Cherednik algebras and CalogeroMoser Cells
The second author was partly supported by the NSF (grant DMS1161999 and DMS1702305) and by a grant from the Simons Foundation (#376202)
Table of contents
Introduction
Reconstruction of Lietheoretic structures from Weyl groups and extension to complex reflection groups
A number of Lietheoretic questions have their answer in terms of the associated Weyl group. Our work is part of a program to reconstruct combinatorial and categorical structures arising in Lietheoretic representation theory from rational Cherednik algebras. Such algebras are associated by Etingof and Ginzburg to more general complex reflection groups, and an aspect of the program is to generalize those combinatorial and categorical structures to complex reflection groups, that will not arise from Lie theory in general.
To be more precise, consider a semisimple complex Lie algebra and let be its Weyl group. Consider also a reductive algebraic group over , with the Lie algebra of . Consider the following:

(Parabolic) Blocks of (deformed) category for , blocks of categories of HarishChandra bimodules.

The set of unipotent characters of , their generic degrees, Lusztig’s Fourier transform matrices.

Unipotent blocks of modular representations of over a field of characteristic prime to .

The Hecke algebra of .

Lattices in the Hecke algebra arising from the KazhdanLusztig basis, Lusztig’s asymptotic algebra .

KazhdanLusztig cells of and left cell representations, families of characters of .

