the one hand, we introduce the
concept of rigid modules here and show that these always lie in a cuspidal family. On the other
hand, we develop a Clifford theory for symplectic leaves. This allows us to deal with Weyl
groups 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 is
the notion of rigid Hc (𝑊 )-modules. We show in Theorem C below that every rigid module
belongs to a cuspidal family. In all examples we consider it turns out that there is at most one
cuspidal family. These two facts allow us to find all cuspidal families.
Definition 4.1. A simple Hc (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, see
e.g. [6]. At 𝑡 = 0, the second author investigated rigid modules in [41]. Recently, they also
played a prominent role in the work [14] of Ciubotaru on Dirac cohomology where they were
called one-𝑊 -type modules. The terminology we adopt comes from the theory of module
varieties, where it is standard. Intuitively, a rigid module is one that cannot be deformed (for
fixed parameter c) to a continuous family of representation; see Lemma 4.9. On the other hand,
if a simple Hc (𝑊 )-module is supported on a symplectic leaf of dimension greater than zero
then one can deform the representation along the leaf. Therefore it is intuitively clear that
rigid modules should be supported at zero dimensional leaves. Showing the precise connection
between rigidity and cuspidality depends on the following theorem.
Theorem 4.2. Let 𝑊 be a complex reflection group. Then no irreducible 𝑊 -module is induced
from a proper parabolic subgroup of 𝑊 , i.e., Ind 𝑊
𝑊 ′ 𝜆 is reducible for all parabolic subgroups
𝑊′ ( 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
𝑛
𝑛
2
𝜒 = 𝑛𝑖=1
𝑛
𝜒
with
𝜒
∈
Irr(𝐺),
then
ℓ(𝜒)
=
𝑖
𝑖
𝑖
𝑖=1 𝑛𝑖 . Note that (𝜒, 𝜒) =
𝑖=1 𝑛𝑖 and
√︀
therefore (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
𝑏𝑃 (𝐺) := min{ℓ(𝜓 𝐺 ) | 𝜓 ∈ Irr(𝑃 )} ,

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C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

𝐺
where 𝜓 𝐺 := Ind 𝐺
𝑃 𝜓. We say that 𝑃 is branching in 𝐺 if 𝑏𝑃 (𝐺) > 1, i.e., 𝜓 is reducible
for all 𝜓 ∈ Irr(𝑃 ). We can now reformulate Theorem 4.2 as saying that all proper parabolic
subgroups 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 −1 = 𝑃 and therefore
𝑃 ∩ 𝑧 𝑃 = 𝑃 . Similarly, we have 𝑧 𝜓 = 𝜓. Hence, (𝜓, 𝑧 𝜓) = (𝜓, 𝜓) = 1 and therefore 𝜓 𝐺 is
not irreducible by [15, 10.25].

Lemma 4.4. Let 𝑁 be a normal subgroup of 𝐺. Let 𝑃 be a subgroup of 𝐺 with branching
index 𝑏𝑃 (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 E 𝐺, see [28, Theorem 19.3], we have
(𝜂 𝐺 , 𝜂 𝐺 ) = (𝜓 𝐺 , 𝜓 𝐺 ) = [𝐼𝐺 (𝜓) : 𝑁 ], where 𝐼𝐺 (𝜓) is the inertia subgroup of 𝜓 in 𝐺. Hence,
ℓ(𝜂 𝐺 ) ≤ [𝐼𝐺 (𝜓) : 𝑁 ]. On the other hand, by Clifford
theory for 𝑁 ∩ 𝑃 E 𝑃 we have
√︀
𝑃
𝑃
𝑃
(𝜂 , 𝜂 )√︀= [𝐼𝑃 (𝜂) : 𝑁 ∩ 𝑃 ]. Hence, ℓ(𝜂 ) ≥ [𝐼𝑃 (𝜂) : 𝑁 ∩ 𝑃 ] and therefore ℓ(𝜂 𝐺 ) ≥
𝑏𝑃 (𝐺) · [𝐼𝑃 (𝜂) : 𝑁 ∩ 𝑃 ]. In total, we must have
[𝐼𝐺 (𝜓) : 𝑁 ]
𝑏𝑃 (𝐺) ≤ √︀
≤ [𝐺 : 𝑁 ] .
[𝐼𝑃 (𝜂) : 𝑁 ∩ 𝑃 ]
Because of our assumption on 𝑏𝑃 (𝐺) this is a contradiction.

Lemma 4.5. Suppose that 𝑁 E 𝐺. Then a subgroup 𝑄 of 𝑁 is branching in 𝑁 if and only if
all its 𝐺-conjugates are branching in 𝑁 .
Proof. This simply follows from the fact that Ind 𝑁𝑔 𝑄 ∘ Con𝑔,𝑄 = Con𝑔,𝑁 ∘Ind 𝑁
𝑄 and that
conjugation Con𝑔,𝑄 with 𝑔 defines a bijection between Irr(𝑄) and Irr(𝑔 𝑄) for all 𝑔 ∈ 𝐺. 
For the proof of Theorem 4.2 we will need the classification of complex reflection groups
due to Shephard and Todd [37], and in particular a description of the parabolic subgroups in the
infinite series 𝐺(𝑚, 𝑚, 𝑛). We quickly recall the definition of these groups. Let 𝑚, 𝑝, 𝑛 ∈ N>0
with 𝑝 dividing 𝑚 and let 𝜁 ∈ C be a primitive 𝑚-th root of unity. Then 𝐺(𝑚, 𝑝, 𝑛) is
the subgroup of GL𝑛 (C) consisting of the generalised permutation matrices with entries in
𝜇𝑚 := ⟨𝜁⟩ such that the product of all non-zero entries is an (𝑚/𝑝)-th root of unity. The
group 𝐺(𝑚, 𝑝, 𝑛) is a normal subgroup of index 𝑝 in 𝐺(𝑚, 1, 𝑛). For a partition 𝜆 of an integer
|𝜆| ≤ 𝑛 let S𝜆 be the corresponding Young subgroup of the symmetric group S|𝜆| . We have
an obvious embedding S𝜆 × 𝐺(𝑚, 𝑚, 𝑛 − |𝜆|) ˓→ 𝐺(𝑚, 𝑚, 𝑛). The following lemma can be
deduced from [39, 3.11].
Lemma 4.6. Up to 𝐺(𝑚, 1, 𝑛)-conjugacy the parabolic subgroups of 𝐺(𝑚, 𝑚, 𝑛) are the standard parabolic subgroups S𝜆 × 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, 1, 𝑛)-conjugates of the above standard parabolic subgroups have to be taken into
account (see [39, 3.11]). For us, however, it is sufficient to know the 𝐺(𝑚, 1, 𝑛)-conjugacy
classes because of Lemma 4.5. By Lemma 4.6 the maximal parabolic subgroups of 𝐺(𝑚, 𝑚, 𝑛)
are up to 𝐺(𝑚, 1, 𝑛)-conjugacy of the form S𝑘 × 𝐺(𝑚, 𝑚, 𝑛 − 𝑘) for 1 ≤ 𝑘 ≤ 𝑛.
Proof of Theorem 4.2. Clearly, we can assume that 𝑊 acts irreducibly on h and that 𝑃 is a
maximal parabolic subgroup. It is well-known (see [29, Corollary 3.24]) that the centre 𝑍(𝑊 )
of 𝑊 is a cyclic group Zℓ = ⟨𝜎⟩. If 𝜎 ̸= 1, then 𝜎 fixes only the origin and so 𝜎 ∈
/ 𝑊 ′ for any
proper parabolic subgroup of 𝑊 . Hence, if |𝑍(𝑊 )| > 1, then the claim holds by Lemma 4.3.
The classification of irreducible complex reflection groups shows that |𝑍(𝑊 )| = 1 implies

G WYN B ELLAMY AND U LRICH T HIEL

15

that 𝑊 ≃ 𝐺(𝑚, 𝑚, 𝑛) for some 𝑚, 𝑛. By Lemma 4.6 and Lemma 4.5 we can assume that
𝑃 = S𝑘 × 𝐺(𝑚, 𝑚, 𝑛 − 𝑘), where 1 ≤ 𝑘 ≤ 𝑛. Let 𝜆 ∈ Irr(𝑃 ).
We assume first that 𝑚 > 1. The module 𝜋𝜆 is isomorphic to 𝜋𝜆′  𝜋𝜇 for some 𝜋𝜆′ ∈ Irr(S𝑘 )
and 𝜋𝜇 ∈ Irr(𝐺(𝑚, 𝑚, 𝑛 − 𝑘)). Note that 𝑃 ⊂ 𝐺(𝑚, 𝑚, 𝑘) × 𝐺(𝑚, 𝑚, 𝑛 − 𝑘) ⊂ 𝑊 .
𝐺(𝑚,𝑚,𝑘) ′
𝜋𝜆 is not irreducible. That is, we may
If 𝑘 > 1 then it suffices to show that Ind S𝑘
assume 𝑘 = 𝑛. The symmetric group S𝑛 is a quotient of 𝐺(𝑚, 𝑚, 𝑛), the morphism given by
′
sending an element to the underlying permutation. Then
[︁ we may consider 𝜋]︁𝜆 as an irreducible
𝐺(𝑚,𝑚,𝑛)
𝐺(𝑚, 𝑚, 𝑛)-module 𝜋𝜆′′ . Clearly 𝜋𝜆′′ |S𝑛 = 𝜋𝜆′ . Hence Ind S𝑛
𝜋𝜆′ : 𝜋𝜆′′ ≥ 1. On the other
𝐺(𝑚,𝑚,𝑛)

hand, dim Ind S𝑛
𝜋𝜆′ = 𝑚𝑛−1 dim 𝜋𝜆′ . Hence it is not irreducible.
In the case 𝑘 = 1, we have 𝑃 = 𝐺(𝑚, 𝑚, 𝑛−1) ⊂ 𝐺(𝑚, 𝑚, 𝑛). Let 𝑄 := 𝐺(𝑚, 1, 𝑛−1) ⊆
𝐺(𝑚, 1, 𝑛) and note that 𝑃 = 𝑄 ∩ 𝐺(𝑚, 𝑚, 𝑛). If we can show that 𝑏𝑄 (𝐺(𝑚, 1, 𝑛)) ≥ 𝑚 + 1,
then Lemma 4.4 shows that 𝑃 is branching in 𝐺(𝑚, 𝑚, 𝑛). But this follows from the branching
rule ([36, Theorem 10]) which shows that, when viewing 𝜆 as an 𝑚-multipartition, we have at
𝐺(𝑚,1,𝑛)
least 𝑚 + 1 constituents in Ind 𝐺(𝑚,1,𝑛−1) 𝜋𝜆 obtained by adding boxes to 𝜆.
Finally, we need to deal with the case 𝑚 = 1, i.e. 𝑊 = S𝑛 . In this case we have 𝑃 =
∑︀ 𝜈
𝜈
𝑛
S𝑘 × S𝑛−𝑘 and it is known that Ind S
S𝑘 ×S𝑛−𝑘 𝜋𝜆  𝜋𝜇 =∑︀ 𝜈 𝑐𝜆,𝜇 𝜋𝜈 , where 𝑐𝜆,𝜇 are the
Littlewood-Richardson coefficients. We need to show that 𝜈 𝑐𝜈𝜆,𝜇 > 1. Presumably, this is
well-known. We will deduce it from the fact that for the Weyl group of Type 𝐵𝑛 we have
(7)

𝑛
Ind 𝐵
𝐵𝑘 ×𝐵𝑛−𝑘 𝜋(𝜆(1) ,𝜆(2) )  𝜋(𝜇(1) ,𝜇(2) ) =

∑︁

(1)

(2)

𝑐𝜈𝜆(1) ,𝜇(1) 𝑐𝜈𝜆(2) ,𝜇(2) 𝜋(𝜈 (1) ,𝜈 (2) ) .

(𝜈 (1) ,𝜈 (2) )

Take 𝜆(1) = 𝜆, 𝜆(2) = ∅, 𝜇(1) = 𝜇 and 𝜇(2) = ∅. Then (7) implies that it suffices to show
𝑛
that Ind 𝐵
𝐵𝑘 ×𝐵𝑛−𝑘 𝜋(𝜆(1) ,𝜆(2) )  𝜋(𝜇(1) ,𝜇(2) ) is not an irreducible 𝐵𝑛 -module. But 𝐵𝑛 contains a
non-trivial central element that does not belong to either 𝐵𝑛−𝑘 or 𝐵𝑘 . This implies by Lemma
4.3 that the induced module is not irreducible.

Proposition 4.7. If 𝐿 is rigid, then 𝐿 ≃ 𝐿c (𝜆) is a Hc (𝑊 )-module (isomorphic to 𝜆 as a
𝑊 -module), for some 𝜆 ∈ Irr 𝑊 .
Proof. If 𝐿 is not a simple Hc (𝑊 )-module, then either the set-theoretic support of 𝐿 as a
C[h]-module is not contained in {0}, or the set-theoretic support of 𝐿 as a C[h* ]-module is
not contained in {0}. Without loss of generality, we assume that the set-theoretic support of
𝐿 as a C[h]-module is not contained in {0}. Thus, there exists some 𝑏 ̸= 0 in h such that
m𝑏 · 𝐿 ̸= 0, where m𝑏 is the maximal ideal defining 𝑏 in h. The stabiliser 𝑊𝑏 of 𝑏 is a proper
subgroup of 𝑊 . Thus, the Bezrukavnikov–Etingof isomorphism, see [2, Theorem 4.3], implies
′
′
that 𝐿 ≃ Ind 𝑊
𝑊𝑏 𝐿 for some Hc′ (𝑊𝑏 )-module 𝐿 . By Theorem 4.2, this implies that 𝐿 is not
rigid. Thus, 𝐿 is a simple Hc (𝑊 )-module. The simple Hc (𝑊 )-modules are of the form 𝐿c (𝜆)
and 𝜆 always appears in the restriction to 𝑊 of 𝐿c (𝜆) with non-zero multiplicity. The result
follows.

Remark 4.8. Proposition 4.7 implies that if 𝐿 is a rigid module then h · 𝐿 = 0 = h* · 𝐿. In
particular, 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 finitedimensional representation theory to say that a simple module 𝐿 of dimension 𝑑 for a finitedimensional algebra 𝐴 is rigid if the set of points 𝑀 in the representation scheme Rep𝑑 (𝐴)
satisfying 𝑀 ≃ 𝐿 is open.
Lemma 4.9. Let 𝐿 be a rigid Hc (𝑊 )-module and set 𝑑 := dim 𝐿. Let Rep𝑑 (Hc (𝑊 )) be the
scheme parameterizing all 𝑑-dimensional representations of Hc (𝑊 ). Let 𝑋 be the set of points
𝑀 in Rep𝑑 (Hc (𝑊 )) such that 𝑀 ≃ 𝐿. Then 𝑋 is a connected component of Rep𝑑 (Hc (𝑊 )).

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C 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 Hc (𝑊 )-modules
over 𝑆 such that the fiber ℱ𝑠0 is isomorphic to 𝐿, for some 𝑠0 ∈ 𝑆. Then it suffices to prove
that h and h* act identically by zero on ℱ. Since C[𝑆] is a domain, its radical is zero, and hence
it suffices to show that h and h* act as zero on every fiber ℱ𝑠 of ℱ for 𝑠 ∈ MaxSpec(C[𝑆]).
We may consider ℱ as a flat family of C[h] o 𝑊 -modules instead and prove the claim in
this setting (repeating the argument for C[h* ] o 𝑊 ). Then the claim is a consequence of
Theorem 4.2 together with the (easy) classification of simple C[h] o 𝑊 -modules. Firstly,
since 𝑆 is connected ℱ𝑠 ≃ 𝐿 as a 𝑊 -module for all 𝑠. This is well-known and follows
for instance from [18, Corollary 1.4]. Therefore, it suffices to show that if 𝑀 is any simple
C[h] o 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
𝑀𝜒 = {𝑚 ∈ 𝑀 | 𝑥 · 𝑚 = 𝜒(𝑥)𝑚, ∀𝑥 ∈ h* } is non-zero. Let 𝑊 ′ ( 𝑊 be the stabilizer of
C[h]o𝑊
𝜒 ∈ h. Since 𝑀 is simple, 𝑀𝜒 is a simple 𝑊 ′ -module and 𝑀 ≃ Ind C[h]o𝑊 ′ 𝑀𝜒 . By Theorem
4.2, 𝑀 is not irreducible. Hence 𝑀 ̸≃ 𝐿 as required.
To deduce the statement of the lemma, take 𝑆 to be any irreducible component (with reduced
scheme structure) of Rep𝑑 (Hc (𝑊 )) containing 𝐿.

