Symplectic Reflection Algebras

Introduction

The algebra of polynomial functions on a symplectic vector space $(V, \omega)$ carries the Poisson bracket of Symplectic Forms and Poisson Brackets, and the bracket is the first term of a deformation: the Weyl algebra of $V$ deforms the symmetric algebra $S(V)$ along the constant bracket, and it is the universal such deformation. A symplectic reflection algebra deforms the smash product $S(V) \rtimes G$, where $G$ is a finite subgroup of the symplectic group generated by symplectic reflections, the elements whose fixed space is a hyperplane; the deformation is by the constant bracket together with the bracket-like contributions of the individual reflections, and it is governed by a scalar $t$, which measures the deformation of the constant bracket, and a $G$-invariant function $c$ on the set of symplectic reflections. The algebras were introduced by Etingof and Ginzburg in 2002, and they carry a triangular decomposition, a PBW theorem proved through a Dunkl embedding, a category of modules whose behaviour at $t = 0$ is that of the orbifold $V/G$ and at $t \neq 0$ is that of the Hecke algebra of the orbifold.

The definition uses only the symplectic form and the finite group, and the article is placed in this category because the form is the input: the odd reflections that generate an orthogonal group give the Clifford algebras of Part I and the rational Cherednik algebras of the companion article, while the rank-two symplectic reflections give the algebras developed here. The representation theory of the symplectic reflection algebras at $t = 0$ is the quantised function theory of the symplectic quotient, and at $t \neq 0$ the algebras are the quantisations of the symplectic resolutions of the quotient singularities, the Hilbert schemes of points on the Kleinian surfaces being the basic example; the quantisation theory of the resolutions belongs to the later part of this category and to the companion literature, and the present article develops the algebras themselves.

No measure, integral, derivative or analytic limit is used. The differential operators that occur in the Dunkl embedding are algebraic: they are the formal partial derivatives on the polynomial ring of $V$, that is, the elements of the Weyl algebra, which is the algebra of the symmetric powers of $V$ and $V^*$ without any analytic completion.

Symplectic Reflections

The Definition

Definition. Let $V$ be a finite-dimensional vector space over a field $K$ of characteristic zero and let $\omega$ be a nondegenerate alternating form on $V$, as in Symplectic Forms and Poisson Brackets. An element $g \in \operatorname{Sp}(V, \omega)$ is a symplectic reflection if $$ \operatorname{rank}(1 - g) = 2, $$ equivalently if the fixed space $H = \ker(1-g) = V^g$ has codimension $2$ in $V$. A finite subgroup $G \subseteq \operatorname{Sp}(V, \omega)$ is a symplectic reflection group if it is generated by its symplectic reflections, and the set of symplectic reflections of $G$ is written $S$.

Proposition. Let $s$ be a symplectic reflection with fixed hyperplane $H$. Then $\operatorname{im}(1-s)$ is a $2$-dimensional subspace $W$ containing no nonzero vector of $H$, the form $\omega$ restricts to a nondegenerate alternating form on $W$, and the fixed hyperplane is $\omega$-orthogonal to $W$.

Proof. Since $s \in \operatorname{Sp}(V,\omega)$, for $h \in H$ and $v \in V$ one has $\omega(h,(1-s)v) = \omega(h,v) - \omega(h,sv) = \omega(h,v) - \omega(s^{-1}h, v) = 0$, so $H$ is $\omega$-orthogonal to $W$. The subspaces $H$ and $W$ are $s$-stable, $V = H + W$ because $v = sv + (1-s)v$ with $sv \in H$, and $\dim H = \dim V - 2$, $\dim W = 2$, so $H \cap W = 0$ by dimensions and $V = H \oplus W$. A vector $w \in W$ orthogonal to $W$ is orthogonal to $V = H \oplus W$ by the orthogonality of $H$ and $W$, hence is zero by the nondegeneracy of $\omega$, so $\omega\vert_W$ is nondegenerate.