Lusztig’s modular categories associated to twosided cells.
It is known or conjectured that the structures above depend only on , viewed as a reflection group. One can hope that (possibly derived) versions of those structures still make sense for a complex reflection group.
Consider the case where a real reflection group. A solution to (i) is provided by Soergel’s bimodules [Soe]. A solution to (ii) was found [BrMa, lusztigexotic, malleexotic]. The combinatorial theory in (v,vi) extends (partly conjecturally) to that setting. Categories as in (vii) were constructed by Lusztig when is a dihedral group [lusztigexotic].
The structures above might make sense for arbitrary ("unequal") parameters, and this is already an open problem for a Weyl group. A partly conjectural theory for (v,vi) has been developed by Lusztig [lusztig], who is developing an interpretation via character sheaves on disconnected groups [lusztigdisconnected].
Hecke algebras are a starting point: they have a topological definition that makes sense for complex reflection groups [BMR], providing a (conjectural) solution to (iv). The Hecke algebras are (conjecturally) deformations of over the space of class functions on supported on reflections.
For certain complex reflection groups ("spetsial"), a combinatorial set (a "spets") has been associated by Broué, Malle and Michel, that plays the role of unipotent characters, and providing an answer to (ii) [malleunipotent, BMM, BMM2]. Generic degrees are associated, building on Fourier transforms generalizing Lusztig’s constructions for Weyl groups. There are generalized induction and restriction functors, and a HarishChandra theory.
When is a cyclic group and for equal parameters, a solution to (vii) has been constructed in [BR]. It is a derived version of a modular category. It gives rise to the Fourier transform defined by Malle [malleunipotent].
In this book, we provide a conjectural extension of (vi) to complex reflection groups.
EtingofGinzburg’s rational Cherednik algebras and CalogeroMoser spaces
Consider a nontrivial finite group acting on a finitedimensional complex vector space and let the complement of the ramification locus of the quotient map . Assume is a reflection group, i.e., is generated by its set of reflections (equivalently: is smooth; equivalently: is an affine space). The quotient variety by the diagonal action of is singular. It is a ramified covering of .
Etingof and Ginzburg have constructed a deformation of this covering [EG]. Here, is a vector space with basis the quotient of by the conjugacy action of . The variety is the CalogeroMoser space. The original covering corresponds to the point .
Etingof and Ginzburg define as the spectrum of the center of the rational Cherednik associated with at . It is a remarkable feature of their work that those important but complicated CalogeroMoser spaces have an explicit description based on noncommutative algebra. We will now explain their constructions.
The rational Cherednik algebra associated to is a flat deformation defined by generators and relations of the algebra over the space of parameters . Its specialization at is the crossed product of the Weyl algebra of by . The Cherednik algebra has a triangular decomposition . Equivalently, it satisfies a PBW Theorem. The algebra has a faithful representation by Dunkl operators on .
Consider the algebra obtained by specializing at and let be its center. It contains as a subalgebra. The CalogeroMoser variety is defined as . The Satake Theorem asserts that multiplication by the averaging idempotent defines an isomorphism of algebras . As a consequence, the morphism is a flat deformation of over .
Galois closure and ramification
The triangular decomposition of the Cherednik algebra at leads to a construction of representation categories similar to that of enveloping algebras of complex semisimple Lie algebras.
The covering , of degree , is not Galois (unless ). Our work is a study of a Galois closure of this covering and of the ramification above the closed subvarieties , and of .
Let be the Galois group of . At , a Galois closure of the covering is given by . This leads to a realization of as a group of permutations of .
This can be reformulated in terms of representations of : the semisimple algebra is not split and is a splitting field. The simple modules are in bijection with . Our work can be viewed as the study of the partition of these modules into blocks corresponding to a given prime ideal of .
CalogeroMoser cells
Let be an irreducible closed subvariety of . We define the cells of as the orbits of the inertia group of . Given a parameter , we study the twosided cells, defined for an irreducible component of the inverse image of . We also study the left cells (where ) and the right cells (where ).
When is a Coxeter group, we conjecture that the cells coincide with the KazhdanLusztig cells, defined by KazhdanLusztig [KaLu] and Lusztig [Lus1], [Lus3]. This depends on the choice of an appropriate in a orbit.