Notice that Lemma 4.9 shows that the set of all rigid modules in Rep𝑑 (Hc (𝑊 )) is open. In
general, the connected component 𝑋 has a very non-trivial scheme structure. This can be seen
from Voigt’s Lemma [18] which implies that
(8)
dim 𝑋 − dim 𝑋red = dim Ext1Hc (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∙Hc (𝑊 ) (𝐿, 𝐿) ≃ ∧∙ 𝑉 ⊗𝑊 EndC (𝐿) ,
where 𝑉 = h ⊕ h* . In particular, it is easy to construct examples of rigid modules where
the right hand side of (8) is strictly positive. Also, the variety Rep𝑑 (Hc (𝑊 )) can have many
connected components. This can be seen, for instance, by considering the case c = 0.
Via the bijection Irr 𝑊 → Irr Hc (𝑊 ) given by Proposition 1.3, the element 𝜆 ∈ Irr 𝑊 is
said to be c-rigid if 𝐿c (𝜆) is a rigid Hc (𝑊 )-module. The following is the main theorem of this
section.
Theorem C. Let 𝑊 be a complex reflection group. If 𝜆 ∈ Irr 𝑊 is c-rigid, then 𝜆 lies in a
cuspidal Calogero–Moser c-family.
Proof. Let ℱ be the Calogero–Moser c-family of 𝐿c (𝜆) and let 𝜒 be the corresponding
point of Xc (𝑊 ). Suppose that ℱ is not cuspidal. Then by Theorem 3.5 there is a parabolic subgroup 𝑊 ′ of 𝑊 , a cuspidal symplectic leaf 𝜒′ of Xc′ (𝑊 ′ ), and an equivalence
∼
Φ𝜒′ ,𝜒 : Hc′ ,𝜒′ (𝑊 ′ )-mod −→ Hc,𝜒 (𝑊 )-mod such that Φ𝜒′ ,𝜒 (𝑀 ) ≃ Ind 𝑊
𝑊 ′ 𝑀 as 𝑊 -modules
′
′
′
′
for all 𝑀 ∈ Hc ,𝜒 (𝑊 )-mod. In particular, there must exist a 𝑊 -module 𝑀 with Ind 𝑊
𝑊′𝑀 ≃
𝐿c (𝜆) ≃ 𝜆. But this is not possible by Theorem 4.2.

Of course, the major advantage of rigid modules is that they are easily detected.
Lemma 4.10. Let 𝜆 : 𝐺 → GL𝑟 (C) be an irreducible representation of 𝑊 . Then 𝐿c (𝜆) is a
rigid module for Hc (𝑊 ) if and only if
∑︁

(9)

c(𝑠)(𝑦, 𝛼𝑠 )(𝛼𝑠∨ , 𝑥)𝜆(𝑠) = 0

𝑠∈Ref(𝑊 )

for all 𝑦 ∈ h and 𝑥 ∈

h* .

Proof. The module 𝐿c (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 𝐿c (𝜆). Hence, 𝐿c (𝜆) is rigid if and only if the

G WYN B ELLAMY AND U LRICH T HIEL

17

̂︀ : 𝑇 (h ⊕ h* ) o 𝑊 → Mat𝑟 (C) with h and h* acting trivially and 𝑊 acting by
representation 𝜆
̂︀
𝜆 descends to Hc (𝑊 ). This is the case if and only if 𝜆([𝑦,
𝑥]) = 0, and this is equivalent to the
asserted equation.


(𝑊 ) = Luscusp
(𝑊 ), i.e. Conjecture B holds for c = 0.
Lemma 4.11. For any 𝑊 we have Ωcusp
0
0
Proof. Recall from Lemma 2.3 that Ω0 (𝑊 ) = Lus0 (𝑊 ) = {Irr 𝑊 }. Furthermore, recall from
𝑊
Example 2.1 that truncated induction is for c = 0 just usual induction, i.e. j𝑊
𝑊𝐼 = Ind 𝑊𝐼 for
a parabolic subgroup 𝑊𝐼 of 𝑊 . Now, if the unique Lusztig family were not cuspidal, then
the 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 the
other hand, all 𝜆 ∈ Irr 𝑊 are rigid for c = 0 by Lemma 4.10. Hence, each 𝜆 ∈ Irr 𝑊 lies in a
cuspidal Calogero–Moser family by Theorem C. As there is just one Calogero–Moser family,
this one is cuspidal.

Remark 4.12. Ciubotaru [14] has recently classified the rigid Hc (𝑊 )-modules for all Weyl
groups and all parameters. We will independently obtain this classification for non-exceptional
Coxeter groups from sections §5 to §8. Ciubotaru furthermore shows for all Weyl groups at
equal parameters—except 𝐸7 —that the rigid modules always lie in a single Calogero–Moser
family, and that this family contains the (unique) cuspidal Lusztig family; for 𝐹4 and 𝐸6 this is
in fact an equality. Using Theorem C, this shows that one direction of Conjecture B also holds
for 𝐹4 and 𝐸6 for equal parameters. However, a classification of the cuspidal symplectic leaves
is 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 ′ ) (Zc (𝑊 )) the
subset of PSpec(Zc (𝑊 )) of Poisson prime ideals p with 𝑊ℒp = (𝑊 ′ ). This set might be empty.
If 𝑊 ⊆ 𝐺 ⊆ 𝑁GL(h) (𝑊 ) is a finite subgroup such that c : Ref(𝑊 ) → C is 𝐺-invariant, then
𝐺 acts on Hc (𝑊 ) by algebra automorphisms. This induces an action of 𝐺 by Poisson algebra
automorphisms on Zc (𝑊 ) and, since 𝑊 acts trivially, this action factors through 𝐺/𝑊 . Applied to a parabolic subgroup 𝑊 ′ of 𝑊 and an arbitrary 𝑊 -invariant function c : Ref(𝑊 ) → C
this shows that Ξ(𝑊 ′ ) acts on Zc′ (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(Zc′ (𝑊 ′ )) such that there is a bijection
1:1

PSpec(𝑊 ′ ) (Zc (𝑊 )) ←→ PMax(Zc′ (𝑊 ′ ))/Ξ(𝑊 ′ ) .
Losev considers in [31] a different completion of the rational Cherednik algebra than the one
used in [2] (which is based on a construction by Bezrukavnikov and Etingof). Therefore we will
now show that Theorem 4.13 still holds in the context of Bezrukavnikov–Etingof completions.
Fix a parabolic subgroup 𝑊 ′ of 𝑊 and let 𝑁 := 𝑁𝑊 (𝑊 ′ ). Let 𝑈 be an affine open subset
′
of h/𝑊 such that 𝑈 ∩ h𝑊
reg /𝑊 is closed, but non-empty, in 𝑈 . Let 𝑉 denote the preimage of
′
′
𝑈 in h. Then 𝑉 is 𝑊 -stable and 𝑉𝑊 ′ = h𝑊
reg ∩ 𝑉 is closed in 𝑉 . Let k denote the 𝑊 -module
′
complement to h𝑊 in h. It is an 𝑁 -module.
′
Let 𝐴 := C[𝑈 ] and set 𝑍 := 𝐴 ⊗C[h]𝑊 Zc (𝑊 ). The prime ideal of 𝐴 defining 𝑈 ∩ h𝑊
reg /𝑊
̂︀ c (𝑊 ) := Spec(𝐴
̂︀q ⊗𝐴 𝑍).
is denoted q. Let 𝐴̂︀q be the completion of 𝐴 along q and set X

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C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS
′

̂︀ c (𝑊 ) should be thought of as the formal neighbourhood of 𝜋 −1 (h𝑊 /𝑊 )
Morally speaking, X
reg
in Xc (𝑊 ). However, since 𝑍 is not a finite 𝐴-module, this is not strictly true.
Let 𝐴′ = C[k/𝑊 ′ × 𝑉𝑊 ′ ] and q′ the prime ideal defining {0} × 𝑉𝑊 ′ in k/𝑊 ′ × 𝑉𝑊 ′ . Then
(︁

)︁

̂︀ ′ (𝑊 ′ , 𝑉 ) := Spec 𝐴
̂︀′ ′ ⊗ ′ Z ′ (𝑊 ′ ) ⊗ C[𝑇 * 𝑉 ′ ] ,
X
c
𝐴
c
𝑊
q
̂︀ ′ (𝑊 ′ , 𝑉 ). The
where 𝑇 * 𝑉𝑊 ′ is the cotangent bundle of 𝑉𝑊 ′ . The group Ξ(𝑊 ′ ) acts on X
c
following is an analogue of the isomorphism Θ in section 3.7 of [7]; a complete proof is given
in [5].

Theorem 4.14. There is an isomorphism of affine Poisson varieties
∼ ̂︀
̂︀ c (𝑊 ) −→
Φ:X
Xc′ (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 ↦→ 𝐴̂︀q ⊗𝐴 p defines a bijection between PSpec(𝑊 ′ ) (Xc (𝑊 )) and the
set of Poisson prime ideals of 𝐴̂︀q ⊗𝐴 𝑍 of height 2 dim k.
′

Proof. First, we must show that 𝐴̂︀q ⊗𝐴 p is prime in 𝐴̂︀q ⊗𝐴 𝑍. Let 𝑌 = 𝑈 ∩h𝑊
reg /𝑊 and denote
′
′
by 𝜋(ℒp ) the closure of 𝜋(ℒp ) ∩ 𝑈 in 𝑈 . Recall that 𝜋(ℒp ) ∩ h𝑊
/𝑊
is
dense in h𝑊
reg
reg /𝑊 .
This implies that 𝜋(ℒp ) ∩ 𝑌 is dense in the closed, irreducible set 𝑌 , i.e. 𝜋(ℒ) ∩ 𝑌 = 𝑌 .
Similarly, 𝜋(ℒp ) ∩ 𝑌 is dense in 𝜋(ℒp ). Thus, 𝜋(ℒp ) = 𝑌 . This implies that q ⊂ p ∩ 𝐴 and
hence 𝑍 · q ⊂ p. Since 𝐴̂︀q is flat over 𝐴, we have a short exact sequence
0 → 𝐴̂︀q ⊗𝐴 p → 𝐴̂︀q ⊗𝐴 𝑍 → 𝐴̂︀q ⊗𝐴 (𝑍/p) → 0.
The order filtration on Hc (𝑊 ) defines an increasing filtration ℱ𝑖 𝑍 on 𝑍 such that each piece
is a coherent 𝐴-module. This restricts to a filtration on p and we have a short exact sequence
0 → ℱ𝑖 p → ℱ𝑖 𝑍 → ℱ𝑖 𝑍/ℱ𝑖 p → 0 of coherent 𝐴-modules. Since tensor products commute
with colimits,
𝐴̂︀q ⊗𝐴 (𝑍/p) = lim 𝐴̂︀q ⊗𝐴 (ℱ𝑖 𝑍/ℱ𝑖 p).
𝑖→∞

But 𝑍 · q ⊂ p implies that
𝐴̂︀q ⊗𝐴 (ℱ𝑖 𝑍/ℱ𝑖 p) = lim

∞←𝑚

(ℱ𝑖 𝑍/ℱ𝑖 p)
𝑚
q (ℱ𝑖 𝑍/ℱ𝑖 p)

= ℱ𝑖 𝑍/ℱ𝑖 p.

Thus, 𝐴̂︀q ⊗𝐴 (𝑍/p) = 𝑍/p is a domain and 𝐴̂︀q ⊗𝐴 p is prime. It is clearly Poisson; see [2,
Lemma 3.5]. Moreover, the fact that 𝐴̂︀q ⊗𝐴 (𝑍/p) = 𝑍/p shows that 𝐴̂︀q ⊗𝐴 p1 = 𝐴̂︀q ⊗𝐴 p2
if and only if p1 = p2 . Lemma 3.3 of loc. cit. says that ht(𝐴̂︀q ⊗𝐴 p) = ht(p), which equals
2rk(𝑊 ′ ). Thus, the map we have written down is injective.
On the other hand, if p′ is a Poisson prime in 𝐴̂︀q ⊗𝐴 𝑍 of height 2 dim k, then Lemmata 3.3
and 3.5 of loc. cit. say that p := p′ ∩ 𝑍 is a Poisson prime of height 2 dim k. Therefore, we just
need to show that 𝑌 ∩ 𝜋(ℒp ) is dense in 𝑌 . The prime p belongs to PSpec(𝑊 ′′ ) (Xc (𝑊 )) for
′′
some parabolic 𝑊 ′′ of 𝑊 of the same rank as 𝑊 ′ . The sets 𝑈 ∩ h𝑊
reg /𝑊 and 𝑌 are disjoint
′′
if 𝑊 ′′ ∈
/ (𝑊 ′ ), which implies that the image in 𝐴̂︀q of the ideal defining 𝑈 ∩ h𝑊
reg /𝑊 is the
whole of 𝐴̂︀q . Therefore the image of p ∩ 𝐴 in 𝐴̂︀q would also be the whole of 𝐴̂︀q if 𝑊 ′′ ∈
/ (𝑊 ′ ).
But since 𝐴̂︀q ⊗𝐴 p is contained in p′ , this cannot happen and thus 𝑊 ′′ ∈ (𝑊 ′ ) as required. 
Proof of Theorem 4.13. Since the symplectic structure on 𝑇 * 𝑉𝑊 ′ is non-degenerate, the only
Poisson prime in C[𝑇 * 𝑉𝑊 ′ ] is the zero ideal. Therefore, every Poisson prime in Zc′ (𝑊 ′ ) ⊗
C[𝑇 * 𝑉𝑊 ′ ] has height at most 2 dim k and the Poisson primes of height 2 dim k are in bijection
with the Poisson maximal ideals of Zc′ (𝑊 ′ ). Repeating the arguments of Lemma 4.15, there

G WYN B ELLAMY AND U LRICH T HIEL

19

′
*
is a bijection between the Poisson primes
in Zc′ (𝑊
[︁
]︁ ) ⊗ C[𝑇 𝑉𝑊 ′ ] of height 2 dim k and the
̂︀ ′ (𝑊 ′ , 𝑉 ) .
Poisson primes of height 2 dim k in C X
c
By Lemma 4.15 and Theorem 4.14, the set PSpec(𝑊 ′ ) (Xc (𝑊 )) is in bijection with the sym̂︀ ′ (𝑊 ′ , 𝑉 )/ Ξ(𝑊 ′ ) of dimension 2(dim h−dim k). Since Ξ(𝑊 ′ ) acts freely on
plectic leaves in X
c
̂︀ ′ (𝑊 ′ , 𝑉 ). Therefore the symplectic leaves in X
̂︀ ′ (𝑊 ′ , 𝑉 )/ Ξ(𝑊 ′ )
𝑉𝑊 ′ it also acts freely on X
c
c
′
of dimension 2(dim h − dim k) are in bijection with the Ξ(𝑊 )-orbits of symplectic leaves in
̂︀ ′ (𝑊 ′ , 𝑉 ) of dimension 2(dim h − dim k). But, as explained above, this is the same as the
X
c
Ξ(𝑊 ′ )-orbits of Poisson maximal ideals Zc′ (𝑊 ′ ).


§4C. Clifford theory
Throughout this section we fix an irreducible complex reflection group (h, 𝑊 ). Moreover, we
assume that there exists a normal subgroup 𝐾 C 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 is
normal 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 ) r 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 algebra
embedding Hc (𝐾) ˓→ Hc (𝑊 ), which is the identity on h and h* . Let Γ = 𝑊/𝐾. As explained
in [4, Section 4.1], the group 𝑊 acts on Hc (𝐾) by conjugation. Thus, it acts on Zc (𝐾). This
action factors through Γ.
We will require the following lemma.
Lemma 4.16. Under the graded
𝑊 -module identification Hc (𝑊 ) = 𝑤∈𝑊 C[h] ⊗ C[h* ] ⊗ 𝑤,
∑︀
every non-zero element 𝑧 = 𝑤∈𝑊 𝑧𝑤 · 𝑤 ∈ Zc (𝑊 ) satisfies 𝑧1 ̸= 0.
⨁︀

Proof. A reformulation of the PBW property is that, under the filtration ℱ𝑖 Hc (𝑊 ) putting h
and h* in degree one, 𝑊 in degree zero and ℱ−1 := 0, the associated graded grℱ Hc (𝑊 ) equals
C[h ⊕ h* ] o 𝑊 . An easy induction on 𝑘 shows that ℱ𝑘 Hc (𝑊 ) = (C[h] ⊗ C[h* ])≤𝑘 ⊗ C𝑊 as a
𝑊 -module, where (C[h] ⊗ C[h* ])≤𝑘 is the sum of all graded pieces of degree at most 𝑘. Then
the short exact sequences 0 → ℱ𝑘−1 → ℱ𝑘 → ℱ𝑘 /ℱ𝑘−1 → 0 can be identified, as short exact
sequences of 𝑊 -modules, with
0 → (C[h] ⊗ C[h* ])≤𝑘−1 ⊗ C𝑊 → (C[h] ⊗ C[h* ])≤𝑘 ⊗ C𝑊 → (C[h] ⊗ C[h* ])𝑘 ⊗ C𝑊 → 0.
∑︀
The image of Zc (𝑊 ) under grℱ equals C[h×h* ]𝑊 . Therefore, 𝑧 = 𝑤∈𝑊 𝑧𝑤 ·𝑤 ∈ ℱ𝑘 rℱ𝑘−1
then its (non-zero!) image in (C[h] ⊗ C[h* ])𝑘 ⊗ C𝑊 belongs to (C[h] ⊗ C[h* ])𝑊
𝑘 . In particular,
𝑧1 ̸= 0.

