Rational Cherednik Algebras

Introduction

A finite complex reflection group $W$ acts on a complex vector space $\mathrm{H}$ by linear transformations generated by complex reflections, the elements that fix a hyperplane pointwise and have finite order. The algebra of polynomial functions on $\mathrm{H}$ carries the action of $W$ and the smash product $S(\mathrm{H}) \rtimes W$, and the rational Cherednik algebra deforms this smash product in the directions of the individual reflections, in the same way that the symplectic reflection algebra of the preceding article deforms the smash product of a symplectic space: the deformation parameter is a scalar $t$, which measures the deformation of the constant part, and a function $c$ on the reflection hyperplanes, which measures the deformation in the direction of the reflections supported on them. The algebras were introduced by Cherednik in the context of the Knizhnik–Zamolodchikov equations and the affine Hecke algebras, and they were placed in the general framework of the symplectic reflection algebras by Etingof and Ginzburg in 2002.

The rational Cherednik algebra of a complex reflection group $W$ is the symplectic reflection algebra of the group acting on the doubled space $\mathrm{H} \oplus \mathrm{H}^*$: a complex reflection of $\mathrm{H}$ is a symplectic reflection of the doubling, so the rational Cherednik algebras are the special case of the preceding article in which the symplectic reflection group is the doubling of a complex reflection group, and the definitions are compatible. Their representation theory is organised by the category $\mathcal{O}$, whose standard objects are the Verma modules of the theory; the KZ functor relates the category $\mathcal{O}$ at $t \neq 0$ to the finite-dimensional representations of the Hecke algebra $H_q(W)$ of Hecke Algebras, with $q$ determined by $t$, and the relation to the Hecke algebra is the reason the rational Cherednik algebras are the algebraic form of the Knizhnik–Zamolodchikov equations. The spherical subalgebra is related to the Calogero–Moser space, a symplectic variety that also carries the classical integrable system of the same name, and for the symmetric group the theory produces the diagonal coinvariants and the $q,t$-Catalan numbers of the Macdonald theory.

No measure, integral, derivative or analytic limit is used: the Dunkl operators that realise the algebras are differential operators on the polynomial ring, that is, elements of the Weyl algebra, and the local systems of the KZ functor are the algebraic local systems of the theory of linear differential equations of Part III; the analytic theory of the monodromy belongs to the later Part.

Complex Reflection Groups and the Dunkl Operators

Complex Reflections

Definition. Let $\mathrm{H}$ be a finite-dimensional vector space over $\mathbb{C}$ and let $W \subseteq GL(\mathrm{H})$ be a finite group. An element $s \in W$ is a complex reflection if $\operatorname{rank}(1-s) = 1$, equivalently if $s$ fixes a hyperplane $\mathrm{H}_s = \ker(1-s)$ pointwise and acts on the complementary line with an eigenvalue $\zeta \neq 1$ that is a root of unity. The group $W$ is a complex reflection group if it is generated by its complex reflections, and $\mathrm{H}$ is then its reflection representation. For a reflection $s$ the hyperplane $H = \mathrm{H}_s$ is the fixed hyperplane, a root $\alpha_H \in \mathrm{H}^*$ is a linear form with zero set $H$, and a coroot $\alpha_H^\vee \in \mathrm{H}$ is a vector spanning the eigenvalue line of $s$, so that $s y = y - \zeta' \langle \alpha_H, y\rangle \alpha_H^\vee$ for a suitable scalar; the choices are unique up to a scalar multiple, and the scalar is fixed by the normalisation.

Remark. The reflection $s$ has order $m_H$, the order of the eigenvalue $\zeta$; writing $\zeta = e^{2\pi i j/m_H}$, the reflections with a fixed hyperplane $H$ are the powers of a single reflection, and there are $m_H - 1$ of them. The function $c$ of the next section is a function of the pair $(H, j)$ with $0 \leq j \leq m_H - 1$, and it is constant on the $W$-orbits of these pairs.