Remark. The proof records the spectral form of a symplectic reflection: $V$ is the $\omega$-orthogonal direct sum $H \oplus W$ of $s$-stable subspaces with $\dim W = 2$, the form $\omega\vert_W$ is nondegenerate, and $s$ acts on $W$ with two eigenvalues $\lambda, \lambda^{-1}$ of product $1$ and neither equal to $1$, since $s$ has no nonzero fixed vector in $W$: the elements of $W$ are the vectors moved by $s$, and the fixed vectors form $H$. In particular a symplectic reflection is the identity on a hyperplane and a twist with eigenvalues $\lambda, \lambda^{-1}$ on the complementary plane.

Definition. Let $s$ be a symplectic reflection. The associated form $\omega_s$ is the alternating form $$ \omega_s(x, y) = \tfrac{1}{2}\,\omega\bigl((1-s)x,\, (1-s)y\bigr). $$ Its radical is the fixed hyperplane $H$ of $s$ and it is nondegenerate on $\operatorname{im}(1-s)$, so it is a form of rank $2$.

Proposition. The form $\omega_s$ is alternating and its radical is $H$; it is $G$-equivariant in the sense that $\omega_{gsg^{-1}}(x,y) = \omega_s(g^{-1}x,g^{-1}y)$; and it is uniquely determined by $H$ and the scalar $\omega_s(w_1,w_2)$ on a symplectic basis of $W$, so the normalisation of $c(s)$ absorbs the choice of $\omega_s$ up to a scalar.

Proof. Alternation is the alternation of $\omega$. If $y \in H$ then $(1-s)y = 0$ and $\omega_s(\cdot,y)=0$, so $H$ is contained in the radical; the radical is $\ker(1-s)$ in the splitting $V = H \oplus W$, since $\omega_s$ restricted to $W$ is nondegenerate because $\omega$ is nondegenerate on $W$ and $(1-s)\vert_W = 1 - s\vert_W$ is invertible on $W$; hence the radical is exactly $H$ and the rank is $2$. Equivariance uses $(1 - gsg^{-1}) = g(1-s)g^{-1}$ and the invariance of $\omega$. The final statement is that an alternating form on the two-dimensional $W$ with the two fixed eigenvalues is determined by one scalar, which is absorbed in the definition of $c(s)$.

Examples of Symplectic Reflection Groups

Example (the two-dimensional case). Let $\dim V = 2$, so that $\operatorname{Sp}(V, \omega) = SL(V)$ and a finite subgroup is a finite subgroup of $SL_2(K)$. For $g \in SL(V)$ the rank of $1-g$ is $2$ unless $g = 1$ and, in characteristic zero, also unless $g$ is a nontrivial unipotent element; for a finite group there are no nontrivial unipotent elements of infinite order and a unipotent element of finite order in characteristic zero is trivial. Hence for a finite subgroup $G \subseteq SL_2(K)$ every nontrivial element is a symplectic reflection, and such a $G$ is a symplectic reflection group. The finite subgroups of $SL_2(\mathbb{C})$ are the binary polyhedral groups, that is, the cyclic groups $\mu_n$, the binary dihedral groups, and the binary tetrahedral, octahedral and icosahedral groups; their quotients by the centre are the finite rotation groups of the sphere, and the corresponding quotient singularities $\mathbb{C}^2/G$ are the Kleinian or Du Val singularities.

Example (the symmetric group). Let $\mathrm{H}$ be the reflection representation of the symmetric group $S_n$, of dimension $n-1$, and let $V = \mathrm{H} \oplus \mathrm{H}^*$ with the standard alternating form $\omega((x,f),(y,g)) = f(y) - g(x)$. Then $S_n$ acts on $V$ by $w \cdot (x,f) = (wx, wf)$, and this action is symplectic. A transposition $s_{ij}$ acts on $\mathrm{H}$ as a complex reflection fixing a hyperplane, and on $V = \mathrm{H}\oplus\mathrm{H}^*$ it fixes the hyperplanes of both summands; the rank of $1 - s_{ij}$ on $V$ is $2$, so transpositions are symplectic reflections, and they generate $S_n$.