Proposition 4.17. The centre Zc (𝑊 ) of Hc (𝑊 ) equals the subalgebra Zc (𝐾)Γ of Zc (𝐾).
Moreover, the embedding Zc (𝑊 ) ˓→ Zc (𝐾) is as Poisson algebras.
Proof. Clearly, Zc (𝑊 ) ∩ Zc (𝐾) ⊆ Zc (𝐾)𝑊 . Therefore, we just need to show that Zc (𝑊 ) ⊂
⨁︀
Zc (𝐾). Fix coset representatives 1 = 𝑤1 , . . . , 𝑤ℓ of 𝐾 in 𝑊 . Then Hc (𝑊 ) = ℓ𝑖=1 Hc (𝐾)𝑤𝑖
∑︀ℓ
as a left Hc (𝐾)-module. Let 𝑧 = 𝑖=1 𝑧𝑖 𝑤𝑖 denote an element in Zc (𝑊 ) with 𝑧𝑖 ∈ Hc (𝐾) for
all 𝑖. We wish to show that 𝑧𝑖 = 0 for 𝑖 ̸= 1. Let 𝑓 ∈ Hc (𝐾). Then
[𝑓, 𝑧] =

ℓ
∑︁

([𝑓, 𝑧𝑖 ] + 𝑧𝑖 (𝑓 − 𝑤𝑖 (𝑓 ))) 𝑤𝑖 .

𝑖=1

Since [𝑓, 𝑧𝑖 ] + 𝑧𝑖 (𝑓 − 𝑤𝑖 (𝑓 )) ∈ Hc (𝐾) for all 𝑖, we must have [𝑓, 𝑧𝑖 ] + 𝑧𝑖 (𝑓 − 𝑤𝑖 (𝑓 )) = 0. In
particular, this implies that 𝑧1 ∈ Zc (𝐾) ∩ Zc (𝑊 ). Without loss of generality, 𝑧1 = 0. But now
it follows from Lemma 4.16 that 𝑧 = 0. Thus, Zc (𝑊 ) = Zc (𝐾)𝑊 .

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C 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 the
construction or simply by noting that the bracket is Γ-invariant and hence restricts to Zc (𝐾)Γ .

Thus, geometrically we have a Poisson morphism 𝜂 : Xc (𝐾) → Xc (𝑊 ) identifying Xc (𝑊 )
with Xc (𝐾)/Γ. It is a finite, surjective map which is generically a Γ-covering. This fits into a
commutative diagram
Xc (𝐾)
ϒc,𝐾

h/𝐾 × h* /𝐾

𝜂

Xc (𝑊 )
ϒc,𝑊

h/𝑊 × h* /𝑊

Lemma 4.18. If ℒ is a leaf of Xc (𝐾), then 𝜂(ℒ) is a finite union of leaves of Xc (𝑊 ).
Proof. Since the stratification of Xc (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, we
may assume that ℒ is closed in Xc (𝐾). Then 𝜂(ℒ) is closed. It is invariant under Hamiltonian
flows if and only if the semi-prime ideal 𝐼(𝜂(ℒ)) is Poisson. But 𝐼(𝜂(ℒ)) = 𝐼(ℒ) ∩ Zc (𝐾)Γ .
Since 𝐼(ℒ) is Poisson and the bracket is invariant under Γ, if 𝑧 ∈ 𝐼(𝜂(ℒ)) ∩ Zc (𝐾)Γ and
ℎ ∈ Zc (𝐾)Γ , then {𝑧, ℎ} ∈ 𝐼(𝜂(ℒ)) ∩ Zc (𝐾)Γ , as required.

Note that, in general, the preimage of a leaf of Xc (𝑊 ) is not a leaf. Let Xc (𝐾)sing be the
singular locus of Xc (𝐾), let Xc (𝐾)sm be the smooth locus and let Xc (𝐾)free be the locus
where Γ acts freely. The following is the geometric counterpart of [4, Lemma 4.12].
Proposition 4.19. The preimage 𝜂 −1 (Xc (𝑊 )sm ) equals Xc (𝐾)sm ∩ Xc (𝐾)free .
Proof. Since Γ preserves the Poisson structure on Xc (𝐾), for each 𝑝 ∈ Xc (𝐾)sm , the group
Γ𝑝 acts symplectically on the tangent space 𝑇𝑝 Xc (𝐾)sm . Thus, (𝑇𝑝 Xc (𝐾)sm )/Γ𝑝 is smooth
if and only if Γ𝑝 = 1. Using the fact that one can linearize the action of a finite group in the
formal neighborhood of any fixed point, this implies that the smooth locus of Xc (𝐾)sm /Γ
equals (Xc (𝐾)sm ∩ Xc (𝐾)free )/Γ. Hence
𝜂 −1 (Xc (𝑊 )sm ) ∩ Xc (𝐾)sm = Xc (𝐾)sm ∩ Xc (𝐾)free .
On the other hand, Xc (𝐾)sing is a union of symplectic leaves ℒ with dim ℒ < dim Xc (𝐾).
Therefore Lemma 4.18 implies that 𝜂(Xc (𝐾)sing ) ⊂ Xc (𝑊 )sing .

The following was stated in [4] in the case Γ is a cyclic group. We give a simple geometric
proof.
Theorem 4.20. Let c : Ref(𝐾) → C be 𝑊 -equivariant.
(a) The group Γ acts on Ωc (𝐾) such that 𝜎 ℱ = {𝜎 𝜆 | 𝜆 ∈ ℱ} for 𝜎 ∈ Γ and ℱ ∈ Ωc (𝐾).
(b) There is a natural bijection between Ωc (𝑊 ) and Ωc (𝐾)/Γ given by
Ωc (𝑊 ) ∋ ℱ ←→ {𝜆 ∈ Irr(𝐾) | 𝜆 ⊂ Res 𝑊
𝐾 𝜇 for some 𝜇 ∈ ℱ } ∈ Ωc (𝐾)/Γ.
Proof. Recall the notation from §1C. We will use the notation and results from [4, §3, §4].
Let Zc (𝑊 ) denote the quotient of Zc (𝑊 ) by the ideal generated by 𝐷(𝑊 )+ , Zc (𝐾) the
̃︀ c (𝐾) the quotient of Zc (𝐾) by
quotient of Zc (𝐾) by the ideal generated by 𝐷(𝐾)+ and Z
̃︀
the ideal generated by 𝐷(𝑊 )+ . We also let Hc (𝐾) denote the quotient of Hc (𝐾) by the ideal
generated by 𝐷(𝑊 )+ . The Satake isomorphism [16, Theorem 3.1] implies that the natural map
̃︀ c (𝐾) is an embedding. The group Γ acts on Z
̃︀ c (𝐾) and Proposition 4.17 now
Zc (𝑊 ) → Z
̃︀ c (𝐾)Γ . Thus,
implies that Zc (𝑊 ) = Z
̃︀ c (𝐾)Γ ˓→ Z
̃︀ c (𝐾)  Zc (𝐾).
Zc (𝑊 ) = Z

G WYN B ELLAMY AND U LRICH T HIEL

21

̃︀ c (𝐾)  Zc (𝐾) is nilpotent. Therefore it identifies the primitive
The kernel of the surjection Z
idempotents in both algebras.
Let
{𝑑𝑖 }𝑖∈Ωc (𝑊 ) , {𝑏′𝑗 }𝑗∈Ωc (𝐾) , {𝑏𝑗 }𝑗∈Ωc (𝐾) ,
̃︀ c (𝐾) and Zc (𝐾). Then Γ acts on {𝑏′ }
denote the primitive idempotents in Zc (𝑊 ), resp. Z
𝑗 𝑗∈Ωc (𝐾)
and the rule
∑︁
𝜎 ′
𝑏′𝑗 ↦→
𝑏𝑗
𝜎∈Γ/StabΓ (𝑏′𝑗 )

defines a bijection
1:1

{𝑑𝑖 }𝑖∈Ωc (𝑊 ) ←→ {𝑏′𝑗 }/Γ.
̃︀ c (𝐾)  Hc (𝐾) and the kernel of this map is generated
There is a natural surjective map H
̃︀ c (𝐾). In particular, the kernel is contained in the
by certain central nilpotent elements in H
̃︀ c (𝐾) and so the map induces a bijection between the simple modules. We can thus
radical of H
consider any simple Hc (𝐾)-module 𝐿c (𝜆) as a simple Hc (𝐾)-module, and to be precise we
̃︀ c (𝜆).
denote this as 𝐿
′
˜
˜ c (𝜆)) ̸= 0. The statements of the theorem then
Now, 𝑏𝑖 · 𝐿c (𝜆) ̸= 0 if and only if (𝜎 𝑏′𝑖 ) · (𝜎 𝐿
follow from the Clifford theoretic fact, compare [4, Proposition 4.7], that
⨁︁
𝜎˜
𝑊
Res 𝐴
𝐿c (𝜇),
𝐴𝐾 𝐿c (𝜆) =
𝜎∈Γ/StabΓ (𝜇)
𝑊
Res 𝐾 𝜆, where 𝐴𝑊

:= Hc (𝑊 )/ Rad Hc (𝑊 ) and 𝐴𝐾 =
for some (any) simple summand 𝜇 of
̃︀ c (𝐾), respectively.
̃︀
̃︀
Hc (𝐾)/ Rad Hc (𝐾) are the maximal semisimple quotients of Hc (𝑊 ) and H

−1
−1
Remark 4.21. Geometrically, Theorem 4.20 is simply saying that ϒ−1
c,𝐾 (0) = 𝜂 (ϒc,𝑊 (0))
is a union of Γ-orbits.

Let Ωc (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 Ωc (𝐾)cusp is Γ-stable and the bijection of Theorem 4.20(a) restricts to an
embedding Ωc (𝐾)cusp /Γ ˓→ Ωc (𝑊 )cusp .
(b) The set Ωc (𝐾)rigid is Γ-stable and the bijection of Theorem 4.20(b) restricts to a
1:1
bijection Ωc (𝑊 )rigid ←→ Ωc (𝐾)rigid /Γ.
Proof. Part (a) follows from Lemma 4.18 which implies that the image of a zero-dimensional
leaf is a zero-dimensional leaf. If 𝐿c (𝜆) is a rigid Hc (𝑊 )-module and 𝜆′ an irreducible
′
summand of Res 𝑊
𝐾 𝜆, then 𝐿c (𝜆 ) is a rigid Hc (𝐾)-module. Conversely, if 𝐿c (𝜇) is a rigid
′
′
Hc (𝐾)-module and 𝜇 an irreducible summand of Ind 𝑊
𝐾 𝜇, then 𝐿c (𝜇 ) is a rigid Hc (𝑊 )module. This implies part (b).

Remark 4.23. The embedding of Proposition 4.22 (1) is not generally a bijection since the
preimage 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 𝐴𝑛 . This is simply the symmetric group S𝑛+1 . It has an 𝑛dimensional irreducible reflection representation. There is just one conjugacy class of reflections
so that our parameter c for rational Cherednik algebras is just a complex number. By Lemma
4.11 we know that Conjecture B holds for c = 0, so we can assume that c > 0.

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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
Xc (𝑊 ) is smooth. Theorem 1.4 now implies that the Calogero–Moser c-families are singletons
and 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 > 0 we have
Conc (𝑊 ) = Irr(𝑊 ). Using Lemma 2.5 we conclude that Conc (𝑊 ) = Irr(𝑊 ) for arbitrary
real c > 0. It then follows that the Lusztig c-families are singletons and using Lemma 2.6 we
furthermore 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 particular
as we now have to deal with a two-dimensional parameter space. We have split the discussion
into several parts, some just dealing with the Calogero–Moser families, some just dealing with
the Lusztig families. At the very end we combine these results to obtain the proof of Theorem A.
§6A.
§6B.
§6C.
§6D.
§6E.
§6F.
§6G.
§6H.
§6I.
§6J.
§6K.
§6L.
§6M.
§6N.
§6O.
§6P.

The group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Reflections and parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
The rational Cherednik algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Symplectic leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Parabolic subgroups attached to symplectic leaves . . . . . . . . . . . . . . . . . . . . . . . . 25
Calogero–Moser families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Simple Hc (𝑊 )-modules in the degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Lusztig families in the non-degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Lusztig families in the degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Calogero–Moser families vs. Lusztig families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Cuspidal Lusztig families in the non-degenerate case . . . . . . . . . . . . . . . . . . . . . . 31
Cuspidal Lusztig families in the degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Rigid modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Cuspidal Lusztig families vs. cuspidal Calogero–Moser families. . . . . . . . . . . .33

§6A. The group
Let 𝑊 be the Weyl group of type 𝐵𝑛 . This group is isomorphic to the group 𝐺(2, 1, 𝑛)
of generalized permutation matrices in GL𝑛 (C) with entries in 𝜇2 := {1, −1} ⊆ C, and this
defines at the same time an irreducible reflection representation of 𝐵𝑛 . Note that 𝑊 = 𝜇𝑛2 o S𝑛 ,
where S𝑛 acts on 𝜇𝑛2 by coordinate permutation. For each 1 ≤ 𝑖 ≤ 𝑛 we have a natural
embedding 𝜀𝑖 of 𝜇2 into 𝑊 , sending 𝑢 ∈ 𝜇2 to the diagonal matrix (1, . . . , 𝑢, . . . , 1) with 𝑢
in the 𝑖-th place. For 1 ≤ 𝑖 < 𝑗 ≤ 𝑛 let 𝑠𝑖𝑗 be the transposition (𝑖, 𝑗) ∈ S𝑛 . For 𝑢 ∈ 𝜇2 set
𝑠𝑖𝑗,𝑢 := 𝑠𝑖𝑗 𝜀𝑖 (𝑢)−1 𝜀𝑗 (𝑢). Note that 𝑠𝑖𝑗,1 = 𝑠𝑖𝑗 . The group 𝑊 is generated by 𝜀1 (−1) and the
transpositions 𝑠𝑖𝑗 .
§6B. Reflections and parabolic subgroups
Let (𝑦1 , . . . , 𝑦𝑛 ) be the standard basis of h := C𝑛 with dual basis (𝑥1 , . . . , 𝑥𝑛 ). For any
1 ≤ 𝑗 ≤ 𝑛 the element 𝜀𝑗 (−1) is a reflection with coroot 𝛼𝑗∨ := 𝑦𝑗 and root 𝛼𝑗 := 2𝑥𝑗 . Also,
∨ := 𝑢𝑦 − 𝑦
for any 𝑢 ∈ 𝜇2 and 1 ≤ 𝑖 < 𝑗 ≤ 𝑛 the element 𝑠𝑖𝑗,𝑢 is a reflection with coroot 𝛼𝑖𝑗,𝑢
𝑖
𝑗
and root 𝛼𝑖𝑗,𝑢 := 𝑢−1 𝑥𝑖 − 𝑥𝑗 = 𝑢𝑥𝑖 − 𝑥𝑗 . These elements are precisely the reflections in 𝑊 .

G WYN B ELLAMY AND U LRICH T HIEL

23

We can now easily compute that
(𝑦𝑘 , 𝛼𝑗 )(𝛼𝑗∨ , 𝑥𝑙 )

(10)

{︂

=

2 if 𝑘 = 𝑗 = 𝑙
0 else

and
if 𝑘, 𝑙 ∈ {𝑖, 𝑗} with 𝑘 = 𝑙
−𝑢 if 𝑘, 𝑙 ∈ {𝑖, 𝑗} with 𝑘 ̸= 𝑙
⎩
0
else.
The conjugacy classes of reflections in 𝑊 are
𝒮0 := {𝑠𝑖𝑗,𝑢 | 𝑢 ∈ 𝜇2 , 1 ≤ 𝑖 < 𝑗 ≤ 𝑛} and 𝒮1 := {𝜀𝑗 (−1) | 1 ≤ 𝑗 ≤ 𝑛} .
We have |𝒮0 | = 𝑛2 − 𝑛 and 𝒮1 = 𝑛. The parabolic subgroups of 𝑊 are, up to conjugacy, of
the form S𝜆 × 𝐵𝑛−|𝜆| for partitions 𝜆 of integers ≤ 𝑛.
(11)

∨
(𝑦𝑘 , 𝛼𝑖𝑗,𝑢 )(𝛼𝑖𝑗,𝑢
, 𝑥𝑙 ) =

⎧
⎨ 1

§6C. Representations
Since 𝑊 = 𝜇𝑛2 o S𝑛 = 𝜇2 ≀ S𝑛 , the irreducible representations of 𝑊 are labeled by bipartitions
𝜆 = (𝜆(0) , 𝜆(1) ) of 𝑛. Let 𝜋𝜆 denote the representation labeled by 𝜆. The trivial representation
of 𝑊 is 𝜋(𝑛,∅) . The representation 𝛾 := 𝜋(∅,𝑛) is a linear character of 𝑊 with 𝛾(𝑠) = 1 for all
𝑠 ∈ 𝒮0 and 𝛾(𝑠) = −1 for 𝑠 ∈ 𝒮1 . We denote by 𝛾𝜋𝜆 the 𝛾-twist of 𝜋𝜆 .
The symmetric group S𝑛 is a quotient of 𝐵𝑛 by sending 𝜀𝑗 (−1) to 1. We can thus consider (irreducible) S𝑛 -modules 𝜋𝜆 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 𝜋𝜆(0)  𝛾𝜋𝜆(1) is an irreducible (𝐵𝑟 × 𝐵𝑛−𝑟 )subrepresentation of 𝜋𝜆 with
𝑛
(12)
𝜋𝜆 = Ind 𝐵
𝐵𝑟 ×𝐵𝑛−𝑟 𝜋𝜆(0)  𝛾𝜋𝜆(1) .
§6D. The rational Cherednik algebra
Fix a 𝑊 -equivariant function c : Ref(𝑊 ) → C and define
𝑐1 := c(𝒮1 ) and 𝜅 := c(𝒮0 ) .
In terms of the Coxeter diagram of type 𝐵𝑛 the weight function c is determined as follows:
𝑐1

𝜅

𝜅

𝜅

𝜅

...
Using equations (10) and (11) we see that the defining relation (1) for Hc (𝑊 ) becomes
[𝑦𝑖 , 𝑥𝑖 ] = −2𝑐1 𝜀𝑖 (−1) − 𝜅

(13)

𝑛
∑︁ ∑︁

𝑠𝑖𝑗,𝑢

𝑢∈𝜇2 𝑗=1
𝑗̸=𝑖

and
(14)

[𝑦𝑖 , 𝑥𝑗 ] = 𝜅

∑︁

𝑢𝑠𝑖𝑗,𝑢 .