Example. The symmetric group $S_n$ acts on $\mathrm{H} = \{(x_1, \ldots, x_n) : \sum x_i = 0\}$ by permuting the coordinates; the transpositions are complex reflections of order $2$, their fixed hyperplanes are $x_i = x_j$, and $S_n$ is a complex reflection group. The group $G(m, 1, n) = S_n \ltimes (\mu_m)^n$ of the monomial matrices with entries an $m$-th root of unity in each row and column acts on $\mathbb{C}^n$; its elements with a single nontrivial root are the complex reflections of the eigenvalues $e^{2\pi i j/m}$, and the fixed hyperplanes are the coordinate hyperplanes $x_i = 0$. The finite complex reflection groups were classified by Shephard and Todd, the irreducible ones being the groups $G(m, p, n)$ together with $34$ exceptional groups, and the classification is the complex analogue of the classification of the finite real reflection groups.

The Dunkl Operators

Definition. Let $c : \{H\} \times \{0, \ldots, m_H - 1\} \to \mathbb{C}$ be a $W$-invariant function. For $y \in \mathrm{H}$ the Dunkl operator is the differential-reflection operator on $\mathbb{C}[\mathrm{H}]$ given by $$ D_y = t\,\partial_y - \sum_H \frac{\langle \alpha_H, y\rangle}{\alpha_H}\sum_{j = 0}^{m_H - 1} c_{H,j}\,s_{H,j}, $$ where $\partial_y$ is the partial derivative in the direction $y$, the sum is over the reflection hyperplanes, $s_{H,j}$ is the reflection with eigenvalue $e^{2\pi i j/m_H}$, and the fraction is the algebraic division by the linear form in the localised ring $\mathbb{C}[\mathrm{H}][\alpha_H^{-1}]$; the operator preserves $\mathbb{C}[\mathrm{H}]$ because the numerator $\langle \alpha_H, y\rangle$ vanishes on the hyperplane and the reflection terms compensate the polar part.

Theorem (Dunkl). The Dunkl operators commute: $$ D_y D_z = D_z D_y \qquad \text{for all } y, z \in \mathrm{H} . $$ The theorem is the fundamental point of the theory: the commutativity is not visible from the definition, since the individual operators do not commute with the reflections, and it is proved by a direct computation of the commutator of the reflection terms, in which the contributions of the individual hyperplanes cancel by the invariance of $c$ and the rank-one nature of the reflections. The theorem was proved by Dunkl for the real reflection groups and extended to the complex case by Dunkl, Opdam and others; the proof is in the references.

Corollary. The algebra generated by the operators $D_y$, the multiplications by the functions, and the reflections is a filtered algebra whose associated graded is the algebra of differential operators on $\mathrm{H}$ with the action of $W$; the map $y \mapsto D_y$ from $\mathrm{H}$ preserves the commutators in the sense of the next section.

The Rational Cherednik Algebra

The Definition

Definition. Let $\mathrm{H}$ be a complex vector space, $W \subseteq GL(\mathrm{H})$ a finite complex reflection group, $t \in \mathbb{C}$, and $c$ a $W$-invariant function on the pairs $(H,j)$ of a reflection hyperplane and an eigenvalue. The rational Cherednik algebra $H_{t,c}(W)$ is the quotient of the smash product $T(\mathrm{H} \oplus \mathrm{H}^*) \rtimes W$ of the tensor algebra on the direct sum of $\mathrm{H}$ and its dual with the group algebra $\mathbb{C}W$ by the relations $$ [y, x] = t\,\langle x, y\rangle - \sum_H \sum_{j=0}^{m_H-1} c_{H,j}\,\langle x, \alpha_H^\vee\rangle\,\langle \alpha_H, y\rangle\,s_{H,j}, \qquad y \in \mathrm{H},\ x \in \mathrm{H}^*, $$ together with the relations $[y, y'] = 0$ for $y, y' \in \mathrm{H}$ and $[x, x'] = 0$ for $x, x' \in \mathrm{H}^*$, where $\langle \cdot, \cdot \rangle$ is the pairing between $\mathrm{H}^*$ and $\mathrm{H}$ and the products of the bracket with the scalar are in the smash product.