Example (wreath products). Let $\Gamma \subseteq SL_2(K)$ be finite and let $G = S_n \ltimes \Gamma^n$ act on $V = (K^2)^n$ by permuting the $n$ planes and acting by $\Gamma$ in each, the form being the direct sum of the standard forms and the action being symplectic. The group $G$ is generated by symplectic reflections: the elements of $\Gamma$ acting in a single plane, the nontrivial ones having rank $2$ on that plane and rank $0$ on the others, together with the transpositions of two planes, which have rank $2$ on the sum of the two planes and rank $0$ elsewhere. The case $\Gamma = \{1\}$ is the symmetric group acting on $(K^2)^n$, the rank-one symplectic reflection group; the case $\Gamma = \{\pm 1\}$ is the hyperoctahedral group $W(B_n) = S_n \ltimes (\mathbb{Z}/2)^n$, generated by the sign changes of a single coordinate and the transpositions, and it sits inside the symplectic group of a symplectic space of dimension $2n$ as a symplectic reflection group. The irreducible symplectic reflection groups have been classified, in the quaternionic form, by Cohen; the classification is the symplectic analogue of the classification of the finite reflection groups.

The Symplectic Reflection Algebra

The Definition

Definition. Let $(V, \omega)$ be a symplectic space over $K$, let $G \subseteq \operatorname{Sp}(V, \omega)$ be a finite symplectic reflection group with set $S$ of symplectic reflections, let $t \in K$ and let $c : S \to K$ be a function that is constant on the conjugacy classes of symplectic reflections. The symplectic reflection algebra $H_{t,c}(V)$ is the quotient of the smash product $T(V) \rtimes G$ of the tensor algebra of $V$ with the group $G$ by the two-sided ideal generated by the relations $$ x\, y - y\, x = t\,\omega(x,y)\, 1 - \sum_{s \in S} c(s)\,\omega_s(x,y)\, s, \qquad x, y \in V, $$ where the elements of $V$ are placed in degree one of $T(V)$ and the group acts by the given representation.

The right-hand side lies in $K \oplus KS$, and the relations are $G$-equivariant: the group acts on $V$ by the symplectic representation, and the function $c$ is invariant, so the ideal is stable under the adjoint action of $G$ and the quotient is an associative algebra with a $G$-action by algebra automorphisms. The algebra is filtered by the degree in $T(V)$, with $G$ in degree zero, and the associated graded algebra is a quotient of $S(V) \rtimes G$.

Example (the trivial group). If $G = \{1\}$ then $S$ is empty and $$ H_{t,0}(V) = T(V) / (xy - yx - t\,\omega(x,y)) , $$ which is the Weyl algebra $A(V)$ for $t \neq 0$ and the symmetric algebra $S(V)$ for $t = 0$. The conventions of the alternating form make $A(V)$ the algebra of the constant Poisson bracket of Symplectic Forms and Poisson Brackets, and the case $G = \{1\}$ is the reason the symplectic reflection algebra is described as a deformation of $S(V) \rtimes G$.

Example (the two-dimensional case). If $\dim V = 2$ and $G \subseteq SL_2(K)$ is finite, the symplectic reflections are the nontrivial elements and the algebra is generated by two elements $x, y$ with the relations $$ x y - y x = t - \sum_{s \neq 1} c(s)\,\omega_s(x,y)\,s . $$ For $G = \mu_n$ generated by a rotation of order $n$ the algebra is the generalised Weyl algebra or the rational Cherednik algebra of type $A_{n-1}$ at the parameter determined by $c$; for $G = \{\pm 1\}$ it is the rational Cherednik algebra of type $A_1$, with the relation $xy - yx = t - 2c\,\sigma$ for the nontrivial element $\sigma$.

The PBW Theorem and the Dunkl Embedding

Theorem (PBW; Etingof–Ginzburg). For every $t \in K$ and every invariant $c$, the multiplication map $$ S(V) \otimes_K K[G] \otimes_K S(V^*) \longrightarrow H_{t,c}(V) $$ is an isomorphism of vector spaces; the algebra has the triangular decomposition $H_{t,c}(V) = S(V) \cdot K[G] \cdot S(V^*)$ in the sense of this isomorphism, and it is a free module over the subalgebra $S(V)$ generated by $V$ and over the subalgebra $S(V^*)$ generated by the linear forms, with bases the corresponding symmetric monomials times the group elements.