𝑢∈𝜇2

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. 𝑐1 ̸= 0 or 𝜅 ̸= 0.
§6E. Isomorphisms
Recall that for any 𝛼 ∈ C* , the algebras Hc (𝑊 ) and H𝛼c (𝑊 ) are isomorphic. Given a
bipartition 𝜆 = (𝜆(0) , 𝜆(1) ), we define 𝜆𝜏 to be (𝜆(1) , 𝜆(0) ). The following proposition follows
from [9, 4.6B].

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C 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(𝑐1 ,𝜅) (𝐵𝑛 ) −→ H(−𝑐1 ,𝜅) (𝐵𝑛 )
with 𝜏 (𝑥) = 𝑥, 𝜏 (𝑦) = 𝑦 and 𝜏 (𝑤) = 𝛾(𝑤)𝑤 for all 𝑥 ∈ h* , 𝑦 ∈ h and 𝑤 ∈ 𝑊 . Moreover,
−1
(a) 𝜏 𝐿(𝑐1 ,𝜅) (𝜆) ≃ 𝐿(−𝑐1 ,𝜅) (𝜆𝜏 ).
(b) 𝜆 and 𝜇 belong to the same Calogero–Moser (𝑐1 , 𝜅)-family if and only if 𝜆𝜏 and 𝜇𝜏
belong to the same Calogero–Moser (−𝑐1 , 𝜅)-family.
(c) 𝜆 is cuspidal, resp. rigid, for H(𝑐1 ,𝜅) (𝑊 ) if and only if 𝜆𝜏 is cuspidal, resp. rigid, for
H(−𝑐1 ,𝜅) (𝑊 ).
In the case 𝜅 = 0 the defining relations (13) and (14) of Hc (𝑊 ) show that we have an
algebra isomorphism Hc (𝑊 ) ≃ H𝑐1 (Z2 )⊗𝑛 o S𝑛 , where S𝑛 naturally acts on the 𝑛-fold tensor
product of the rational Cherednik algebra at 𝑐1 for the cyclic group of order 2. From this we
get an isomorphism of Poisson varieties Xc (𝑊 ) ≃ 𝑆 𝑛 (X𝑐1 (Z2 )), where 𝑆 𝑛 denotes the 𝑛-th
symmetric power. Since 𝑐1 ̸= 0, the Calogero–Moser space X𝑐1 (Z2 ) is a smooth symplectic
surface 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 singletons
by Theorem 1.4 and none of them is cuspidal by Lemma 3.7. Using the relation between
Calogero–Moser spaces and representation varieties of deformed preprojective algebras, Martino has determined in his Ph.D thesis [34, Section 5] for precisely which parameters the
Calogero–Moser space is smooth and gave a parametrization of the symplectic leaves.1 To
simplify notations we set [𝑎, 𝑏] := {𝑎, . . . , 𝑏} and denote by ±[𝑎, 𝑏] the set [−𝑏, −𝑎] ∪ [𝑎, 𝑏] for
integers 𝑎 ≤ 𝑏. Note that ±[0, 𝑏] = [−𝑏, 𝑏].
Theorem 6.2 (Martino). Let c = (𝜅, 𝑐1 ).
(a) Xc (𝑊 ) is singular if and only if 𝜅 = 0 or 𝑐1 = 𝑚𝜅 for some 𝑚 ∈ ±[0, 𝑛 − 1].
(b) If 𝜅 = 0, then the symplectic leaves of Xc (𝑊 ) are parameterised by the set 𝒫(𝑛) of
partitions of 𝑛. For 𝜆 ∈ 𝒫(𝑛), the corresponding leaf ℒ𝜆 has dimension 2ℓ(𝜆), where
ℓ(𝜆) is the length of 𝜆.
(c) If 𝑐1 = 𝑚𝜅, with 𝜅 ̸= 0, then there is a bijection 𝑘 ↦→ ℒ𝑘 ,
1:1

{symplectic leaves ℒ of Xc (𝑊 ) } ←→ {𝑘 ∈ N≥0 | 𝑘(𝑘 + 𝑚) ≤ 𝑛} .
Moreover, dim ℒ𝑘 = 2(𝑛 − 𝑘(𝑘 + 𝑚)).
We say that c is singular if Xc (𝑊 ) is singular. Moreover, we call singular parameters with
𝜅 ̸= 0 non-degenerate and those with 𝜅 = 0 degenerate. By the formulas for the dimensions
of the symplectic leaves we can immediately deduce when zero-dimensional leaves (and thus
cuspidal Calogero–Moser families) exist.
Corollary 6.3. The space Xc (𝑊 ) has a zero-dimensional symplectic leaf if and only if 𝑐1 = 𝑚𝜅
for some 𝑚 ∈ ±[0, 𝑛 − 1] such that 𝑛 = 𝑘(𝑘 + 𝑚) for some 𝑘 > 0. In this case there is a
unique 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 − 1] there is at most one 𝑘 ≥ 0 with 𝑛 = 𝑘(𝑘 + 𝑚).
1In [34] the parameters are named (𝑐 , 𝑐 ) instead of (𝑐 , 𝜅).
𝛾
1
1

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25

§6G. Parabolic subgroups attached to symplectic leaves
We would like to parameterise the symplectic leaves of Xc (𝐵𝑛 ) by conjugacy classes of
parabolic subgroups and work out the geometric ordering.
Lemma 6.5. If 𝑐1 = 𝑚𝜅 for some 𝑚 ∈ ±[0, 𝑛−1], then the leaf ℒ𝑘 is labeled by the conjugacy
class of the parabolic 𝐵𝑘(𝑘+𝑚) and
ℒ𝑘 ≺ ℒ𝑘′

⇐⇒

(𝐵𝑘(𝑘+𝑚) ) ≤ (𝐵𝑘′ (𝑘′ +𝑚) )

⇐⇒

𝑘 ≥ 𝑘′ .

Proof. If ℒ𝑘 is labeled by the parabolic 𝑊 ′ then Xc′ (𝑊 ′ ) contains at least one zero-dimensional
leaf and 𝑊 ′ must have rank 𝑛 − 12 dim ℒ𝑘 = 𝑘(𝑘 + 𝑚). Since 𝜅 ̸= 0, the parabolic must be
of the form 𝐵𝑚 for some 𝑚. Hence 𝑊 ′ = 𝐵𝑘(𝑘+𝑚) . It is a consequence of the proof of [34,
Proposition 5.7] that ℒ𝑘 ≺ ℒ𝑘′ if and only if 𝑘 ≥ 𝑘 ′ .

In the degenerate case 𝜅 = 0, recall from §6E that there is an isomorphism of Poisson
varieties Xc (𝐵𝑛 ) {︁≃ 𝑆 𝑛 (X𝑐1 (Z2 )). Then ℒ𝜆 =}︁𝑆 𝜆 (X𝑐1 (Z2 )), where 𝑆 𝜆 (𝑋) is the image in
∑︀ℓ(𝜆)
𝑆 𝑛 (𝑋) of the set
𝑖=1 𝜆𝑖 · 𝑥𝑖 | 𝑥𝑖 ̸= 𝑥𝑗 ∈ 𝑋 . This implies that ℒ𝜆 is labeled by the class
of the parabolic subgroup S𝜆 = S𝜆1 × · · · × S𝜆ℓ(𝜆) and
ℒ𝜆 ≺ ℒ𝜇 ⇔ (S𝜆 ) ≤ (S𝜇 ).
−1
Moreover, in this case, if ϒc (0) = {𝑝, 𝑞} for H𝑐1 (Z2 ), where 𝑝 =
𝑞 = Supp 𝐿𝑐1 (sgnZ2 ), then in Xc (𝐵𝑛 ) we have
(15)
ϒ−1
c (0) = {𝑛1 · 𝑝 + 𝑛2 · 𝑞 | 𝑛1 + 𝑛2 = 𝑛, 𝑛𝑖 ≥ 0}.

Supp 𝐿𝑐1 (1Z2 ) and

The point 𝑛1 · 𝑝 + 𝑛2 · 𝑞 belongs to the leaf ℒ(𝑛1 ,𝑛2 ) .
§6H. Calogero–Moser families
The Calogero–Moser families in type 𝐵𝑛 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 recall
the description given in [35] now.
Let 𝜆 = (𝜆1 , 𝜆2 , . . .) be a partition. We think of 𝜆 as a stack of boxes, left justified, with the
bottom row containing 𝜆1 boxes, the next row containing 𝜆2 boxes and so forth. The content
ct() of a box  = (𝑖, 𝑗) ∈ 𝜆 is defined to be 𝑗 − 𝑖. We consider the group ring Z[C] of the
additive group C and write 𝑥𝛼 for the element corresponding to 𝛼 ∈ C. The residue of 𝜆 is the
element
∑︁
Res 𝜆 (𝑥) :=
𝑥ct() ∈ Z[Z] ⊆ Z[C] .
∈𝜆

Just as in [10, §3A], we define for a triple m = (𝑚0 , 𝑚1 , 𝑚′ ) of complex numbers (the charge),
and a bipartition 𝜆 = (𝜆(0) , 𝜆(1) ), the charged residue as
′
′
𝑚0
Res m
Res 𝜆(0) (𝑥𝑚 ) + 𝑥𝑚1 Res 𝜆(1) (𝑥𝑚 ) ∈ Z[C] .
𝜆 (𝑥) := 𝑥
The following theorem is [35, Theorem 5.5]. The additional parameters (ℎ, 𝐻0 , 𝐻1 ) used in
loc. cit. are given by ℎ = −𝜅, 𝐻0 = −𝑐1 , and 𝐻1 = 𝑐1 .
Theorem 6.6 (Martino). Two bipartitions 𝜆 and 𝜇 lie in the same Calogero–Moser c-family if
^
^ := (0, 𝑐1 , −𝜅).
and only if Res c𝜆^ (𝑥) = Res c𝜇
(𝑥) with respect to the charge c
§6I. Simple Hc (𝑊 )-modules in the degenerate case
In the degenerate case 𝜅 = 0 it is possible to determine the structure of the simple Hc (𝑊 )modules 𝐿c (𝜆) as 𝑊 -modules.
Lemma 6.7. If 𝜅 = 0, then 𝜆 = (𝜆(0) , 𝜆(1) ) ∈ 𝒫2 (𝑛) and 𝜇 = (𝜇(0) , 𝜇(1) ) ∈ 𝒫2 (𝑛) lie in the
same Calogero–Moser c-family if and only if |𝜆(1) | = |𝜇(1) |. In particular, there are 𝑛 + 1

26

C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

deg
deg with
Calogero–Moser families ℱ0;𝑛
, . . . , ℱ𝑛;𝑛
deg
ℱ𝑖;𝑛
= {𝜆 ∈ 𝒫2 (𝑛) | |𝜆(0) | = 𝑖} .

Proof. In the case 𝜅 = 0 we have
∑︁
∑︁
Res c𝜆^ (𝑥) =
1ct + 𝑥𝑐1
1ct = |𝜆(0) | + 𝑥𝑐1 |𝜆(1) | = 𝑛 − |𝜆(1) | + 𝑥𝑐1 |𝜆(1) | .
∈𝜆(0)

∈𝜆(1)



The claim follows directly from Theorem 6.6.

deg
is labeled by the class of the
Proposition 6.8. Assume that 𝜅 = 0. Then the family ℱ𝑖;𝑛
∼

deg
parabolic S𝑖 × S𝑛−𝑖 ⊂ 𝐵𝑛 and we have a bijection Irr(S𝑖 × S𝑛−𝑖 ) −→ ℱ𝑖;𝑛
, sending the
(0)
(1)
pair of partitions (𝜆 , 𝜆 ) to itself (thought of as a bipartition) such that
𝑛
𝐿c (𝜆(0) , 𝜆(1) ) ≃ Ind 𝐵
S𝑖 ×S𝑛−𝑖 𝜋𝜆(0)  𝜋𝜆(1)

as 𝑊 -modules.
Proof. Since Hc (𝐵𝑛 ) ≃ (H𝑐1 (Z2 )⊗2 ) o S𝑛 in this case (see §6E), we have
𝑛
𝐿c (𝜆(0) , 𝜆(1) ) ≃ Ind 𝐵
(H𝑐

⊗2 )o(S ×S
𝑖
𝑛−𝑖 )
1 (Z2 )

𝐿𝑐1 (1Z2 ) ⊗ 𝐿𝑐1 (sgnZ2 ) ⊗ (𝜋𝜆(0)  𝜋𝜆(1) ) ,

Since 𝑐1 ̸= 0, both 𝐿𝑐1 (1Z2 ) and 𝐿𝑐1 (sgnZ2 ) are isomorphic to the regular representation as
(0)
(1)
Z2 -modules. Recall that we have described ϒ−1
c (0) in (15). If 𝑖 = |𝜆 |, so that 𝑛 − 𝑖 = |𝜆 |,
then the support of 𝐿c (𝜆(0) , 𝜆(1) ) is 𝑖 · 𝑝 + (𝑛 − 𝑖) · 𝑞, which lies on the the leaf labeled by the
parabolic S𝑖 × S𝑛−𝑖 . The result follows.

§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 can
restrict to the so-called integral case where 𝑐1 is an integral multiple of 𝜅.
Proposition 6.9. Suppose that 𝜅 > 0. If there is no 𝑚 ∈ N with 𝑐1 = 𝑚𝜅, then Conc 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 has
been shown by Lusztig [33, Proposition 22.25] when 𝑐1 is rational. We reduce the general
case 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 s𝜆 , see [23, Theorem 10.5.2] and
[33, Lemma 22.12], shows that there exist finitely many integers 𝑟1𝜆 , 𝑟2𝜆 , . . ., 𝑠𝜆1 , 𝑠𝜆2 , . . ., with
(𝑟𝑖 , 𝑠𝑖 ) ̸= (𝑟𝑗 , 𝑠𝑗 ) for 𝑖 ̸= 𝑗, and rational numbers 𝑓1𝜆 , 𝑓2𝜆 , . . . such that
∑︁
𝜆
𝜆
s =
𝑓 𝜆 𝑞 2(𝑟𝑖 𝜅+𝑠𝑖 𝑐1 ) .
𝜆

𝑖

𝑖

Recall that 𝜅 = 1 and 𝑐1 ∈
/ Q. We claim that a𝜆 = min{𝑟𝑖𝜆 𝜅 + 𝑠𝜆𝑖 𝑐1 | 𝑖 = 1, . . .}. Note that
this is not the case in general since there might be some cancellation between the 𝑓𝑖𝜆 when
𝑟𝑖𝜆 𝜅 + 𝑠𝜆𝑖 𝑐1 = 𝑟𝑗𝜆 𝜅 + 𝑠𝜆𝑗 𝑐1 for some 𝑖 ̸= 𝑗. However, in our case the fact that 𝜅 = 1 and 𝑐1 is
irrational implies that 𝑟𝑖𝜆 𝜅 + 𝑠𝜆𝑖 𝑐1 = 𝑟𝑗𝜆 𝜅 + 𝑠𝜆𝑗 𝑐1 if and only if 𝑟𝑖𝜆 = 𝑟𝑗𝜆 and 𝑠𝜆𝑖 = 𝑠𝜆𝑗 , i.e. 𝑖 = 𝑗.
The claim follows.
The definition of j-induction and constructible representations makes it clear that if we are
given two parameters c and c′ such that
(16)
a𝜆 = a𝜇 ⇔ a𝜆′ = a𝜇′
for all irreducible representations 𝜆 and 𝜇 of all parabolic subgroups of 𝑊 , then Conc 𝑊 =
Conc′ 𝑊 . Since there are only finitely many 𝑟𝑖𝜆 and 𝑠𝜆𝑗 as 𝜆 ranges over all irreducible representations of all parabolic subgroups of 𝑊 , one can easily choose a rational number 𝑐′1 > 0

G WYN B ELLAMY AND U LRICH T HIEL

27

with |𝑐1 − 𝑐′1 | very small and 𝑐′1 not an integer such that
𝑟𝑖𝜆 + 𝑠𝜆𝑖 𝑐1 < 𝑟𝑗𝜇 + 𝑠𝜇𝑗 𝑐1 ⇔ 𝑟𝑖𝜆 + 𝑠𝜆𝑖 𝑐′1 < 𝑟𝑗𝜇 + 𝑠𝜇𝑗 𝑐′1
for all 𝜆, 𝜇 and 𝑖, 𝑗. In particular, for c′ = (𝑐′1 , 1) equation (16) holds. Moreover, since 𝑐′1 is
rational, every constructible representation in Conc′ 𝑊 is irreducible by [33, Proposition 22.25].
Hence Conc 𝑊 = Irr 𝑊 , too.