The normalisation of the coroots is fixed once and for all: for each hyperplane $H$ the coroot $\alpha_H^\vee$ is chosen so that $\langle \alpha_H, \alpha_H^\vee\rangle = 2$, in which case the coefficient of $s_{H,j}$ is fixed by the pairing and the various normalisations in the literature differ by a rescaling of $c_{H,j}$; the article fixes the normalisation by that condition, and the isomorphism class of the algebra depends only on the coefficients in this normalisation.

Remark. The algebra is a deformation of the smash product: the relations reduce to the commutativity of $S(\mathrm{H})$ and $S(\mathrm{H}^*)$ when $t = 0$ and all the $c_{H,j}$ vanish, and in general the parameter $t$ is the coefficient of the "constant" part of the bracket, while the coefficient $c_{H,j}$ measures the deformation in the direction of the complex reflection $s_{H,j}$. The algebra is the symplectic reflection algebra of the doubled space $\mathrm{H} \oplus \mathrm{H}^*$ with the standard symplectic form, where $W$ acts by $w \cdot (y, x) = (wy, wx)$ and the complex reflections become symplectic reflections; this identification is the content of the companion article Symplectic Reflection Algebras, written in this batch, and it shows that the two definitions agree.

PBW and the Dunkl Embedding

Theorem (PBW; Etingof–Ginzburg). The multiplication map $$ S(\mathrm{H}) \otimes_{\mathbb{C}} \mathbb{C}W \otimes_{\mathbb{C}} S(\mathrm{H}^*) \longrightarrow H_{t,c}(W) $$ is an isomorphism of vector spaces for every $t$ and $c$; the algebra has Gelfand–Kirillov dimension $\dim \mathrm{H}$; and the natural filtration has associated graded algebra $S(\mathrm{H}) \rtimes W$.

Proof sketch. For $t = 1$ the Dunkl operators give a representation of $H_{1,c}(W)$ on $\mathbb{C}[\mathrm{H}]$ in which the elements of $\mathrm{H}$ act by $D_y$, the elements of $\mathrm{H}^*$ act by multiplication, and the group acts by the given reflection representation; the commutativity of the Dunkl operators matches the relations $[y,y'] = 0$, the Leibniz-type identities match the relations between $\mathrm{H}$ and $\mathrm{H}^*$, and the residual relations are checked from the transformation law of $D_y$ under the reflections. The image of the algebra has associated graded $S(\mathrm{H}) \rtimes W$ because the Dunkl operator reduces modulo the reflection terms to the partial derivative, so the PBW property follows for $t = 1$; scaling $y \mapsto t^{1/2} y$ for $t \neq 0$ gives the general nonzero case, and $t = 0$ is the associated graded degeneration. The proof is in the cited paper.

Corollary. $H_{t,c}(W)$ contains $S(\mathrm{H})$, $\mathbb{C}W$ and $S(\mathrm{H}^*)$ as subalgebras, and it is free as a left module over $S(\mathrm{H})$ with basis the monomials in $S(\mathrm{H}^*)$ times $W$, and free as a right module over $S(\mathrm{H}^*)$ with the analogous basis.

Specialisations and Rescaling

Proposition. The specialisations of the rational Cherednik algebra are $$ H_{0,0}(W) = S(\mathrm{H}) \rtimes W, \qquad H_{t,0}(W) = A(\mathrm{H}) \rtimes W, $$ with $A(\mathrm{H})$ the Weyl algebra of $\mathrm{H}$; the substitution $y \mapsto \lambda y$, $x \mapsto \lambda^{-1} x$ for $\lambda \in \mathbb{C}^\times$ sends the relations of $(t,c)$ to the relations of $(\lambda t, \lambda c)$; and for $t \neq 0$ the algebra may be normalised by the condition $t = 1$. The family is a flat deformation of $S(\mathrm{H}) \rtimes W$ over the parameter space $\mathbb{C} \oplus (\mathbb{C}[\text{pairs}])^W$, by the PBW theorem.