Proof sketch. Etingof and Ginzburg embed $H_{1,c}(V)$ into the algebra of differential-reflection operators on $V$: on the polynomial ring $K[V] = S(V^*)$ the elements of $V$ act as the first-order operators $$ D_x = \partial_x - \sum_{s \in S} c(s)\,\frac{\omega_s(x,\alpha_s)}{\alpha_s}\, s , $$ where $\alpha_s$ is a linear form whose zero set is the fixed hyperplane of $s$ and the fraction is the algebraic division by a linear form in the localised ring; the identities $D_x D_y - D_y D_x = \omega(x,y) - \sum_s c(s)\omega_s(x,y)s$ are verified from the transformation law of the $D_x$ under the reflections, and the embedding of $S(V^*)$ by multiplication closes the algebra. Since the operators $D_x$ reduce modulo the reflection terms to the partial derivatives, the associated graded of the image is $S(V) \rtimes G$, which gives the PBW property for $t = 1$; the substitution $x \mapsto t^{1/2}x$ for the elements of $V$ rescales the relation to give every $t \neq 0$, and the case $t = 0$ is the associated graded case. The details are in the cited paper.

Corollary. The algebra $H_{t,c}(V)$ has Gelfand–Kirillov dimension $\dim V$ and the same Hilbert series as $S(V) \rtimes G$; the natural filtration is exhausting and separated; and the algebra is a finite module over its subalgebra $S(V)$ and over its subalgebra $S(V^*)$.

Remark. The Dunkl embedding is the symplectic analogue of the classical Dunkl operator of the rational Cherednik theory, and it is the reason the symplectic reflection algebra is a "rational Cherednik"-type algebra: the deformations of the symmetric algebra are realised by reflection-differential operators whose coefficients are the rank-two forms $\omega_s$ divided by the linear forms cutting out the hyperplanes of the reflections.

The Parameters and Their Rescaling

Definition. The parameter space of the symplectic reflection algebra is $$ \mathrm{P} = K \oplus \bigl(K[S]\bigr)^G, $$ the second factor being the invariants of the permutation representation on the set of symplectic reflections, that is, the functions constant on conjugacy classes. The pair $(t,c)$ has a rescaling: for $\lambda \in K^\times$ the substitution $x \mapsto \lambda x$ on $V$ sends the relations of $(t,c)$ to the relations of $(\lambda^2 t, \lambda^2 c)$, so the pair $(t,c)$ and $(\lambda^2 t, \lambda^2 c)$ give isomorphic algebras, and the pair $(t,c)$ may be normalised by fixing $t = 1$ when $t \neq 0$.

Proposition. The algebra $H_{t,c}(V)$ depends only on the pair $(t,c)$ up to the rescaling of the definition, and the specialisations are: $$ H_{0,0}(V) = S(V) \rtimes G, \qquad H_{t,0}(V) = A(V) \rtimes G, $$ with $A(V)$ the Weyl algebra; the family $\{H_{t,c}\}$ is a flat deformation of $S(V) \rtimes G$ over the parameter space, in the sense that $H_{t,c}$ has a filtration with associated graded $S(V) \rtimes G$ for all $(t,c)$.

Proof. The statements about the specialisations are the definitions. Flatness is the PBW theorem: the associated graded is $S(V) \rtimes G$ independently of $(t,c)$, so the family is a deformation of the associated graded, and the independence of the deformation parameter is the content of the PBW property.

The Centre and the Structure of the Algebra

Theorem. The centre of the symplectic reflection algebra satisfies $$ \operatorname{gr} Z\bigl(H_{0,c}(V)\bigr) = S(V)^G, \qquad Z\bigl(H_{0,0}(V)\bigr) = S(V)^G, \qquad Z\bigl(H_{t,c}(V)\bigr) = K \quad \text{for } t \neq 0 \text{ and } (t,c) \text{ outside a countable union of hyperplanes} . $$ The first two statements are the theorem of Etingof and Ginzburg for the case $t = 0$: the standard filtration of $H_{0,c}$ induces a filtration of the centre whose associated graded is the ring $S(V)^G$ of the invariant polynomials, and at $c = 0$ the centre is exactly that ring, the invariants being central for every $c$. The third is their triviality statement for the deformed level: the algebra $H_{t,0} = A(V) \rtimes G$ is simple with trivial centre for $t \neq 0$, and semi-continuity gives the triviality of the centre outside a countable union of hyperplanes of parameters.