We can thus restrict to the case 𝑐1 = 𝑚𝜅 for some 𝑚 ∈ N, which by Lemma 2.5 is the same
as c = (𝑚, 1). The Lusztig families in this case have been described by Lusztig [33, §22] using
the notion of symbols. We review the notion of symbols for general integral parameters.
We assume that 𝜅 > 0 and that c = (𝑐1 , 𝜅) ≥ 0 is integral.
We can uniquely write 𝑐1 = 𝑚𝜅 + 𝑟 for some 𝑚, 𝑟 ∈ N≥0 with 𝑟 < 𝜅. Fix an arbitrary
integer 𝑁 > 0. A symbol for 𝐵𝑛 with respect to 𝑁 at parameter c is a list of the form
(︂
)︂
𝛽1 𝛽2 · · · · · · · · · 𝛽𝑁 +𝑚−1 𝛽𝑁 +𝑚
(17)
𝑆=
,
𝛾1 𝛾2 · · · 𝛾𝑁
where 0 ≤ 𝛽1 < · · · < 𝛽𝑁 +𝑚 are congruent to 𝑟 modulo 𝜅 and 0 ≤ 𝛾1 < · · · < 𝛾𝑁 are
divisible by 𝜅, such that
(︃ )︃
∑︁
∑︁
𝑚
2
(18)
𝛽𝑖 +
𝛾𝑗 = 𝑛𝜅 + 𝜅𝑁 + 𝑁 (𝑐1 − 𝜅) + 𝜅
+ 𝑟𝑚 .
2
𝑖
𝑗
𝑁
𝑁 +1
Let Sy𝑁
c;𝑛 denote the set of all such symbols. We have an embedding Syc;𝑛 ˓→ Syc;𝑛 sending
a symbol 𝑆 as above to the symbol
(︂
)︂
𝑟 𝛽1 + 𝜅 𝛽2 + 𝜅 · · ·
···
𝛽𝑁 +𝑚 + 𝜅
(19)
𝑆[1] =
.
0 𝛾1 + 𝜅 𝛾 2 + 𝜅 · · · 𝛾𝑁 + 𝜅
For 𝑖 ∈ N we denote by 𝑆[𝑖] the 𝑖-fold composition of the above map applied to 𝑆 and call this
the shift of 𝑆 by 𝑖. Let Syc;𝑛 be the direct limit of the Sy𝑁
c;𝑛 with respect to the above maps. We
(0)

(1)

say that 𝑁 is large enough for a bipartition 𝜆 = (𝜆(0) , 𝜆(1) ) of 𝑛 if 𝜆𝑁 +𝑚+1 = 0 = 𝜆𝑁 +1 . We
(︀𝛽 )︀
𝑁
then define the corresponding symbol Sy𝑁
c;𝑛 (𝜆) = 𝛾 ∈ Syc;𝑛 via
(︁

(20)

)︁

(0)
𝛽𝑖 := 𝜅 𝜆𝑁 +𝑚−𝑖+1 + 𝑖 − 1 + 𝑟 for 𝑖 ∈ [1, 𝑁 + 𝑚]

(︁

(1)
𝛾𝑗 := 𝜅 𝜆𝑁 −𝑗+1 + 𝑗 − 1

)︁

for 𝑗 ∈ [1, 𝑁 ] .

If 𝑁 is large enough for all bipartitions of 𝑛, e.g., 𝑁 ≥ 𝑛, the map 𝜆 ↦→ Sy𝑁
c;𝑛 (𝜆) defines a
𝑁
bijection between the set 𝒫2 (𝑛) of bipartitions of 𝑛 and Syc;𝑛 . For a symbol 𝑆 we then denote
by 𝜋𝑆 the representation of 𝑊 labeled by the bipartition corresponding to 𝑆. The content ct(𝑆)
of a symbol 𝑆 ∈ Sy𝑁
c;𝑛 is the multiset of its entries, i.e., the list of entries with repetitions but
∑︀
ignoring positions. We can, and will, equally well write the content as a polynomial 𝑖≥0 𝑛𝑖 𝑥𝑖 ,
where 𝑛𝑖 denotes the multiplicity of the entry 𝑖 in 𝑆. It is clear from the definition of a symbol
that it has at least 𝑁 + 𝑚 distinct entries and the multiplicity of an entry in a symbol is at
most 2.
Example 6.10. Let 𝜆 =

(︁

,

)︁

and (𝑐1 , 𝜅) = (1, 1). Then
(︂

)︂

0 1 3 5
=
∈ Sy3(1,1);4 .
0 1 3
(︂
)︂
1 3
This symbol is in fact the shift of
∈ Sy1(1,1);4 by 2.
1
Sy3(1,1);4 (𝜆)

28

C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

Theorem 6.11 (Lusztig). Let c = (𝑐1 , 𝜅) ≥ 0 with 𝜅 > 0 and 𝑐1 = 𝑚𝜅 for some 𝑚 ∈ N. Then
two bipartitions 𝜆 and 𝜇 lie in the same Lusztig c-family if and only if ct(Sy𝑁
(𝑚,1);𝑛 (𝜆)) =
𝑁
ct(Sy(𝑚,1);𝑛 (𝜇)) for 𝑁 sufficiently large.
Proof. Because of Lemma 2.5 we can assume that c = (𝑚, 1). Then the description of the
c-constructible characters in [33, Proposition 22.24] along with [33, Lemma 22.22] proves the
claim.

§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) ) ∈ 𝒫2 (𝑛) and 𝜇 = (𝜇(0) , 𝜇(1) ) ∈ 𝒫2 (𝑛) lie in
the same Lusztig c-family if and only if |𝜆(1) | = |𝜇(1) |. In particular, there are precisely 𝑛 + 1
deg
deg with
Lusztig families ℱ0;𝑛
, . . . , ℱ𝑛;𝑛
deg
ℱ𝑖;𝑛
= {𝜆 ∈ 𝒫2 (𝑛) | |𝜆(0) | = 𝑖} .

§6L. Calogero–Moser families vs. Lusztig families
We can now prove Theorem 2.4 for type 𝐵𝑛 .
Corollary 6.13. For type 𝐵𝑛 and any parameter c ≥ 0 the Lusztig c-families are equal to the
Calogero–Moser c-families.
Proof. If 𝜅 = 0, the claim follows from Lemma 6.7 and Lemma 6.12. Now, assume that 𝜅 ̸= 0.
If 𝑐1 ̸= 𝑚𝜅 for all 𝑚 ∈ N≥0 , then we know from Theorem 6.2(a) and Proposition 6.9 that
both the Calogero–Moser c-families and the Lusztig c-families are singletons. So, suppose
that 𝑐1 = 𝑚𝜅 for some 𝑚 ∈ N≥0 . Because of Lemma 2.5 and Lemma 3.8 we can assume that
𝑁
𝜅 = 1. It follows from [10, Proposition 3.4] that ct(Sy𝑁
c;𝑛 (𝜆)) = ct(Syc;𝑛 (𝜇)) for 𝑁 large
^
enough if and only if Res c𝜆^ (𝑥) = Res c𝜇
(𝑥). By Theorem 6.6 and Theorem 6.11 this shows that
Ωc (𝑊 ) = Lusc (𝑊 ).

§6M. Cuspidal Lusztig families in the non-degenerate case
In the non-singular case, Proposition 6.9 and Lemma 2.6 immediately imply the following
result.
Lemma 6.14. If 𝜅 > 0 and there is no 𝑚 ∈ N with 𝑐1 = 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 𝑐1 = 𝑚𝜅 = 𝑚 for some 𝑚 ∈ N≥0 .
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 unequal
parameters. 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
𝑁
we denote by Sy𝑁
c;𝑛 (ℱ) the set of symbols Syc;𝑛 (𝜆) with 𝜆 ∈ ℱ. Using the combinatorics of
symbols, we can explicitly determine the size of ℱ.

G WYN B ELLAMY AND U LRICH T HIEL

29

Lemma 6.15. Let ℱ ∈ Lusc (𝑊 ). Let 𝑖≥0 𝑛𝑖 𝑥𝑖 be the content of one (any) 𝑆 ∈ Sy𝑁
(ℱ).
(︀2𝑘ℱ +𝑚)︀ c;𝑛
Set 𝑘ℱ := 𝑁 − |{𝑖 | 𝑛𝑖 = 2}|. Then 𝑘ℱ ≥ 0, 𝑘ℱ (𝑘ℱ + 𝑚) ≤ 𝑛, and |ℱ| =
.
𝑘ℱ
∑︀

Proof. Let 𝑆 ∈ Sy𝑁
c;𝑛 (ℱ) and set 𝑘 := 𝑘ℱ . The multiplicity of an entry in 𝑆 is at most equal
to 2 and 𝑆 has at least 𝑁 + 𝑚 distinct entries. Since 𝑆 has exactly 2𝑁 + 𝑚 entries with
multiplicity, this immediately shows that 𝑘 ≥ 0. 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
𝑁
{𝑖 | 𝑛𝑖 = 2} defines a unique symbol in Sy𝑁
c;𝑛 (ℱ), and in this way all symbols of Syc;𝑛 (ℱ) are
(︀𝑁 +𝑘+𝑚−(𝑁 −𝑘))︀ (︀2𝑘+𝑚)︀
obtained. The number of such sets is equal to
=
.
𝑘
𝑁 −(𝑁 −𝑘)
It remains to show that 𝑘(𝑘 + 𝑚) ≤ 𝑛. Since 𝑆 ∈ Sy𝑁
c;𝑛 , equation (18) says that
(︃ )︃
∑︁

(21)

𝛽𝑖 +

∑︁

𝑖

2

𝛾𝑗 − 𝑁 − 𝑁 (𝑚 − 1) −

𝑗

𝑚
2

=𝑛,

where the 𝛽𝑖 and 𝛾𝑗 are the entries of 𝑆. Hence, it suffices to show that the left hand side
is at least as big as 𝑘(𝑘 + 𝑚). Recall that 𝑁 − 𝑘 is equal to the number of pairs (𝑖, 𝑗) such
that 𝛽𝑖 = 𝛾𝑗 . Since 𝛽𝑖 , 𝛾𝑗 ≥ 0, the expression on the left is minimal if 𝛽𝑖 = 𝛾𝑖 = 𝑖 − 1 for
𝑖 = 1, . . . , 𝑁 − 𝑘 and the remaining 2𝑘 + 𝑚 entries are in {𝑁 − 𝑘, . . . , 𝑁 + 𝑘 + 𝑚 − 1}. Then
the left hand side of equation (21) becomes
𝑁
−𝑘
∑︁

2(𝑖 − 1) +

𝑁 +𝑘+𝑚−1
∑︁

(︃ )︃

𝑚
2

2

𝑖 − 𝑁 − 𝑁 (𝑚 − 1) −

𝑖=𝑁 −𝑘

𝑖=1

= 𝑘(𝑘 + 𝑚) .


Definition 6.16. A symbol 𝑆 ∈ Sy𝑁
c;𝑛 with content
for all 𝑖 = 0, 1, . . .

∑︀

𝑖≥0 𝑛𝑖 𝑥

𝑖

is called cuspidal if 𝑛𝑖 ≥ 𝑛𝑖+1

If 𝑆 is a cuspidal symbol then 𝑆[1] is also cuspidal.
Suppose that 𝑛 = 𝑘(𝑘 + 𝑚) for some 𝑘 ∈ N>0 . Then we have the box partition (𝑘 𝑘+𝑚 ) of
𝑛. If 𝜆 is any partition such that ℓ(𝜆) ≤ 𝑘 + 𝑚 and 𝜆1 ≤ 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 − 𝜆𝑘+𝑚+1−𝑗 ≥ 𝑖}
= 𝑘 + 𝑚 + 1 − min{𝑗 ∈ [1, 𝑘 + 𝑚] | 𝑘 − 𝑖 ≥ 𝜆𝑗 } .
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
cusp
:= {(𝜆, 𝜆† ) | ℓ(𝜆) ≤ 𝑘 + 𝑚, 𝜆1 ≤ 𝑘} .
ℱ𝑘,𝑚
Example 6.17. If 𝑛 = 6 and 𝑚 = 1, we can write 𝑛 = 𝑘(𝑘 + 𝑚) with 𝑘 = 2 and get
{︃
cusp
ℱ2,1

=

(︁

(︁

)︁ (︁

, ∅,

,

)︁ (︁

,

,

,

)︁

)︁

(︃

)︃

, ∅ ,

,
(︃

,

(︁

)︁

,

)︃ (︃

,

,

(︃

)︃

,

,

)︃

,

,

(︁

,
,

)︁

}︃

.

If 𝑛 = 3 and 𝑚 = 2, we can write 𝑛 = 𝑘(𝑘 + 𝑚) with 𝑘 = 1 and get
{︃
cusp
ℱ1,2

=

(∅ ,

),(

,

),

(︁

,

)︁

(︃

,

)︃}︃

, ∅

.

Lemma 6.18. Suppose that 𝑛 = 𝑘(𝑘 + 𝑚). The content of the symbol Sy𝑘c;𝑛 (𝜆, 𝜆† ) is equal
∑︀
cusp
cusp
to 2𝑘+𝑚−1
𝑥𝑖 for any (𝜆, 𝜆† ) ∈ ℱ𝑘,𝑚
. In particular, Sy𝑘c;𝑛 (𝜆, 𝜆† ) is cuspidal and ℱ𝑘,𝑚
is a
𝑖=0
(︀2𝑘+𝑚)︀
cusp
Lusztig c-family with |ℱ𝑘,𝑚 | =
.
𝑘

30

C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

cusp
Proof. First, note that 𝑁 = 𝑘 is large enough for any (𝜆, 𝜆† ) ∈ ℱ𝑘,𝑚
. The symbol Sy𝑘c;𝑛 (𝜆, 𝜆† ) =
(︀𝛽 )︀
𝛾 is then given by

𝛽𝑖 := 𝜆𝑘+𝑚−𝑖+1 + 𝑖 − 1 for 𝑖 ∈ [1, 𝑘 + 𝑚]
𝛾𝑗 := 𝜆†𝑘−𝑗+1 + 𝑗 − 1
for 𝑗 ∈ [1, 𝑘] .
Our assertion about the content of this symbol is equivalent to showing that the symbol contains
the entry 0, all entries are bounded above by 2𝑘 + 𝑚 − 1, and that 𝛽𝑖 ̸= 𝛾𝑗 for all 𝑖, 𝑗. First, we
have
𝛽1 = 𝜆𝑘+𝑚 and 𝛾1 = 𝜆†𝑘 = 𝑘 + 𝑚 + 1 − min{𝑗 ∈ [𝑘 + 𝑚 | 0 ≥ 𝜆𝑗 } = 𝑘 + 𝑚 − ℓ(𝜆) .
We immediately see that either 𝛽1 = 0 or 𝛾1 = 0. On the other hand, we have
𝛽𝑘+𝑚 = 𝜆1 + 𝑘 + 𝑚 − 1 and 𝛾𝑘 = 𝜆†1 + 𝑘 − 1 = 2𝑘 + 𝑚 − min{𝑗 ∈ [1, 𝑘 + 𝑚] | 𝑘 > 𝜆𝑗 } .
This shows that the entries of the symbol are at most equal to 2𝑘 + 𝑚 − 1. Showing that 𝛽𝑖 ̸= 𝛾𝑗
for all 𝑖 ∈ [1, 𝑘 + 𝑚] and 𝑗 ∈ [1, 𝑚] is equivalent to showing that 𝛽𝑘+𝑚−𝑖+1 ̸= 𝛾𝑘−𝑗+1 for all
𝑖 ∈ [1, 𝑘 + 𝑚] and 𝑗 ∈ [1, 𝑚]. Now,
𝛽𝑘+𝑚−𝑖+1 = 𝛾𝑘−𝑗+1 ⇔ 𝜆𝑖 + 𝑚 − 𝑖 = 𝜆†𝑗 + 𝑘 − 𝑗
(22)
⇔ 𝜆𝑖 − 𝑖 = 𝑘 + 1 − min{𝑙 ∈ [1, 𝑘 + 𝑚] | 𝑘 − 𝑗 ≥ 𝜆𝑙 } − 𝑗 .
Suppose that 𝑘 − 𝑗 ≥ 𝜆𝑖 . Then min{𝑙 | 𝑘 − 𝑗 ≥ 𝜆𝑙 } ≤ 𝑖 and we get from equation (22) the
estimate 𝜆𝑖 − 𝑖 ≥ 𝑘 + 1 − 𝑖 − 𝑗. This implies 𝜆𝑖 > 𝑘 − 𝑗, contradicting the assumption. On the
other hand, if 𝑘 − 𝑗 ≤ 𝜆𝑖 , we similarly deduce the estimate 𝜆𝑖 < 𝑘 − 𝑗, again a contradiction.
Hence, 𝛽𝑖 ̸= 𝛾𝑗 for all 𝑖, 𝑗.
cusp
The number of elements in ℱ𝑘,𝑚
equals the number of sub-partitions 𝜆 of (𝑘 𝑘+𝑚 ), and this
(︀2𝑘+𝑚)︀
cusp
number is equal to
. We have just seen that ℱ𝑘,𝑚
is contained in a single Lusztig family
𝑘
ℱ. Since the multiplicity of each entry
in
the
content
we
have just computed is equal to 1, it
(︀
)︀
cusp
follows from Lemma 6.15 that |ℱ| = 2𝑘+𝑚
.
Hence,
ℱ

𝑘,𝑚 = ℱ is a Lusztig family.
𝑘
The symbols in Lemma 6.18 are in fact the minimal representatives of cuspidal symbols.
Lemma 6.19. Suppose that 𝑆 ∈ Sy𝑁
c;𝑛 is a cuspidal symbol. Then 𝑛 = 𝑘(𝑘 + 𝑚) for some 𝑘
cusp
and 𝑆 ∈ ℱ𝑘,𝑚 .
Proof. Recall that 𝑆 cuspidal means that 𝑛𝑖 ≥ 𝑛𝑖+1 for 𝑖 = 0, 1, . . .. Since there is at least one
𝑖 such that 𝑛𝑖 ̸= 0, we have 𝑛0 ̸= 0. As in Lemma 6.15, let 𝑘 := 𝑁 − |{𝑖 | 𝑛𝑖 = 2}|. Because
𝑆 is cuspidal, the symbol 𝑆 ′ := 𝑆[−(𝑁 − 𝑘)] ∈ Sy𝑘c;𝑛 is well-defined. By definition of shift,
∑︀
the content of this symbol is equal to 2𝑘+𝑚−1
𝑥𝑖 . Equation (18) for 𝑆 ′ says that
𝑖=0
𝑛=

2𝑘+𝑚−1
∑︁

(︃ )︃

𝑚
2

2

𝑖 − 𝑘 − 𝑘(𝑚 − 1) −

𝑖=0

= 𝑘(𝑘 + 𝑚) .

cusp
by Lemma 6.18 and Theorem 6.11.
Hence, 𝑆 ∈ ℱ𝑘,𝑚


′

𝑁
′
For a symbol 𝑆 ∈ Sy𝑁
c;𝑛 as in (17) we define the symbol 𝑆 ∈ Syc;𝑛 for certain 𝑁 as follows.
Choose 𝑡 ≥ max{𝛽𝑁 +𝑚 , 𝛾𝑁 }. Note that 𝑡 ≥ 𝑚 since 𝑆 has at least 𝑁 + 𝑚 distinct entries.
Now, the first row of 𝑆 is the set {0, 1, . . . , 𝑡} r {𝑡 − 𝛾1 , . . . , 𝑡 − 𝛾𝑁 } and the second row is
−𝑚
{0, 1, . . . , 𝑡} r {𝑡 − 𝛽1 , . . . , 𝑡 − 𝛽𝑁 +𝑚 }. By [33, 22.8], the symbol 𝑆 belongs to Sy𝑡+1−𝑁
c;𝑛
and by [33, Lemma 22.18] we have 𝜋𝑆 ⊗ sgn𝑊 = 𝜋𝑆 .