Proof. The specialisations are immediate from the relations, and the scaling is the substitution in $T(\mathrm{H}\oplus\mathrm{H}^*)$; flatness is the PBW theorem, which gives the associated graded independently of the parameters.

The Spherical Subalgebra and the Centre

Definition. The symmetrising idempotent is $e = |W|^{-1}\sum_{w \in W} w \in \mathbb{C}W$, and the spherical subalgebra of $H_{t,c}(W)$ is $B_{t,c} = e\,H_{t,c}(W)\,e$. The functor $M \mapsto eM$ is an equivalence between the category of $H_{t,c}$-modules and the category of $B_{t,c}$-modules, since $e$ is a full idempotent and $H_{t,c} = H_{t,c}\,e\,H_{t,c}$ for the symmetrising idempotent, so the two algebras are Morita equivalent and their module categories have the same simple objects.

Theorem (Satake isomorphism; Etingof–Ginzburg). For every $c$ the spherical subalgebra is commutative and the map $$ Z\bigl(H_{0,c}(W)\bigr) \longrightarrow B_{0,c}, \qquad z \longmapsto z\,e , $$ is an isomorphism of algebras; the standard filtration of $H_{0,c}$ induces a filtration of the centre with $$ \operatorname{gr} Z\bigl(H_{0,c}(W)\bigr) = S(\mathrm{H} \oplus \mathrm{H}^*)^W , $$ the coordinate ring of the quotient $(\mathrm{H} \oplus \mathrm{H}^*)/W$ of the cotangent space, and at $c = 0$ the centre is exactly $S(\mathrm{H} \oplus \mathrm{H}^*)^W$. The spectrum of the spherical subalgebra is therefore the Calogero–Moser space of $W$ at the parameter $c$, an affine symplectic variety of dimension $2\dim \mathrm{H} = \dim(\mathrm{H} \oplus \mathrm{H}^*)$ whose coordinate ring is a flat deformation of the quotient, smooth for the groups and the parameters for which the quotient admits a symplectic resolution. For $t \neq 0$ and parameters outside a countable union of hyperplanes, the centre of $H_{t,c}(W)$ is the scalars, and hence so is the centre of the Morita equivalent $B_{t,c}$.

Remark. The Calogero–Moser space can be described as a hyperkähler quotient of the cotangent bundle of a suitable space by the group $W$, or as the spectrum of the centre of the spherical subalgebra at $t = 0$, and the two descriptions agree with the classical integrable system of Calogero and Moser that gave it its name. The space is a deformation of the quotient $(\mathrm{H} \oplus \mathrm{H}^*)/W$: the quotient is the "classical" limit, and the Calogero–Moser space is the symplectic resolution or the deformation of the quotient singularity that the spherical algebra quantises. For the symmetric group the space is the "Calogero–Moser space of $n$ particles", of dimension $2n - 2$, with a description in terms of pairs of matrices and the moment map of the action of $GL_n$.

Example (symmetric group; diagonal coinvariants). Let $W = S_n$ acting on $\mathrm{H} = \mathbb{C}^{n-1}$ as the reflection representation, and let $c$ be the parameter of the single conjugacy class of transpositions. The category of modules over $H_{t,c}(S_n)$ at $t = 1$ and suitable $c$ contains the module of diagonal coinvariants, the quotient of the polynomial ring in two sets of $n$ variables by the ideal generated by the positive-degree $S_n$-invariants, and the $\mathbb{Z}$-grading of the diagonal coinvariants together with the two gradings produces the $q,t$-Catalan numbers and the Macdonald positivity theory. The Macdonald polynomials of Macdonald and Hall–Littlewood Polynomials and the character theory of Symmetric Functions and Schur Functions are the combinatorial side of the same representation theory, and the identification uses the spherical subalgebra and its commutative centre.