We analyze in detail the case where is cyclic (and ): this is the only case where we have a complete description of the objects studied in this book. We also provide a detailed study of the case of a Weyl group of type : the Galois group is a Weyl group of type and we show that the CalogeroMoser cells coincide with the KazhdanLusztig cells. Our approach for is based on a detailed study of and of the ramification of the covering, without constructing explicitly the variety .
Etingof and Ginzburg have introduced a deformation of the Euler vector field. We show that is the Galois group of its minimal polynomial. This element plays an important role in the study of ramification, but is not enough to separate cells in general.
Families and cell representations
We construct a bijection between the set of twosided cells and the set of blocks of the restricted rational Cherednik algebra (the specialization of at ). This latter set is in bijection with . Given a simple module , there is an indecomposable representation of Verma type (a “babyVerma” module) of with a unique simple quotient [Gor1]. The partition into blocks of those modules gives a partition of into CalogeroMoser families, which are conjecturally related to the families of the Hecke algebra of (cf [Gor2, GoMa, Bel5, Mar1, Mar2]). We show that, in a given CalogeroMoser family, the matrix of multiplicities has rank , a property conjectured by Ulrich Thiel. Families satisfy a semicontinuity property with respected to specialization of the parameter. We show that families are minimal subsets that are unions of families for a generic parameter and on whose Verma modules the Euler element takes constant values. More precisely, this second statement is related to a set of hyperplanes of where the families change. Those hyperplanes are related to the affine hyperplanes where the category [ggor] for the specialization at of is not semisimple. These are, in turn, related to the components of the locus of parameters where the Hecke algebra of is not semisimple.
We introduce a notion of simple cell module associated with a left cell. We conjecture that the multiplicity of such a simple module in a Verma module is the same as the multiplicity of in the cell representation of KazhdanLusztig, when is a Coxeter group. We study twosided cells associated with a smooth point of : in that case, Gordon has shown that the corresponding block contains a unique Verma module. We show that the multiplicity of any simple cell module in that Verma module is (for a left cell contained in the given twosided cell).
There is a unique irreducible representation of with minimal invariant in each CalogeroMoser family. In Lusztig’s theory, this is a characterization of special representations (case of equal parameters). We show that in each left cell representation there is also a unique irreducible representation of with minimal invariant, and this can change inside a family. The corresponding result has been proved recently (case by case) in the setting of Lusztig’s theory by the first author [bonnafeb]. It is an instance of the CalogeroMoser theory shedding some light on the KazhdanLusztig and Lusztig theory.
Description of the chapters
We review in Chapter \thechapter the basic theory of complex reflection groups: invariant theory, rationality of representations, fake degrees. We close that chapter with the particular case of real reflection groups endowed with the choice of a real chamber, i.e., finite Coxeter groups. All along the book, we devote special sections to the case of Coxeter groups when particular features arise in their case.
Chapters \thechapter and LABEL:chapter:cherednik0 are devoted to the basic structure theory of rational Cherednik algebras, following Etingof and Ginzburg [EG]. The definition of generic Cherednik algebras is given in Chapter \thechapter, followed by the fundamental PBW Decomposition Theorem. We introduce the polynomial representation via Dunkl operators and prove its faithfulness. We introduce next the spherical algebra and prove some basic properties, including the double endomorphism Theorem. We also introduce the Euler element and gradings, filtrations, and automorphisms.
Chapter LABEL:chapter:cherednik0 is devoted to the Cherednik algebra at . An important result is the Satake isomorphism between the center of the Cherednik algebra and the spherical subalgebra. We discuss localizations and cases of Morita equivalence between the Cherednik algebra and its spherical subalgebra. We provide some complements: filtrations, symmetrizing form and Hilbert series.
Our original work starts in Part LABEL:part:extension: it deals with the covering and its ramification.