Example 6.20. Let 𝜆 =

(︁

,

)︁

Sy3(1,1);4 (𝜆)

and (𝑐1 , 𝜅) = (1, 1). Recall from Example 6.10 that
(︂

)︂

0 1 3 5
=
∈ Sy3(1,1);4 .
0 1 3

G WYN B ELLAMY AND U LRICH T HIEL

31

Choosing 𝑡 = 5 we get
Sy3(1,1);4 (𝜆)

(︂

)︂

(︁
0 1 3
=
= Sy2(1,1);4
1 3

)︁

,

.

Indeed,
⎛
(︁

,

)︁

⎞

⊗ ⎝∅ ,
⏟

⎠=

(︁

,

)︁

.

⏞

= sgn𝑊

Theorem 6.21. There exists a cuspidal Lusztig family if and only if there is a 𝑘 > 0 such that
cusp
𝑛 = 𝑘(𝑘 + 𝑚). In this case there is a unique cuspidal family, and it is equal to ℱ𝑘,𝑚
.
Proof. Because of the transitivity of j-induction, Lusztig families of 𝐵𝑛 induced from some
parabolic subgroup are also induced from some maximal parabolic subgroup. These subgroups
are all of the form 𝐵𝑙 × S𝑛−𝑙 for some 0 ≤ 𝑙 ≤ 𝑛 − 1. The restriction of the parameter c to
S𝑛−𝑙 is equal to 𝜅 > 0 and so the Lusztig families of S𝑛−𝑙 are singletons by §5. A Lusztig
family of 𝐵𝑙 × S𝑛−𝑙 is thus of the form {𝜋𝑆  𝜋𝜆 | 𝜋𝑆 ∈ ℱ} for some Lusztig family ℱ
of 𝐵𝑙 and some fixed 𝜆 ∈ 𝒫(𝑛 − 𝑙). Since any given number can only appear at most twice
in 𝑆, either the set of 𝑛 − 𝑙 largest entries in 𝑆 is well-defined or there is a choice of two
possible "largest 𝑛 − 𝑙-entries". Notice that this depends only on the content of 𝑆. By adding
1 to the 𝑛 − 𝑙 largest entries, we get either a new symbol 𝑆 ′ or two new symbols 𝑆 𝐼 and
𝑛
𝑆 𝐼𝐼 . Then, as explained in [33, Section 22.15], j𝐵
𝐵𝑙 ×S𝑛−𝑙 𝜋𝑆  sgn𝑊 equals 𝜋𝑆 ′ or 𝜋𝑆 𝐼 ⊕ 𝜋𝑆 𝐼𝐼 .
In the latter case, the j-induction of 𝜋𝑆  𝜋𝜆 is not irreducible and so j-induction
does not
∑︀
induce a Lusztig family of 𝐵𝑛 . We thus assume we are in the former case. Let 𝑖≥0 𝑛𝑖 𝑥𝑖 be
the content of 𝑆 ′ . Here 𝑛𝑖 ∈ {0, 1, 2} and 𝑛𝑖 = 0 for 𝑖 ≫ 0. Assume that there exists some
𝑖 such that 𝑛𝑖 > 𝑛𝑖−1 . There are two possibilities, either 𝑛𝑖−1 = 0 or (𝑛𝑖−1 , 𝑛𝑖 ) = (1, 2).
Consider first the former. We let 𝑙 be defined such that the 𝑛 − 𝑙 largest numbers in 𝑆 are
{𝑛𝑖 · 𝑖, 𝑛𝑖+1 · (𝑖 + 1), . . .}. Here, 𝑛𝑖 · 𝑖 means that 𝑖 occurs with multiplicity 𝑛𝑖 . Since 𝑛𝑖−1 = 0,
we can remove 1 from each of the 𝑛 − 𝑙 largest numbers and still have a well-defined symbol 𝑆 ′′ .
𝑛
Moreover, j𝐵
𝐵𝑙 ×S𝑛−𝑙 𝜋𝑆 ′′ ⊗ sgn𝑊 = 𝜋𝑆 ′ . This applies to all symbols in the family to which 𝑆
belongs. Hence this family is not cuspidal. The other case is where (𝑛𝑖−1 , 𝑛𝑖 ) = (1,∑︀
2). In this
case it suffices to show that 𝜋𝑆 = 𝜋𝑆 ⊗ sgn𝑊 is not cuspidal. If the content of 𝑆 is 𝑖≥0 𝑛𝑖 𝑥𝑖 ,
∑︀
then the content of 𝑆 equals 𝑖≥0 (2 − 𝑛𝑡−𝑖 )𝑥𝑖 for some 𝑡 ≫ 0. Thus, there exists some 𝑗 such
that (𝑛𝑗−1 , 𝑛𝑗 ) equals (0, 1) 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 𝑛𝑖 > 𝑛𝑖−1 . When 𝑛𝑖 ≤ 𝑛𝑖−1 for all 𝑖,
cusp
Lemma 6.19 implies that 𝜋𝑆 belongs to the family ℱ𝑘,𝑚
for some 𝑘 with 𝑛 = 𝑘(𝑘 + 𝑚). If ℱ ′
is a family in Irr(𝐵𝑙 × S𝑛−𝑙 ) for some 𝑙 < 𝑛, then Lemma 6.15 implies that |ℱ ′ | < |ℱ|. Hence
ℱ cannot be induced and must be cuspidal.

§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 families
deg
are labeled ℱ𝑖;𝑛
for 𝑖 = 0, . . . , 𝑛.
∼

deg
deg
Lemma 6.22. For any 𝑖, tensoring with the sign character yields a bijection ℱ𝑖;𝑛
−→ ℱ𝑛−𝑖;𝑛
.

Proof. If 𝜆 = (𝜆(0) , 𝜆(1) ) ∈ 𝒫2 (𝑛), then 𝜋𝜆 ⊗ sgn𝑊 = 𝜋((𝜆(1) )* ,(𝜆(0) )* ) by [23, Theorem
deg
deg
5.5.6(c)]. Hence, if 𝜋𝜆 ∈ ℱ𝑖;𝑛
, then 𝜋𝜆 ⊗ sgn𝑊 ∈ ℱ𝑛−𝑖;𝑛
. As sgn𝑊 is an automorphism on
Irr 𝑊 , the claim follows.


32

C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

Proposition 6.23. If 𝜅 = 0 but 𝑐1 ̸= 0, there are no cuspidal Lusztig c-families.
deg
𝑛
Proof. We claim that j𝐵
𝐵𝑖 ×S𝑛−𝑖 induces a bijection between the Lusztig family ℱ𝑖;𝑖 × Irr S𝑛−𝑖
deg
deg
of the parabolic subgroup 𝐵𝑖 × S𝑛−𝑖 of 𝐵𝑛 and ℱ𝑖;𝑛
for 0 ≤ 𝑖 < 𝑛. This shows that all ℱ𝑖;𝑛
deg
deg by Lemma 6.22, this proves the claim.
with 𝑖 < 𝑛 are induced, and since ℱ0;𝑛
⊗ sgn𝑊 = ℱ𝑛;𝑛
So, assume that 0 ≤ 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 ∈ 𝒫2 (𝑖) is a𝜇 = 𝑐1 |𝜇(1) |. Moreover, in the degenerate
case, the restriction of the parameter to S𝑛−𝑖 is zero so that the a-invariants 𝑎𝜈 are zero for all
𝜈 ∈ 𝒫(𝑛 − 𝑖) by Example 2.1. Hence,
𝑛
j𝐵
𝐵𝑖 ×S𝑛−𝑖 𝜋𝜇  𝜋𝜈 =

∑︁

𝑛
⟨Ind 𝐵
𝐵𝑖 ×S𝑛−𝑖 𝜋𝜇  𝜋𝜈 , 𝜋𝜆 ⟩𝜋𝜆

𝜆∈𝒫2 (𝑛)
|𝜆(1) |=|𝜇(1) |
deg
for 𝜇 ∈ 𝒫2 (𝑖) and 𝜈 ∈ 𝒫(𝑛 − 𝑖). We will show that if (𝜇, 𝜈) ∈ ℱ𝑖;𝑖
× Irr S𝑛−𝑖 , then
𝑛
(23)
j𝐵
𝐵𝑖 ×S𝑛−𝑖 𝜋𝜇  𝜋𝜈 = 𝜋(𝜈,𝜇(1) ) .
From this equation, the claim follows immediately. Since S𝑛−𝑖 is a subgroup of 𝐵𝑛−𝑖 , we get
the following relation using the branching rules:

𝐵 ×𝐵

𝐵𝑛
𝑖
𝑛−𝑖
𝑛
Ind 𝐵
𝐵𝑖 ×S𝑛−𝑖 𝜋𝜇  𝜋𝜈 = Ind 𝐵𝑖 ×𝐵𝑛−𝑖 Ind 𝐵𝑖 ×S𝑛 𝜋𝜇  𝜋𝜈
𝑖

=

𝑛
Ind 𝐵
𝐵𝑖 ×𝐵𝑛−𝑖

(︁

𝐵

𝜋𝜇  Ind S𝑛−𝑖
𝜋
𝑛−𝑖 𝜈

)︁

⎛

⎞

𝑛
⎝
= Ind 𝐵
𝐵𝑖 ×𝐵𝑛−𝑖 𝜋𝜇 

∑︁

𝑐𝜈𝛼 𝜋𝛼 ⎠

𝛼∈𝒫2 (𝑛−𝑖)

∑︁

=

∑︁

(0)

(1)

𝑐𝜆𝜇(0) ,𝛼(0) 𝑐𝜆𝜇(1) ,𝛼(1) 𝑐𝜈𝛼 𝜋𝜆 .

𝜆∈𝒫2 (𝑛) 𝛼∈𝒫2 (𝑛−𝑖)

Note that the sum runs only over those 𝛼 with 𝛼(0) , 𝛼(1) ⊆ 𝜈 and |𝜈| = |𝛼(0) | + |𝛼(1) |, and
similarly only over those 𝜆 which satisfy
(24)
𝜇(𝑗) , 𝛼(𝑗) ⊆ 𝜆(𝑗)
and
(25)
|𝜆(𝑗) | = |𝜇(𝑗) | + |𝛼(𝑗) |
for 𝑗 ∈ {1, 2}. To show (23) we need to show that among those 𝜆 ∈ 𝒫2 (𝑛) occurring in this
sum such that |𝜆(1) | = |𝜇(1) | we have
(0)
(1)
𝑐𝜆𝜇(0) ,𝛼(0) 𝑐𝜆𝜇(1) ,𝛼(1) 𝑐𝜈𝛼
(0)

{︃

=

1 if 𝜆 = (𝜈, 𝜇(1) ), 𝜇 = (∅, 𝜇(1) ), 𝛼 = (𝜈, ∅)
0 else.

(1)

So, suppose that 𝑐𝜆𝜇(0) ,𝛼(0) 𝑐𝜆𝜇(1) ,𝛼(1) 𝑐𝜈𝛼 ̸= 0. By (24) we have 𝜇(1) ⊆ 𝜆(1) , which implies that
(1)

(1)

𝜇𝑘 ≤ 𝜆𝑘 for all 𝑘. Hence, as |𝜆(1) | = |𝜇(1) | by assumption, we must have 𝜇(1) = 𝜆(1) . In
combination with (25) this shows that 𝛼(1) = ∅.
By definition of the Littlewood–Richardson coefficients, the coefficient 𝑐𝜈𝛼 = 𝑐𝜈𝛼(0) ,∅ is equal
to the coefficient of the symmetric polynomial 𝑠𝜈 in the product 𝑠𝛼(0) · 𝑠∅ = 𝑠𝛼(0) · 1 = 𝑠𝛼(0) .
Hence,
{︃
1 if 𝜈 = 𝛼(0)
𝜈
𝑐𝛼 =
0 else
and therefore we must have 𝜈 = 𝛼(0) . With the same argumentation we see that
(1)

(1)

𝑐𝜆𝜇(1) ,𝛼(1) = 𝑐𝜆𝜆(1) ,∅ = 1 .

G WYN B ELLAMY AND U LRICH T HIEL

33

deg
Since 𝜇 ∈ ℱ𝑖;𝑖
, we have 𝜇(0) = ∅. Hence,
(0)
𝑐𝜆𝜇(0) ,𝛼(0)

=

(0)
𝑐𝜆∅,𝜈

{︃

=

1 if 𝜆(0) = 𝜈
0 else

so that 𝜆(0) = 𝜈. This proves the claim.



§6O. Rigid modules
We will now show that rigid modules exist precisely in the cuspidal cases and describe them
explicitly. In this section, we consider again an arbitrary complex parameter c.
Theorem 6.24. There is a rigid Hc (𝑊 )-module if and only if c is cuspidal, i.e., 𝑐1 = 𝑚𝜅 for
some 𝑚 ∈ ±[0, 𝑛 − 1] with 𝑛 = 𝑘(𝑘 + 𝑚) for some 𝑘 > 0. In this case there are exactly two
rigid Hc (𝑊 )-modules, namely 𝐿c (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 Hc (𝑊 )-module
if c is cuspidal. So, assume that 𝑐1 = 𝑚𝜅 for some 𝑚 ∈ ±[0, 𝑛 − 1] with 𝑛 = 𝑘(𝑘 + 𝑚) for
some 𝑘 > 0. By Proposition 6.1, we can assume that 𝑚 ∈ [0, 𝑛 − 1]. Let 𝜆 = (𝜆(0) , 𝜆(1) ) be a
bipartition of 𝑛. By Lemma 4.10 and equations (13) and (14) the representation 𝜋𝜆 is c-rigid if
and only if
(26)

2𝑚𝜀𝑖 (−1) · 𝑣 +

𝑛
∑︁

(𝑠𝑖𝑗 + 𝑠𝑖𝑗,−1 ) · 𝑣 = 0 ,

∀ 𝑣 ∈ 𝜋𝜆 , 𝑖 = 1, . . . , 𝑛 ,

𝑗=1
𝑗̸=𝑖

and
(27)

(𝑠𝑖𝑗 − 𝑠𝑖𝑗,−1 ) · 𝑣 = 0 ,

∀ 𝑣 ∈ 𝜋𝜆 , 𝑖 ̸= 𝑗 .

|𝜆(0) |. Take 𝑣

Let 𝑟 :=
to be a non-zero vector in the irreducible (𝐵𝑟 ×𝐵𝑛−𝑟 )-subrepresentation
𝜋𝜆(0) ⊗ 𝛾𝜋𝜆(1) inducing 𝜋𝜆 ; see equation (12). Suppose that 𝑟 ∈
/ {0, 𝑛}. Then we can find 𝑖 < 𝑗
with 𝑖 ≤ 𝑟 and 𝑟 < 𝑗. Due to this choice, we have 𝜀𝑖 (−1) · 𝑣 = 𝑣 and 𝜀𝑗 (−1) · 𝑣 = −𝑣 as we
twist by 𝛾 in the second component. Hence,
𝑠𝑖𝑗,−1 · 𝑣 = 𝑠𝑖𝑗 𝜀𝑖 (−1)𝜀𝑗 (−1) · 𝑣 = −𝑠𝑖𝑗 · 𝑣 .
Equation (27) thus says that 2𝑠𝑖𝑗 · 𝑣 = 0 and therefore already 𝑣 = 0. This is a contradiction, so
we must have 𝑟 ∈ {0, 𝑛}. Assume that 𝑟 = 𝑛. Then 𝜋𝜆 = 𝜋𝜆(0) . Now, equation (27) says that
𝑛
∑︁

𝑠𝑖𝑗 · 𝑣 = −𝑚𝑣 ,

∀ 𝑣 ∈ 𝜋𝜆(0) , 𝑖 = 1, . . . 𝑛 .