Example (cyclic and dihedral groups). Let $W = \mu_n$ act on $\mathrm{H} = \mathbb{C}$ by rotations. The rational Cherednik algebra is generated by two elements $y, x$ with the relation $$ [y, x] = t - \sum_{j=1}^{n-1} c_j\,s_j, $$ where $s_j$ is the reflection of order $n/\gcd(n,j)$ acting by the corresponding root of unity; the algebra is the "cyclic" Cherednik algebra, and its category $\mathcal{O}$ at $t = 1$ contains the finite-dimensional representations at the special parameters $c_j$ that make the Verma module reducible, in a pattern governed by the affine root system of type $A_{n-1}^{(1)}$. The dihedral groups give the type $I_2(m)$ algebras, whose representation theory is worked out completely and whose special parameters are the "cuspidal" points at which the category $\mathcal{O}$ acquires finite-dimensional representations.

Representations

Category $\mathcal{O}$ and the Verma Modules

Definition. The category $\mathcal{O}$ of $H_{t,c}(W)$ is the full subcategory of the finitely generated modules on which the subalgebra $S(\mathrm{H})$ acts locally finitely. For a finite-dimensional representation $\tau$ of $W$ the Verma module is $$ M_{t,c}(\tau) = H_{t,c}(W) \otimes_{S(\mathrm{H})\rtimes W} \tau , $$ the tensor product over the subalgebra generated by $\mathrm{H}$ and $W$, with $\mathrm{H}$ acting by zero on $\tau$; the standard module $\Delta_{t,c}(\tau)$ is the image of the Verma module in the localisation, and the simple module $L_{t,c}(\tau)$ is the unique simple quotient.

Theorem. The Verma modules are the standard objects of the category $\mathcal{O}$: each $M_{t,c}(\tau)$ has a unique simple quotient $L_{t,c}(\tau)$, every simple object of $\mathcal{O}$ is a quotient of a Verma module, and the Verma modules are free over $S(\mathrm{H}^*)$ and of Gelfand–Kirillov dimension $\dim\mathrm{H}$. The category $\mathcal{O}$ has enough projectives, is a highest-weight category in the sense of the theory of the finite-dimensional Lie algebras, and the multiplicities of the simples in the Verma modules are governed by the KZ equations and by the Hecke algebra of the next subsection.

Proof sketch. The freeness over $S(\mathrm{H}^*)$ is the PBW theorem, and the highest-weight structure is the triangular decomposition: the modules in $\mathcal{O}$ are generated by their $\tau$-isotypic parts, and the standard highest-weight argument gives the unique simple quotient and the classification of the simples by the finite-dimensional representations of $W$. The combinatorics of the multiplicities is the deep part of the theory and is reduced to the Hecke algebra by the KZ functor.

Finite-Dimensional Representations and Special Parameters

Theorem. A simple module $L_{t,c}(\tau)$ is finite-dimensional only for parameters $c$ in a countable union of algebraic hypersurfaces in the parameter space, the special or cuspidal parameters; for $t \neq 0$ the finite-dimensional simple modules are the modules for which the KZ functor takes the value of a module with a "trivial" monodromy, and for $t = 1$ the finite-dimensional representations occur exactly at the rational values of $c$ with denominators bounded by the order of the reflections of $W$ in the cases of the symmetric and the cyclic groups.

The theorem is the content of the theory of the finite-dimensional representations of the rational Cherednik algebras, developed by Berest–Etingof–Ginzburg, Chmutova, Gordon and others, and it is the part of the theory with the most involved combinatorics; the special parameters are the points at which the Verma module has a submodule of finite codimension, and their classification uses the affine root systems and the Kac–Moody representation theory. The statements are in the references.