We introduce in Chapter LABEL:chapter:galoisCM some of our basic objects of study, namely the Galois closure of the covering and its Galois group. At parameter , the corresponding data is very easily described, and its embedding in the family depends on a choice. We explain this in §LABEL:subsection:specialization_galois_0, and show that this allows an identification of the generic fiber of with . We show in §LABEL:section:galois_euler that the extension is generated by the Euler element, and that the corresponding result at the level of rings is true if and only if is generated by a single reflection. We discuss in §LABEL:section:deploiement the decomposition of the algebra as a product of matrix algebras over . In other parts of §LABEL:chapter:galoisCM, we discuss gradings and automorphisms, and construct an order element of when all reflections of have order and . The last part §LABEL:se:geometrieCM is a geometrical translation of the previous constructions.
We introduce CalogeroMoser cells in Chapter LABEL:chapter:cellulesCM. They are defined in §LABEL:section:definition_cellules as orbits of inertia groups on and shown in §LABEL:subsection:cellules_et_blocs to coincide with blocks of the Cherednik algebra. We study next the ramification locus and smoothness. We give two more equivalent definitions of CalogeroMoser cells: via irreducible components of the base change by of the Galois cover (§LABEL:se:geometryCMcells) and via lifting of paths (§LABEL:se:cellstopology).
Part LABEL:part:verma is the heart of the book. It discusses CalogeroMoser cells associated with the ramification at , and , and relations with representations of the Cherednik algebras, as well as (conjectural) relations with Hecke algebras.
We start in Chapter LABEL:se:representations with a discussion of graded representations of Cherednik algebras. They form a highest weight category (in the sense of Appendix §LABEL:ap:hw). We discuss the standard objects, the Verma modules, and the Euler action on them.
Chapter LABEL:ch:Hecke is devoted to Hecke algebras. We recall in §LABEL:se:definitionsHecke the definition of Hecke algebras of complex reflection groups and some of its basic (partly conjectural) properties. We introduce a "cyclotomic" version, where the Hecke parameters are powers of a fixed indeterminate. We explain in §LABEL:se:KZ the construction of the KnizhnikZamolodchikov functor [ggor] realizing the category of representations of the Hecke algebra as a quotient of a (nongraded) category for the Cherednik algebra at . Thanks to the double endomorphism Theorem, the semisimplicity of the Hecke algebra is equivalent to that of the category . We present in §LABEL:section:representationshecke Malle’s splitting result for irreducible representations of Hecke algebras and we consider central characters. We discuss in §LABEL:se:Heckefamilies the notion of Hecke families. We finish in §LABEL:section:celluleskl with a brief exposition of the theory of KazhdanLusztig cells of elements of and of families of characters of and cellular characters.
Chapter LABEL:chapter:bebeverma is devoted to the representation theory of restricted Cherednik algebras and to CalogeroMoser families. We recall in §LABEL:se:represtricted and §LABEL:section:familles_CM some basic results of Gordon [gordon] on representations of restricted Cherednik algebras and CalogeroMoser families. Graded representations give rise to a highest weight category, as noticed by Bellamy and Thiel [BelTh], and we follow that approach. We show in §LABEL:section:dim_graduee the existence of a unique representation with minimal invariant in each family and generalize results of [gordon] on graded dimensions. We discuss in §LABEL:section:geometrie_CM the relation between the geometry of CalogeroMoser at and the CalogeroMoser families. The final section §LABEL:se:blocksCMfamilies relates CalogeroMoser families with blocks of category at and with blocks of Hecke algebras.
In Chapter LABEL:chapter:bilatere, we get back to the Galois cover and study twosided cells. We construct a bijection between the set of twosided cells and the set of families.
We continue in Chapter LABEL:chapter:gauche with the study of left (and right) cells and we define CalogeroMoser cellular characters. We analyze in §LABEL:section:choixgauche the choices involved in the definition of left cells, using Verma modules. We study the relevant decomposition groups, and we reinterpret left cells as blocks of a suitable specialization of the Cherednik algebra. We finish in §LABEL:se:backtocellular with basic properties relating cellular characters and left cells and we give an alternative definition of cellular characters as the socle of the restriction of a projective module. We also show that a cellular character involves a unique irreducible representation with minimal invariant.
Chapter LABEL:ch:decmat brings decomposition matrices in the study of cells. We show in §LABEL:se:isobabyVerma that the decomposition matrix of baby Verma modules in a block has rank , as conjectured by Thiel. In §LABEL:se:T=1to0, we prove that cellular characters are sums with nonnegative coefficients of characters of projective modules of the Hecke algebra over appropriate base rings.