𝑗=1
𝑗̸=𝑖

∑︀

In other words, 𝑗̸=𝑖 𝑠𝑖𝑗 acts by a scalar on 𝜋𝜆(0) . This holds in particular for 𝑖 = 1, and
now a standard result (see [17, Lemma 2.4]) implies that 𝜆(0) = (𝑙𝑏 ) is a rectangle
for some
∑︀
positive integers 𝑙, 𝑏 with 𝑙𝑏 = 𝑛. The 𝑛-th Jucys–Murphy element 𝑧𝑛 = 𝑗<𝑛 𝑠𝑗𝑛 acts on
𝜋(𝑙𝑏 ) by multiplication by 𝑙 − 𝑏 since every standard tableaux on (𝑎𝑏 ) must have 𝑛 in the top
corner. Thus, 𝑙𝑏 = 𝑛 and 𝑙 + 𝑚 = 𝑏, so 𝑛 = 𝑙(𝑙 + 𝑚). Because of Remark 6.4 we must
have 𝑙 = 𝑘, proving the claim. If 𝑟 = 0, then 𝜋𝜆 = 𝛾𝜋𝜆(1) and the same argument shows that
𝜆(1) = ((𝑘 + 𝑚)𝑘 ).

Remark 6.25. The proof of Theorem 6.24 can be adapted to all the groups 𝐺(ℓ, 1, 𝑛) = Zℓ ≀S𝑛 .
§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 𝐵𝑛 .
Theorem 6.26. Assume that c ≥ 0. Then

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C 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 if
it 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
cusp
(unique) cuspidal family ℱ𝑘,𝑚
.

§7. Type 𝐷
The group 𝐷𝑛 is a normal subgroup of 𝐵𝑛 of index two. By setting c = (0, 𝜅), we get an
embedding Hc (𝐷𝑛 ) ˓→ Hc (𝐵𝑛 ). Thus, we are in the situation of section §4C. We assume that
𝜅 ̸= 0. Recall that the irreducible representations of 𝐷𝑛 are essentially given by unordered
bipartitions 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 𝐷𝑛 -module. If 𝜆 = 𝜇, then there are two
non-isomorphic simple modules {𝜆}1 and {𝜆}2 labeled by 𝜆. These modules are defined by
𝜋(𝜆,𝜇) |𝐷𝑛 = 𝜋{𝜆,𝜇} for 𝜆 ̸= 𝜇 ,
and 𝜋(𝜆,𝜆) |𝐷𝑛 = 𝜋{𝜆}1 ⊕ 𝜋{𝜆}2 .
Lemma 7.1. If there exists 𝑘 such that 𝑛 = 𝑘 2 then there is a unique rigid Hc (𝐷𝑛 )-module,
which is 𝐿c ({(𝑘 𝑘 ), ∅}). Otherwise, there are no rigid modules.
Proof. By Theorem 6.24, if 𝑛 = 𝑘 2 for some 𝑘, then the modules 𝐿c ((𝑘 𝑘 ), ∅) and 𝐿c (∅, (𝑘 𝑘 ))
are the two rigid modules for Hc (𝐵𝑛 ) and if there exists no 𝑘 such that 𝑛 = 𝑘 2 , then there
exists no rigid modules. Therefore, Proposition 4.22 implies that, in this latter case, there exist
no rigid modules for Hc (𝐷𝑛 ). Moreover, in the case 𝑛 = 𝑘 2 , the rigid Hc (𝐷𝑛 )-modules are
precisely the modules of the form 𝐿c (𝜆), where 𝜋𝜆 is an irreducible 𝐷𝑛 -submodule of 𝜋((𝑘𝑘 ),∅)
or 𝜋(∅,(𝑘𝑘 )) . But both of these 𝐵𝑛 -modules restrict to the irreducible 𝐷𝑛 -module 𝜋{(𝑘𝑘 ),∅} . 
Theorem 7.2. Assume that 𝜅 ̸= 0. The symplectic leaves of Xc (𝐷𝑛 ) are in bijection with the
set {𝑘 ≥ 1 | 𝑘 2 ≤ 𝑛}, such that
(a) dim ℒ𝑘 = 2(𝑛 − 𝑘 2 ),
(b) the leaf ℒ𝑘 is labeled by the conjugacy class of the parabolic subgroup 𝐷𝑘2 ,
(c) ℒ𝑘 ≺ ℒ𝑘′ if and only if (𝐷𝑘2 ) ≤ (𝐷(𝑘′ )2 ), if and only if 𝑘 ≥ 𝑘 ′ .
Proof. By Lemma 7.1 and Theorem C, there exists at least one zero-dimensional leaf in Xc (𝐷𝑛 )
when 𝑛 = 𝑘 2 . But we know that there is exactly one zero-dimensional leaf in Xc (𝐵𝑛 ) when
𝑛 = 𝑘 2 . Thus, Lemma 4.18 implies that Xc (𝐷𝑛 ) contains exactly one zero-dimensional leaf
when 𝑛 = 𝑘 2 . Since Xc (𝐵𝑛 ) contains no zero-dimensional leaves when 𝑛 ̸= 𝑘 2 , Lemma 4.18
implies that Xc (𝐷𝑛 ) also contains no zero-dimensional leaves in this case.
Now, we apply Theorem 4.13. By Lemma 4.6, the proper parabolic subgroups of 𝐷𝑛 are all
conjugate to a subgroup of the form 𝐷𝑚 × S𝜆 , where 0 ≤ 𝑚 < 𝑛 and 𝜆 is a partition of 𝑛 − 𝑚.
We denote by c′ the restriction of c to 𝐷𝑚 × S𝜆 . Let us consider when Xc′ (𝐷𝑚 × S𝜆 ) =
Xc′ (𝐷𝑚 ) × Xc (S𝜆 ) admits a zero dimensional leaf. Since 𝜅 ̸= 0, there is a zero-dimensional
leaf in Xc′ (S𝜆 ) if and only if S𝜆 = {1}, i.e. if 𝜆 = (1𝑛−𝑚 ). In this case Xc (S𝜆 ) is a point.
Moreover, Xc′ (𝐷𝑚 ) has a zero-dimensional leaf if and only if there exists a 𝑘 such that 𝑚 = 𝑘 2 .
Thus, either 𝑚 = 𝑘 2 and 𝜆 = (1𝑛−𝑚 ), in which case there is exactly one zero-dimensional leaf
in Xc′ (𝐷𝑚 × S𝜆 ), or there are no zero-dimensional leaves in Xc′ (𝐷𝑚 × S𝜆 ). Hence Theorem
4.13 implies the statements of the theorem.

By [4, Corollary 6.10], the Calogero–Moser families for Hc (𝐷𝑛 ), with c ̸= 0, are described as follows. If 𝜆 is a partition of 𝑛/2, then the two representations {𝜆}1 and {𝜆}2 each
form 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 ≥ 0.

G WYN B ELLAMY AND U LRICH T HIEL

35

(a) The Lusztig c-families for 𝐷𝑛 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 (𝑐1 , 𝜅) = (0, 1).
The first part of the theorem follows from Corollary 6.13, [33, Section 22.26], and the above
description of the Calogero–Moser families. As shown in [32, Section 8.1], there is a unique
cuspidal Lusztig family when 𝑛 = 𝑘 2 and none otherwise. In the case 𝑛 = 𝑘 2 , it is the unique
family containing the symbol
(︂
)︂
0, 2, . . . , 2𝑘 − 2
(28)
.
1, 3, . . . , 2𝑘 − 1
The content of this symbol is

∑︀2𝑘−1 𝑖
𝑖=0 𝑥 , which is the same as the content of the symbol
)︂
(︂

𝑆=

𝑘, 𝑘 + 1, . . . , 2𝑘 − 1
0,
1,
..., 𝑘 − 1

.

This is the symbol of ((𝑘 𝑘 ), ∅) in Sy𝑘(1,0);𝑛 , which implies that the cuspidal Lusztig family
corresponding to the content of the symbol (28) is the same as the Calogero–Moser family containing {(𝑘 𝑘 ), ∅}. By Lemma 7.1 and Theorem 7.2, this is the unique cuspidal Calogero–Moser
family.


§8. Type 𝐼2 (𝑚)
In this section we treat the case of dihedral groups. We show that almost all representations
of the restricted rational Cherednik algebra are rigid. From this we easily obtain the proof of
Theorem A. We note that the results here together with [42, Appendix B] give a complete
description of the representation theory of restricted rational Cherednik algebras for dihedral
groups at all parameters.
§8A. The group
Throughout, we assume that 𝑚 ≥ 5 and choose a primitive 𝑚-th root of unity 𝜁 ∈ C. Let 𝑊
be the Coxeter group of type 𝐼2 (𝑚). This is the dihedral group of order 2𝑚. It has two natural
presentations, namely the Coxeter presentation ⟨𝑠, 𝑡 | 𝑠2 = 𝑡2 = (𝑠𝑡)𝑚 = 1⟩ and the geometric
presentation ⟨𝑠, 𝑟 | 𝑟𝑚 = 1, 𝑠2 = 1, 𝑠−1 𝑟𝑠 = 𝑟−1 ⟩ with a generating rotation 𝑟 := 𝑠𝑡 for the
symmetries of a regular 𝑚-gon.
§8B. Representations
The representation theory of 𝑊 depends on the parity of 𝑚. In the following we use the same
notation 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
{1}, {𝑟±1 }, {𝑟±2 }, . . . , {𝑟±(𝑚−1)/2 }, {𝑟𝑙 𝑠 | 0 ≤ 𝑙 ≤ 𝑚 − 1} ,
and so the total number of conjugacy classes is (𝑚 + 3)/2. There are two irreducible onedimensional representations: the trivial one 1𝑊 and the sign representation 𝜀 : 𝑊 → C
with
𝜀(𝑠) = −1 , 𝜀(𝑡) = −1 , 𝜀(𝑟) = 1 .
The remaining (𝑚 + 3)/2 − 2 = (𝑚 − 1)/2 irreducible representations 𝜙𝑖 , 1 ≤ 𝑖 ≤ (𝑚 − 1)/2,
are all two-dimensional and are given by
)︂
(︂
)︂
(︂
)︂
(︂ 𝑖
0 1
0 𝜁 −𝑖
𝜁
0
𝜙𝑖 (𝑠) =
, 𝜙𝑖 (𝑡) := 𝑖
, 𝜙𝑖 (𝑟) =
.
1 0
𝜁
0
0 𝜁 −𝑖
We denote the character of 𝜙𝑖 by 𝜒𝑖 .

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C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

If 𝑚 is even, then the conjugacy classes of 𝑊 are
{1} , {𝑟±1 } , {𝑟±2 } , . . . , {𝑟±𝑚/2 } ,
{𝑟2𝑘 𝑠 | 0 ≤ 𝑘 ≤ (𝑚/2) − 1} , {𝑟2𝑘+1 𝑠 | 0 ≤ 𝑘 ≤ (𝑚/2) − 1} ,
and so the total number of conjugacy classes is (𝑚 + 6)/2. There are four irreducible onedimensional representations: the trivial one 1𝑊 , the sign representation 𝜀, and two further
representations 𝜀1 , 𝜀2 with
𝜀(𝑠) = −1 , 𝜀(𝑡) = −1 , 𝜀(𝑟) = 1 ,
𝜀1 (𝑠) = 1 ,
𝜀1 (𝑡) = −1 , 𝜀1 (𝑟) = −1 ,
𝜀2 (𝑠) = −1 , 𝜀2 (𝑡) = 1 ,
𝜀2 (𝑟) = −1 .
The remaining (𝑚 + 6)/2 − 4 = (𝑚 − 2)/2 irreducible representations 𝜙𝑖 , 1 ≤ 𝑖 ≤ (𝑚 − 2)/2,
are all two-dimensional and are defined as in case 𝑚 is odd. Again, we denote the character of
𝜙𝑖 by 𝜒𝑖 .
§8C. Reflections and parameters
The two-dimensional faithful irreducible representation 𝜙1 of 𝑊 is a reflection representation
in which precisely the elements 𝑠𝑙 := 𝑟𝑙 𝑠 for 0 ≤ 𝑙 ≤ 𝑚 − 1 act as reflections. We will always
fix this representation as the reflection representation of 𝑊 . Let (𝑦1 , 𝑦2 ) be the standard basis
of h := C2 and let (𝑥1 , 𝑥2 ) be the dual basis. We can easily verify that roots and coroots for the
reflections 𝑠𝑙 are given by
𝛼𝑠𝑙 = 𝑥1 − 𝜁 −𝑙 𝑥2 and 𝛼𝑠∨𝑙 = 𝑦1 − 𝜁𝑦2 .
With this we see that the Cherednik coefficients (𝑦𝑖 , 𝑥𝑗 )𝑠𝑙 = −(𝑦𝑖 , 𝛼𝑠𝑙 )(𝛼𝑠∨𝑙 , 𝑥𝑗 ) are:
(𝑦1 , 𝑥1 )𝑠𝑙 = −1 , (𝑦1 , 𝑥2 )𝑠𝑙 = 𝜁 −𝑙 , (𝑦2 , 𝑥1 )𝑠𝑙 = 𝜁 𝑙 , (𝑦2 , 𝑥2 )𝑠𝑙 = −1 .
If 𝑚 is odd, there is just one conjugacy class of reflections in 𝑊 , namely the one of 𝑠 which is
{𝑠𝑙 | 0 ≤ 𝑙 ≤ 𝑚 − 1}. If 𝑚 is even, there are two conjugacy classes of reflections in 𝑊 , namely
𝑚
the one of 𝑠 which is {𝑠2𝑙 | 0 ≤ 𝑙 ≤ 𝑚
2 − 1} and the one of 𝑡 which is {𝑠2𝑙+1 | 0 ≤ 𝑙 ≤ 2 − 1}.
Note that
(︃
)︃
0 𝜁 𝑖𝑙
𝜙𝑖 (𝑠𝑙 ) =
.
𝜁 −𝑖𝑙 0
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 families
and rigid representations. To simplify notation we denote by ℱ the set of two-dimensional
irreducible characters of 𝑊 . To allow a presentation which is independent of the parity of 𝑚
we set
{︂
{𝜙𝑖 | 1 < 𝑖 ≤ (𝑚 − 1)/2} = ℱ ∖ {𝜙1 }
if 𝑚 is odd
ℛ :=
{𝜙𝑖 | 1 < 𝑖 < (𝑚 − 2)/2} = ℱ ∖ {𝜙1 , 𝜙(𝑚−2)/2 } if 𝑚 is even.
We make the convention that we ignore 𝜀1 and 𝜀2 whenever 𝑚 is odd.
Theorem 8.1. The (cuspidal) Calogero–Moser families and rigid representations of Hc (𝑊 ) are
as listed in Table 2.
In the next three sections we will prove this theorem.

G WYN B ELLAMY AND U LRICH T HIEL

37

Parameters

CM families

𝑎, 𝑏 ̸= 0 and 𝑎 ̸= ±𝑏 {1}, {𝜀}, {𝜀1 }, {𝜀2 }, ℱ
𝑎 = 0 and 𝑏 ̸= 0
{1, 𝜀2 }, {𝜀, 𝜀1 }, ℱ
𝑎 ̸= 0 and 𝑏 = 0
{1, 𝜀1 }, {𝜀, 𝜀2 }, ℱ
𝑎 = 𝑏 ̸= 0
{1}, {𝜀}, {𝜀1 , 𝜀2 } ∪ ℱ
𝑎 = −𝑏 ̸= 0
{𝜀1 }, {𝜀2 }, {1, 𝜀} ∪ ℱ

rigid
cuspidal
representations CM families
ℛ
ℱ
ℛ
ℱ
ℛ
ℱ
𝜀1 , 𝜀2 , 𝜙|ℱ | , ℛ {𝜀1 , 𝜀2 } ∪ ℱ
1, 𝜀, 𝜙1 , ℛ
{1, 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 Hc (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 these
families is given: two irreducible characters 𝜆 and 𝜇 of 𝑊 lie in the same Euler c-family if and
only if
(︂
)︂
∑︁
𝜆(𝑥) 𝜇(𝑥)
c(𝑥)
−
=0.
𝜆(1)
𝜇(1)
𝑥∈Ref(𝑊 )
This formula is in our case equivalent to
(︂
)︂
(︂
)︂
𝜆(𝑠) 𝜇(𝑠)
𝜇(𝑡)
𝜆(𝑡)
𝑎
−
+𝑏
−
=0.
𝜆(1) 𝜇(1)
𝜆(1) 𝜇(1)
From this it is easy to deduce that the Euler families are as in Table 2. In [3] the first author has
proven 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 the
parity of 𝑚.
Proposition 8.2. Assume that 𝑚 is odd. The following holds:
(a) The representations 1, 𝜀, and 𝜙1 are not rigid.
(b) The representation 𝜙𝑖 is a rigid representation for all 1 < 𝑖 ≤ (𝑚 − 1)/2.
Proof. The representation 𝜙𝑖 for 1 ≤ 𝑖 ≤ (𝑚 − 1)/2 is rigid if and only if
0=𝑎

𝑚−1
∑︁

(𝑦𝑘 , 𝑥𝑗 )𝑠𝑙 𝜙𝑖 (𝑠𝑙 )

𝑙=0

for all 𝑘, 𝑗 ∈ {1, 2}. As 𝑎 ̸= 0, this is equivalent to
𝑚−1
∑︁

(𝑦𝑘 , 𝑥𝑗 )𝑠𝑙 𝜙𝑖 (𝑠𝑙 ) = 0

(29)

𝑙=0

for all 𝑘, 𝑗 ∈ {1, 2}. Note that
𝑚−1
∑︁

(𝜁 𝑞 )𝑙 =

𝑚 if 𝑞 ∈ 𝑚Z
0 else.