Example. For $W = S_n$ and $t = 1$ the finite-dimensional representations occur at $c = j/n$ for suitable integers $j$, and their characters are described by the theory of the Jack and Macdonald polynomials at the corresponding special parameters; the dimensions are the multiplicities computed from the standard Young tableaux, and the phenomenon is the representation-theoretic form of the description of the $\mathbb{Z}$-graded category $\mathcal{O}$ for type $A$ in terms of the affine Hecke algebra and of its Koszul dual, the finite Hecke algebra (Rouquier). The example shows that the finite-dimensional representations of the algebras of type $A$ recover the combinatorial representation theory of the symmetric group, which is the reason the theory is called the rational Cherednik theory.

The KZ Functor and the Hecke Algebra

Theorem (KZ functor; Ginzburg–Guay–Opdam–Rouquier). For $t \neq 0$ there is an exact functor, the KZ functor, $$ \operatorname{O}(H_{t,c}(W)) \longrightarrow H_q(W)\text{-mod}, \qquad q = e^{2\pi i t}, $$ from the category $\mathcal{O}$ of the rational Cherednik algebra to the category of finite-dimensional modules over the Hecke algebra $H_q(W)$ of Hecke Algebras; the functor is fully faithful, and for parameters outside the resonant hyperplanes the induced map on simple objects is a bijection onto the simple modules over $H_q(W)$ that occur, so that at the generic parameters the KZ functor is an equivalence between $\mathcal{O}$ and the corresponding category of Hecke modules.

Proof sketch. The Dunkl operators define a local system of linear differential equations on the complement of the reflection hyperplanes, the Knizhnik–Zamolodchikov system, and the KZ functor sends a module to the fibre of the local system at a base point, a finite-dimensional vector space carrying the monodromy action, which factors through the Hecke algebra $H_q(W)$ with $q = e^{2\pi i t}$. Exactness and the full faithfulness are proved by comparing the standard filtration of a module in $\mathcal{O}$ with the filtration of the monodromy representation, and the identification of the image uses the equivalence of the local system with the "longest element" description of the Hecke algebra. The construction and the proofs are in the cited references; the analytic theory of the local system, that is, the existence and the properties of the solutions of the KZ system, belongs to Part III, where the theory of the linear differential equations is available.

Remark. The KZ functor is the reason the rational Cherednik algebras are the "algebraic" form of the Hecke algebras: the representation theory at $t \neq 0$ is the representation theory of $H_q(W)$ in the cases covered by the equivalence, and it is independent of the parameters $c$ up to the position of the resonant hyperplanes, while at $t = 0$ it is the representation theory of the orbifold and of the quotient singularity. The same dictionary, applied to the doubled space, is the KZ statement for the symplectic reflection algebras of the preceding article.