Chapter LABEL:ch:Gaudin shows, following a suggestion of Etingof, that cells can be interpreted in terms of spectra of certain Gaudintype operators. This provides a topological approach to cells and cellular characters.
We analyze in Chapter LABEL:chapter:bb the cells associated to a smooth point of a CalogeroMoser space in . This is based on the use of the action and the resulting attracting sets. We show first, without smoothness assumptions, that irreducible components of attracting sets parametrize the cellular characters.
The next two chapters are devoted to conjectures. Chapter LABEL:part:coxeter discusses the motivation of this book, namely the expected relation between CalogeroMoser cells and KazhdanLusztig cells when is a Coxeter group. We start in §LABEL:chapter:hecke with Martino’s conjecture that CalogeroMoser families are unions of Hecke families. §LABEL:section:conjectures and §LABEL:chapter:arguments state and discuss our main conjecture. We give some cases where the conjecture on cellular characters does hold and give some evidence for the conjecture on cells.
Chapter LABEL:ch:conjgeometry gives a conjecture on the cohomology ring and the equivariant cohomology ring of CalogeroMoser spaces, extending the description of EtingofGinzburg in the smooth case. We also conjecture that irreducible components of the fixed points of finite order automorphisms on the CalogeroMoser space are CalogeroMoser spaces for reflection subquotients.
Part IV is based on the study of particular cases. Chapter LABEL:chapitre_nul presents the theory for the parameter .
Chapter LABEL:chapitre:rang_1 is devoted to the case of of dimension . We give a description of the objects introduced earlier, in particular the Galois closure . We show that generic decomposition groups can be very complicated, for particular values of the parameter.
Chapter LABEL:chapitre:b2 analyzes the case of a finite Coxeter group of type . We determine in §LABEL:section:quotient_B2 the ring of diagonal invariants and the minimal polynomial of the Euler element. We continue in §LABEL:section:Q_B2 with the determination of the corresponding deformed objects. The CalogeroMoser families are then easily found. We move next to the determination of the Galois group . Section §LABEL:se:cellsB2 is the more complicated study of ramification and the determination of the CalogeroMoser cells. We finish in §LABEL:sec:fixedb2 with a discussion of fixed points of the action of groups of roots of unity, confirming Conjecture FIX.
We have gathered in six Appendices some general algebraic considerations, few of which are original.
Appendix LABEL:appendice:filtration is a brief exposition of filtered modules and filtered algebras. We analyze in particular properties of an algebra with a filtration that are consequences of the corresponding properties for the associated graded algebra. We discuss symmetric algebras in §LABEL:se:symmalg.
Appendix LABEL:chapter:galoisrappels gathers some basic facts on ramification theory for commutative rings around decomposition and inertia groups. We recollect some properties of Galois groups and discriminants, and close the chapter with a topological version of the ramification theory and its connection with the commutative rings theory.
Appendix LABEL:appendice_graduation is a discussion of some aspects of the theory of graded rings. We consider general rings in §LABEL:intro_graduation. We next discuss in §LABEL:section:graduation_integrale gradings in the setting of commutative ring extensions. We finally consider gradings and invariants rings in §LABEL:section:GR.
We present in Appendix LABEL:appendice:_blocs some results on blocks and base change for algebras finite and free over a base. We discuss in particular central characters and idempotents, and the locus where the block decomposition changes.
Appendix LABEL:appendice:invariant deals with finite group actions on rings (commutative or not), and compare the crossproduct algebra and the invariant ring. We consider in particular the module categories and the centers.
Appendix LABEL:ap:hw provides a generalized theory of highest weight categories over commutative rings. We discuss in particular base change (§LABEL:se:basechange), Grothendieck groups (§LABEL:hw:K0 and §LABEL:se:completedK0), decomposition maps (§LABEL:se:hwdecmap) and blocks (§LABEL:se:hwblocks). A particular class of highest weight categories arises from graded algebras with a triangular decomposition (§LABEL:se:Appendixtriangular), generalizing [ggor] to noninner gradings.
Before the index of notations, we have included in "Prime ideals and geometry" some diagrams summarizing the commutative algebra and geometry studied in this book.
We would like to thank G. Bellamy, P. Etingof, I. Gordon, M. Martino and U. Thiel for their help and their suggestions.
Gunter Malle has suggested many improvements on a preliminary version of this book, and has provided very valuable help with Galois theory questions: we thank him for all of this.