{︂

𝑙=0

Using the Cherednik coefficients computed in §8C, equation (29) is for 𝑘 = 1 = 𝑗 and for
𝑘 = 2 = 𝑗 equivalent to
𝑚−1
∑︁
𝑙=0

(︃

0

𝜁 −𝑖𝑙

𝜁 𝑖𝑙
0

)︃

= 0 ⇐⇒

𝑚−1
∑︁

𝑚−1
∑︁

𝑙=0

𝑙=0

(𝜁 𝑖 )𝑙 = 0 and

(𝜁 −𝑖 )𝑙 = 0

38

C 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 ∈
/ 𝑚Z. Since 1 ≤ 𝑖 ≤
(𝑚 − 1)/2, this is always satisfied. For 𝑘 = 1 and 𝑗 = 2 equation (29) is equivalent to
𝑚−1
∑︁

(︃

𝜁

−𝑙

𝜁 −𝑖𝑙

𝑙=0

⇐⇒

0

𝑚−1
∑︁

𝜁 𝑖𝑙
0

)︃

= 0 ⇐⇒

(𝜁 𝑖−1 )𝑙 = 0 and

𝑚−1
∑︁

𝜁 𝑖𝑙−𝑙 = 0 and

𝑚−1
∑︁

𝑙=0
𝑚−1
∑︁

𝑙=0

𝜁 −𝑖𝑙−𝑙 = 0

𝑙=0

(𝜁 −𝑖−1 )𝑙 = 0

𝑙=0

⇐⇒ 𝑖 − 1 ∈
/ 𝑚Z and 𝑖 + 1 ∈
/ 𝑚Z .
Due to 1 ≤ 𝑖 ≤ (𝑚 − 1)/2 the condition 𝑖 + 1 ∈
/ 𝑚Z 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 equivalent
to
(︃
)︃
𝑚−1
𝑚−1
𝑚−1
∑︁
∑︁
∑︁
0 𝜁 𝑖𝑙
𝑙
𝑖𝑙+𝑙
𝜁
=
0
⇐⇒
𝜁
=
0
and
𝜁 −𝑖𝑙+𝑙 = 0
−𝑖𝑙
𝜁
0
𝑙=0

⇐⇒

𝑚−1
∑︁

𝑙=0
𝑚−1
∑︁

𝑙=0

𝑙=0

(𝜁 𝑖+1 )𝑙 = 0 and

𝑙=0

(𝜁 −𝑖+1 )𝑙 = 0

⇐⇒ 𝑖 + 1 ∈
/ 𝑚Z and 𝑖 − 1 ∈
/ 𝑚Z .
Again, the condition 𝑖 + 1 ∈
/ Z is always satisfied and 𝑖 − 1 ∈
/ 𝑚Z holds if and only if 𝑖 ̸= 1.
This proves the claim.

Proposition 8.3. Assume that 𝑚 is even. The following holds:
(a) For any 1 < 𝑖 < (𝑚 − 2)/2 the representation 𝜙𝑖 is rigid.
(b) The representation 𝜙1 is rigid if and only if 𝑎 = −𝑏.
(c) The representation 𝜙(𝑚−2)/2 is rigid if and only if 𝑎 = 𝑏.
(d) The representations 1 and 𝜀 are rigid if and only if 𝑎 = −𝑏.
(e) The representations 𝜀1 and 𝜀2 are rigid if and only if 𝑎 = 𝑏.
Proof. This follows from a similar direct computation as in the proof of Proposition 8.2. We
omit the details here.

§8G. Cuspidal Calogero–Moser families
For 𝑚 ≥ 5 the first author has shown in [2, §5.5] that independent of the parameter c there
is exactly one cuspidal Calogero–Moser family. It thus remains to identify this family. Since
𝑚 ≥ 5 we have ℛ =
̸ ∅, and as ℛ is always contained in the Calogero–Moser family which
in Table 2 is claimed to be cuspidal, it follows from Theorem C that this family is indeed the
unique cuspidal one.
§8H. Lusztig families
From now on we assume that c ≥ 0. The Lusztig families of 𝑊 are listed in Table 3 which is
taken from [22, 1.7.3].
Parameters
Lusztig families
𝑏=𝑎>0
{1𝑊 }, {𝜀}, {𝜀1 , 𝜀2 } ∪ ℱ
𝑏 > 𝑎 > 0 or 𝑎 > 𝑏 > 0 {1𝑊 }, {𝜀}, {𝜀1 }, {𝜀2 }, ℱ
𝑏>𝑎=0
{1𝑊 , 𝜀1 }, {𝜀, 𝜀2 }, ℱ
𝑎>𝑏=0
{1𝑊 , 𝜀2 }, {𝜀, 𝜀1 }, ℱ
Table 3. Lusztig families

G WYN B ELLAMY AND U LRICH T HIEL

39

Comparison with the Calogero–Moser families immediately yields the proof of Theorem 2.4
for dihedral groups:
Corollary 8.4. For any c ≥ 0 the Lusztig c-families are equal to the Calogero–Moser cfamilies.
§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 𝑃1 := ⟨𝑠⟩
and the group 𝑃2 := ⟨𝑡⟩, which are both Coxeter groups of type A1 . Let 𝜓𝑖 be the non-trivial
irreducible character of 𝑃𝑖 and note that this is the sign representation of this Coxeter group. It
is not hard to compute that
∑︁
∑︁
Ind 𝑊
𝜒𝑗 , Ind 𝑊
𝜒𝑗 + 𝛿1𝑊 ,
𝑃1 1𝑃1 = 1𝑊 + 𝜀1 +
𝑃1 𝜓1 = 𝜀 + 𝜀2 +
𝑗

Ind 𝑊
𝑃2 1𝑃2 = 1𝑊 + 𝜀2 +

𝑗

∑︁

𝜒𝑗 ,

Ind 𝑊
𝑃2 𝜓2 = 𝜀 + 𝜀1 +

𝑗

∑︁

𝜒 + 𝛿1𝑊 ,

𝑗

where
{︂

0 if 𝑚 is even
1 if 𝑚 is odd
and the sums are taken over all two-dimensional characters.
Lusztig’s a-functions a𝜒 of the irreducible characters 𝜒 of 𝑊 with respect to c are listed in
Table 4 which is taken from [22, 1.3.7], where the last row follows by symmetry. Using [22,
𝛿 :=

Parameters
𝑏=𝑎>0
𝑏>𝑎≥0
𝑎>𝑏≥0

𝜙𝑖 1𝑊
𝑎
0
𝑏
0
𝑎
0

𝜀
𝑚𝑎
𝑚
(𝑎
+ 𝑏)
2
𝑚
(𝑎
+ 𝑏)
2

𝜀1
𝑎
𝑎
𝑚
(𝑎
−
𝑏) + 𝑏
2

𝜀2
𝑎
𝑚
(𝑏
−
𝑎) + 𝑎
2
𝑏

Table 4. The a-function

1.3.3] we see that the a-functions for the irreducible characters of the parabolic subgroups with
respect to the restriction of c to these groups are as in Table 5. From these tables we can deduce
𝜒
a𝜒

1 𝑃1
0

𝜓1 1𝑃2
𝑏
0

𝜓2
𝑎

Table 5. The a-function for the parabolic subgroups 𝑃𝑖 .

that Lusztig’s j-induction is as in Table 6.
Parameters
𝑏=𝑎>0
𝑏>𝑎>0
𝑏>𝑎=0
𝑎>𝑏>0
𝑎>𝑏=0

j𝑊
j𝑊
𝑃1 1 𝑃1
𝑃1 𝜓1
∑︀
1𝑊
𝜀2 + 𝑗 𝜒 𝑗
∑︀
1𝑊
𝜒
∑︀𝑗 𝑗
1 𝑊 + 𝜀1
𝑗 𝜒𝑗
1𝑊
𝜀2
1𝑊
𝜀2

j𝑊
j𝑊
𝑃2 1𝑃2
𝑃2 𝜓2
∑︀
1𝑊
𝜀1 + 𝑗 𝜒 𝑗
1𝑊
𝜀1
1𝑊
𝜀
∑︀ 1
1𝑊
𝜒
∑︀𝑗 𝑗
1𝑊 + 𝜀2
𝑗 𝜒𝑗

Table 6. j-induction.

Using the table of j-inductions we can now easily determine the cuspidal Lusztig families.
Lemma 8.5. Let c ≥ 0. There is a unique cuspidal Lusztig family. This family is equal to
{𝜀1 , 𝜀2 } ∪ ℱ if 𝑏 = 𝑎, and otherwise it is equal to ℱ.

40

C USPIDAL C ALOGERO –M OSER AND L USZTIG FAMILIES FOR C OXETER GROUPS

Proof. The Lusztig families of the parabolic subgroup 𝑃𝑖 are {1𝑃𝑖 } and {𝜓𝑖 } if 𝑏 ̸= 0, respectively 𝑎 ̸= 0, and they are {1𝑃𝑖 , 𝜓𝑖 } if 𝑏 = 0, respectively 𝑎 = 0. The claim follows now easily
from the definition of cuspidality using the table of j-inductions.

Comparison with the cuspidal Calogero–Moser families completes the proof of Theorem A.

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]

[16]
[17]
[18]

[19]
[20]
[21]
[22]
[23]

[24]
[25]
[26]
[27]

H. Barcelo and E. Ihrig. Lattices of parabolic subgroups in connection with hyperplane arrangements. In: J.
Algebraic Combin. 9.1 (1999), pp. 5–24.
G. Bellamy. Cuspidal representations of rational Cherednik algebras at 𝑡 = 0. In: Math. Z. 269.3-4 (2008),
pp. 609–627.
G Bellamy. Generalized Calogero–Moser spaces and rational Cherednik algebras. PhD thesis. University of
Edinburgh, 2010.
G. Bellamy. The Calogero-Moser partition for 𝐺(𝑚, 𝑑, 𝑛). In: Nagoya Math. J. 207 (2012), pp. 47–77.
G. Bellamy and O. Chalykh. Motivic invariants of Calogero–Moser spaces. In preparation.
Y. Berest, P. Etingof, and V. Ginzburg. Finite-dimensional representations of rational Cherednik algebras. In:
Int. Math. Res. Not. 19 (2003), pp. 1053–1088.
R. Bezrukavnikov and P. Etingof. Parabolic induction and restriction functors for rational Cherednik
algebras. In: Selecta Math. (N.S.) 14.3-4 (2009), pp. 397–425.
C. Bonnafé. Constructible characters and 𝑏-invariants. In: arXiv:1302.2721 (2013).
C. Bonnafé and R. Rouquier. Cellules de Calogero-Moser. In: arXiv 1302.2720 (2013).
M. Broué and S. Kim. Familles de caractères des algèbres de Hecke cyclotomiques. In: Adv. Math. 172.1
(2002), pp. 53–136.
M. Broué, G. Malle, and R. Rouquier. Complex reflection groups, braid groups, Hecke algebras. In: J. Reine
Angew. Math. 500 (1998), pp. 127–190.
K. A. Brown and I. Gordon. Poisson orders, symplectic reflection algebras and representation theory. In: J.
Reine Angew. Math. 559 (2003), pp. 193–216.
K. A. Brown and I. G. Gordon. The ramification of centres: Lie algebras in positive characteristic and
quantised enveloping algebras. In: Math. Z. 238.4 (2001), pp. 733–779.
D. Ciubotaru. One-𝑊 -type modules for rational Cherednik algebra and cuspidal two-sided cells. In:
arXiv:1503.07890 (2015).
C. W. Curtis and I. Reiner. Methods of representation theory. Vol. I. With applications to finite groups and
orders, Pure and Applied Mathematics, A Wiley-Interscience Publication. New York: John Wiley & Sons
Inc., 1981, pp. xxi+819.
P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and deformed HarishChandra homomorphism. In: Invent. Math. 147.2 (2002), pp. 243–348.
P. Etingof and S. Montarani. Finite dimensional representations of symplectic reflection algebras associated
to wreath products. In: Represent. Theory 9 (2005), 457–467 (electronic).
P. Gabriel. Finite representation type is open. In: Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 10. Carleton Univ., Ottawa, Ont., 1974, 23
pp. Carleton Math. Lecture Notes, No. 9.
M. Geck. Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal
parameters. In: Represent. Theory 6 (2002), pp. 1–30.
M. Geck. Left cells and constructible representations. In: Represent. Theory 9 (2005), 385–416 (electronic).
M. Geck and L. Iancu. Ordering Lusztig’s families in type 𝐵𝑛 . In: J. Algebraic Combin. 38.2 (2013), pp. 457–
489.
M. Geck and N. Jacon. Representations of Hecke algebras at roots of unity. Vol. 15. Algebra and Applications.
London: Springer-Verlag London Ltd., 2011, pp. xii+401.
M. Geck and G. Pfeiffer. Characters of finite Coxeter groups and Iwahori-Hecke algebras. Vol. 21. London
Mathematical Society Monographs. New Series. New York: The Clarendon Press Oxford University Press,
2000, pp. xvi+446.
V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier. On the category 𝒪 for rational Cherednik algebras. In:
Invent. Math. 154.3 (2003), pp. 617–651.
I. G. Gordon. Baby Verma modules for rational Cherednik algebras. In: Bull. London Math. Soc. 35.3 (2003),
pp. 321–336.
I. G. Gordon and M. Martino. Calogero-Moser space, restricted rational Cherednik algebras and two-sided
cells. In: Math. Res. Lett. 16.2 (2009), pp. 255–262.
R. R. Holmes and D. K. Nakano. Brauer-type reciprocity for a class of graded associative algebras. In: J.
Algebra 144.1 (1991), pp. 117–126.

G WYN B ELLAMY AND U LRICH T HIEL
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]

41

B. Huppert. Character theory of finite groups. Vol. 25. de Gruyter Expositions in Mathematics. Walter de
Gruyter & Co., Berlin, 1998, pp. vi+618.
G. I. Lehrer and D. E. Taylor. Unitary reflection groups. Vol. 20. Australian Mathematical Society Lecture
Series. Cambridge: Cambridge University Press, 2009, pp. viii+294.
E. Liboz. Orderings on Calogero-Moser partition of imprimitive groups. In: J. Algebra 417 (2014), pp. 116–
144.
I. Losev. Completions of symplectic reflection algebras. In: Selecta Math. (N.S.) 18.1 (2012), pp. 179–251.
G. Lusztig. Characters of reductive groups over a finite field. Vol. 107. Annals of Mathematics Studies.
Princeton University Press, Princeton, NJ, 1984, pp. xxi+384.
G. Lusztig. Hecke algebras with unequal parameters. Vol. 18. CRM Monograph Series. Providence, RI:
American Mathematical Society, 2003, pp. vi+136.
M. Martino. Symplectic reflection algebras and Poisson geometry. PhD thesis. University of Glasgow, 2006.
M. Martino. Blocks of restricted rational Cherednik algebras for 𝐺(𝑚, 𝑑, 𝑛). In: J. Algebra 397 (2014),
pp. 209–224.
I. A. Pushkarev. On the representation theory of wreath products of finite groups and symmetric groups. In:
Journal of Mathematical Sciences 96.5 (1999), pp. 3590–3599.
G. C. Shephard and J. A. Todd. Finite unitary reflection groups. In: Canadian J. Math. 6 (1954), pp. 274–304.
R. Steinberg. Differential equations invariant under finite reflection groups. In: Trans. Amer. Math. Soc. 112
(1964), pp. 392–400.
D. E. Taylor. Reflection subgroups of finite complex reflection groups. In: J. Algebra 366 (2012), pp. 218–234.
U. Thiel. A counter-example to Martino’s conjecture about generic Calogero-Moser families. In: Algebr.
Represent. Theory 17.5 (2014), pp. 1323–1348.
U. Thiel. On restricted rational Cherednik algebras. Dissertation, TU Kaiserslautern. 2014.
U. Thiel. Restricted rational Cherednik algebras. In: Representation theory: Recent developments and
perspectives from a priority programme. To appear, see http://arxiv.org/abs/1603.05230. 2016.
A. Weinstein. The local structure of Poisson manifolds. In: J. Differential Geom. 18.3 (1983), pp. 523–557.