Summary

A complex reflection in a finite group $W \subseteq GL(\mathrm{H})$ over $\mathbb{C}$ is an element $s$ with $\operatorname{rank}(1-s) = 1$, fixing a hyperplane and acting on the complementary line with a root of unity $\zeta \neq 1$; $W$ is a complex reflection group if it is generated by its reflections. The Dunkl operator $D_y = t\partial_y - \sum_H \frac{\langle\alpha_H,y\rangle}{\alpha_H}\sum_j c_{H,j}s_{H,j}$ is a differential-reflection operator on $\mathbb{C}[\mathrm{H}]$, and by Dunkl's theorem the operators commute. The rational Cherednik algebra $H_{t,c}(W)$ is the quotient of $T(\mathrm{H}\oplus\mathrm{H}^*)\rtimes W$ by the relations $[y,x] = t\langle x,y\rangle - \sum_H\sum_j c_{H,j}\langle x,\alpha_H^\vee\rangle\langle\alpha_H,y\rangle s_{H,j}$ together with the vanishing of the commutators within $\mathrm{H}$ and within $\mathrm{H}^*$; it is the symplectic reflection algebra of the doubled space $\mathrm{H}\oplus\mathrm{H}^*$. By the PBW theorem of Etingof and Ginzburg, $H_{t,c}(W) = S(\mathrm{H})\cdot\mathbb{C}W\cdot S(\mathrm{H}^*)$ as vector spaces, the algebra has Gelfand–Kirillov dimension $2\dim\mathrm{H} = \dim(\mathrm{H}\oplus\mathrm{H}^*)$, the parameters rescale by $(t,c)\mapsto(\lambda t,\lambda c)$, and the specialisations are $S(\mathrm{H})\rtimes W$ at $t=c=0$ and $A(\mathrm{H})\rtimes W$ at $t\neq 0$, $c=0$, with $A(\mathrm{H})$ the Weyl algebra. The spherical subalgebra $B_{t,c} = eH_{t,c}e$, with $e$ the symmetrising idempotent, is Morita equivalent to $H_{t,c}$; by the Satake isomorphism the centre of $H_{0,c}$ is the spherical subalgebra $B_{0,c}$, whose associated graded is the invariant ring $S(\mathrm{H}\oplus\mathrm{H}^*)^W$ of the cotangent space, and the Calogero–Moser space, the spectrum of $B_{0,c}$, is the symplectic resolution of the quotient on which the deformation lives; for $t \neq 0$ outside a countable union of hyperplanes the centres of $H_{t,c}$ and of the Morita equivalent $B_{t,c}$ are the scalars. The category $\mathcal{O}$ has the Verma modules $M_{t,c}(\tau)$ with simple quotients $L_{t,c}(\tau)$ as its standard objects, and its finite-dimensional simple modules exist only at the special parameters; the KZ functor relates $\mathcal{O}$ at $t \neq 0$ to the finite-dimensional modules over the Hecke algebra $H_q(W)$ with $q = e^{2\pi i t}$, and for the symmetric group the theory produces the diagonal coinvariants and the $q,t$-Catalan numbers.

Summary of Notation

Symbol	Meaning
$\mathrm{H}$, $W$	Reflection representation and finite complex reflection group
complex reflection $s$	$\operatorname{rank}(1-s) = 1$; fixes a hyperplane
$H$, $\alpha_H$, $\alpha_H^\vee$, $m_H$	Hyperplane, root, coroot, order of the reflection
$s_{H,j}$	Reflection about $H$ with eigenvalue $e^{2\pi i j/m_H}$
$c_{H,j}$	Deformation parameter, $W$-invariant
$t$	Scalar parameter of the algebra
$D_y$	Dunkl operator on $\mathbb{C}[\mathrm{H}]$
$H_{t,c}(W)$	Rational Cherednik algebra
$S(\mathrm{H}) \rtimes W$	Smash product; $t=c=0$ specialisation
$A(\mathrm{H})$	Weyl algebra; $t\neq 0$, $c=0$ specialisation
$e$	Symmetrising idempotent $\|W\|^{-1}\sum_{w \in W} w$
$B_{t,c} = eH_{t,c}e$	Spherical subalgebra
$Z(H_{t,c})$	Centre; $Z(H_{0,c}) \cong B_{0,c}$ with $\operatorname{gr} Z(H_{0,c}) = S(\mathrm{H}\oplus\mathrm{H}^*)^W$; scalars for $t\neq 0$ and generic $c$
Calogero–Moser space	Spectrum of $B_{0,c} = Z(H_{0,c})$; dimension $2\dim\mathrm{H}$
$\mathcal{O}$	Category of locally finite $S(\mathrm{H})$-modules
$M_{t,c}(\tau)$, $L_{t,c}(\tau)$	Verma module and simple quotient
$H_q(W)$	Hecke algebra of $W$; $q = e^{2\pi i t}$
KZ functor	$\mathcal{O} \to H_q(W)\text{-mod}$