Commentary. — This book contains an earlier version of our work [cm1]. The structure of the text has changed and the presentation of classical results on Cherednik algebras is now mostly selfcontained. There are a number of new results (see for instance Chapters LABEL:se:representations and LABEL:chapter:bebeverma) based on appropriate highest weight category considerations (Appendix LABEL:ap:hw is new), and a new topological approach (Chapter LABEL:ch:Gaudin on Gaudin algebras in particular).
Chapitre I Reflection groups and Cherednik algebras
Chapter \thechapter Notations
\thechapter.1 Integers
We put .
\thechapter.2 Modules
Let be a ring. Given a subset of , we denote by the twosided ideal of generated by . Given an module, we denote by the intersection of the maximal proper submodules of . We denote by the category of modules and by the category of finitely generated modules and we put , where denotes the Grothendieck group of an exact category . We denote by the category of finitely generated projective modules. Given an abelian category, we denote by its full subcategory of projective objects.
Given , we denote by (or simply ) its class in .
We denote by the set of isomorphism classes of simple modules. Assume is a finitedimensional algebra over the field . We have an isomorphism . If is semisimple, we have a bilinear form on given by . When is split semisimple, provides an orthonormal basis.
Let be a finite group and assume is a field. We denote by (or simply by ) the set of irreducible characters of over . When , there is a bijection of linear characters of with values in is denoted by (or ) . We have an embedding , and equality holds if and only if is abelian and contains all th roots of unity, where is the exponent of . . The group
\thechapter.3 Gradings
\thechapter.3.a
Let be a ring and a set. We denote by the free module with basis . We sometimes denote elements of as formal sums: , where .
\thechapter.3.b
Let be a monoid. We denote by (or ) the monoid algebra of over . Its basis of elements of is denoted by .
A graded module is a module with a decomposition (that is the same as a comodule over the coalgebra ). Given , we denote by the graded module given by . We denote by the additive category of graded modules such that is a free module of finite rank for all . Given , we put
We have defined an isomorphism of abelian groups . This construction provides a bijection from the set of isomorphism classes of objects of to . Given with , we define the graded module by . We have .
We say that a subset of a graded module is homogeneous if every element of is a sum of elements in for various elements .
\thechapter.3.c
A graded module is a graded module. We put . If for (for example, if is graded), then is an element of the ring of Laurent power series : this is the Hilbert series of . Similarly, if for , then .
When has finite rank over , we define the weight sequence of as the unique sequence of integers . such that
A bigraded module is a graded module. We put and , so that for . When is graded, we have .
When is a graded ring and is a finitely generated graded module, we denote by (or simply ) its class in the Grothendieck group of the category of finitely generated graded modules. Note that is a module, with .
\thechapter.3.d
Assume is a commutative ring. There is a tensor product of graded modules given by . When the fibers of the multiplication map are finite, the multiplication in provides with a ring structure, the tensor product preserves , and .
A graded algebra is a algebra with a grading such that .
Chapter \thechapter Reflection groups
All along this book, we consider a fixed characteristic field , a finitedimensional vector space of dimension and a finite subgroup of . We will write for . We denote by
\thechapter.4 Determinant, roots, coroots
We denote by the determinant representation of
We have a perfect pairing between and its dual
Given , we choose and such that
or equivalently
Note that, since has characteristic , all elements of are diagonalizable, hence
(\thechapter.4.1) 
Given and we have
(\thechapter.4.2) 
and
(\thechapter.4.3) 
\thechapter.5 Invariants
We denote by (respectively ) the symmetric algebra of (respectively ). We identify it with the algebra of polynomial functions on (respectively ). The action of on induces an action by algebra automorphisms on and and we will consider the graded subalgebras of invariants and . The coinvariant algebras and are the graded finitedimensional algebras
ShephardToddChevalley’s Theorem asserts that the property of to be generated by reflections is equivalent to structural properties of . We provide here a version augmented with quantitative properties (see for example [broue, Theorem 4.1]). We state a version with , while the same statements hold with replaced by .
Let us define the sequence degrees of as the weight sequence of (cf §\thechapter.3.C). of
Theorem \thechapter.5.1 (ShephardTodd, Chevalley).

The algebra is a polynomial algebra generated by homogeneous elements of degrees . We have

The module is free of rank .

The module is free of rank . So, .
Remark \thechapter.5.2.
Note that when , there is a skewlinear isomorphism between the representations and of , hence the sequence of degrees for the action of on is the same as the one for the action of on . In general, note that the representation of can be defined over a finite extension of , which can be embedded in : so, the equality of degrees for the actions on and holds for any .
This equality can also be deduced from Molien’s formula [broue, Lemma 3.28].
Let . Since , we deduce that . A generator is given by the image of : this provides an isomorphism .
The composition
factors through an isomorphism . We denote by the composition
We refer to §LABEL:se:symmalg for basic facts on symmetric algebras.
Proposition \thechapter.5.3.
is a symmetrizing form for the algebra .
Proof.
We need to show that the morphism of graded modules
is an isomorphism. By the graded Nakayama lemma, it is enough to do so after applying . We have , where is the projection onto the homogeneous component of degree . This is a symmetrizing form for [broue, Theorem 4.25], hence is an isomorphism. ∎
Note that the same statements hold for replaced by .
\thechapter.6 Hyperplanes and parabolic subgroups
Notation. We fix an embedding of the group of roots of unity of in . When the class of is in the image of this embedding, we denote by the corresponding element of .
We denote by the set of reflecting hyperplanes of :
There is a surjective equivariant map . Given a subset of , we denote by the pointwise stabilizer of :
Given , we denote by the order of the cyclic subgroup of . We denote by the generator of with determinant . This is a reflection with hyperplane . We have
The following lemma is clear.
Lemma \thechapter.6.1.
and are conjugate in if and only if and are in the same orbit and .
Given a orbit of hyperplanes of , we denote by the common value of the for . Lemma \thechapter.6.1 provides a bijection from to the set of pairs where and .
We denote by the set of pairs with and .
Let . Define the discriminant . The following result shows that points outside reflecting hyperplanes have trivial stabilizers [broue, Theorem 4.7].
Theorem \thechapter.6.2 (Steinberg).
Given , the group is generated by its reflections. As a consequence, and .
\thechapter.7 Irreducible characters
The rationality property of the reflection representation of is classical.
Proposition \thechapter.7.1.
Let be a subfield of containing the traces of the elements of acting on . Then there exists a submodule of such that .
Proof.
Assume first is irreducible. Let be a simple module such that for some integer . Let . Since has only one nontrivial eigenvalue on , it also has only one nontrivial eigenvalue on . Let be the eigenspace of acting on for the nontrivial eigenvalue. This is an dimensional subspace of , stable under the action of the division algebra . Since that division algebra has dimension over and has a module that has dimension over , we deduce that . The proposition follows by taking for the image of by an isomorphism .
Assume now is arbitrary. Let be a decomposition of the module , where is irreducible for be the subgroup of of elements acting trivially on . The group is a reflection group on . The discussion above shows there is a submodule of such that . Let be a submodule of such that . Let . We have and : this proves the proposition. ∎ . Let
The following rationality property of all representations of complex reflection groups is proven using the classification of those groups [benard, bessis].
Theorem \thechapter.7.2 (Benard, Bessis).
Let be a subfield of containing the traces of the elements of acting on . Then the algebra is split semisimple. In particular, is split semisimple.
\thechapter.8 Hilbert series
\thechapter.8.a Invariants
The algebra admits a standard bigrading, by giving to the elements of the bidegree and to those of the bidegree . We clearly have
(\thechapter.8.1) 
Using the notation of Theorem \thechapter.5.1(a), we get also easily that
(\thechapter.8.2) 
On the other hand, the bigraded Hilbert series of the diagonal invariant algebra is given by a formula à la Molien
(\thechapter.8.3) 
whose proof is obtained word by word from the proof of the usual Molien formula.
\thechapter.8.b Fake degrees
We identify with : given a finite dimensional graded module, we make the identification
It is clear that is the evaluation at of and that
Let denote the unique family of elements of such that
(\thechapter.8.4) 
Definition \thechapter.8.5.
The polynomial is called the fake degree of . Its valuation is denoted by and is called the invariant of .
The fake degree of satisfies
(\thechapter.8.6) 
Note that
(\thechapter.8.7) 
(here, denotes the dual character of , that is, ). Note also that, if denotes the trivial character of , then
and 
We deduce:
Lemma \thechapter.8.8.
The elements and are not zero divisors in .
Remark \thechapter.8.9.
Note that
is a zero divisor in (as soon as ).
We can now give another formula for the Hilbert series :
Proposition \thechapter.8.10.
Proof.
Let be a stable graded complement to in . Since is a free module, we have isomorphisms of graded modules
Similarly, if is a stable graded complement of in , then we have isomorphisms of graded modules
In other words, we have isomorphisms of graded modules
We deduce an isomorphism of bigraded vector spaces
By (\thechapter.8.4) and (\thechapter.8.7), we have
So the formula follows from the fact that . ∎
To conclude this section, we gather in a same formula Molien’s Formula (\thechapter.8.3) and Proposition \thechapter.8.10:
\thechapter.9 Coxeter groups
Let us recall the following classical equivalences:
Proposition \thechapter.9.1.
The following assertions are equivalent:

There exists a subset of such that is a Coxeter system.

as modules.

There exists a invariant nondegenerate symmetric bilinear form .

There exists a subfield of and a stable vector subspace of such that and embeds as a subfield of .
Whenever one (or all the) assertion(s) of Proposition \thechapter.9.1 is (are) satisfied, we say that is a Coxeter group. In this case, the text will be followed by a gray line on the left, as below.
Assumption, choice. From now on, and until the end of §\thechapter.9, we assume that is a Coxeter group. We fix a subfield of that embeds as a subfield of and a stable vector subspace of such that . We also fix a connected component of . We denote by the set of such that has real codimension in . So, is a Coxeter system. This notation will be used all along this book, provided that is a Coxeter group.
The following is a particular case of Theorem \thechapter.7.2.
Theorem \thechapter.9.2.
The algebra is split. In particular, the characters of are real valued, that is, for all character of .
Recall also the following.
Lemma \thechapter.9.3.
If