Remark. The two cases are the two ends of the deformation. At $t = 0$ the algebra is a deformation of the smash product whose centre is a deformation of the invariant ring $S(V)^G = K[V]^G = K[V/G]$, the coordinate ring of the symplectic quotient orbifold, the associated graded being that ring and the centre being exactly it at $c = 0$; the algebra can be viewed as a noncommutative resolution or a quantisation of the orbifold. For $t \neq 0$ and parameters outside a countable union of hyperplanes the centre collapses to the scalars, and the algebra is a "quantum" object: it has no classical limit with a Poisson structure of its own, and its representation theory is concentrated in the finite-dimensional or the category-$\mathcal{O}$ part rather than in the centre.

Theorem. The subalgebras $S(V)$ and $S(V^*)$ are polynomial algebras in $\dim V$ indeterminates, the group algebra $KG$ is a subalgebra, and the multiplication maps induce the triangular decomposition; $H_{t,c}(V)$ is free as a left $S(V)$-module and as a right $S(V^*)$-module, with bases respectively the monomials in $S(V^*)$ times $G$ and the monomials in $S(V)$ times $G$.

Proof. The PBW theorem identifies $H_{t,c}$ with $S(V) \otimes KG \otimes S(V^*)$ as a vector space, and the subalgebra generated by $V$ is $S(V)$ because the relations express the commutator of two elements of $V$ in the span of $1$ and $KS$, which does not involve additional generators; the same argument applies to $S(V^*)$, whose generators are the linear forms. The basis statement is then the PBW isomorphism read in a single factor.

Example (the Kleinian case). Let $\dim V = 2$ and let $G \subseteq SL_2(\mathbb{C})$ be finite. Then $S(V)^G = \mathbb{C}[x,y]^G$ is the coordinate ring of the Kleinian singularity $\mathbb{C}^2/G$, and $H_{0,c}(V)$ is a deformation of the skew group ring with centre a deformation of $\mathbb{C}[x,y]^G$, the invariant ring itself occurring at $c = 0$; the algebra $H_{0,c}$ is a "noncommutative resolution" of the singularity for suitable $c$, and its category of modules provides the McKay correspondence between the resolutions of $\mathbb{C}^2/G$ and the representations of $G$.

Representations

The Polynomial Representation and Category $\mathcal{O}$

Definition. The polynomial representation or Dunkl representation of $H_{t,c}(V)$ is the vector space $K[V] = S(V^*)$ on which $S(V^*)$ acts by multiplication and the elements of $V$ act by the Dunkl operators $D_x$ of the PBW theorem. The category $\mathcal{O}$ of $H_{t,c}(V)$ is the full subcategory of the finitely generated modules on which $S(V)$ acts locally finitely; its objects are the "highest weight" modules, and the standard modules are the Verma modules $$ M_{t,c}(\tau) = H_{t,c}(V) \otimes_{S(V) \rtimes G} \tau $$ for a finite-dimensional representation $\tau$ of $G$, on which the elements of $V$ act as zero.

Proposition. The Verma modules $M_{t,c}(\tau)$ have a unique irreducible quotient $L_{t,c}(\tau)$ when the parameter is not resonant; the module $M_{t,c}(\tau)$ is free over $S(V^*)$ and has Gelfand–Kirillov dimension $\dim V$; the polynomial representation is $M_{0,0}(\mathrm{triv})$ and it is irreducible for generic $c$; the finite-dimensional representations of $H_{t,c}$ exist only for special parameters.

Proof sketch. The freeness is the PBW theorem; the existence and uniqueness of the irreducible quotient is the usual highest weight argument for a filtered algebra with a triangular decomposition, using that the cyclic vector generates over $S(V^*)$, and the genericity statement is the standard analysis of the Verma module for the associated "rational" parameters, in which the only obstructions come from finitely many hyperplanes in the parameter space.

The KZ Functor and the Hecke Algebra

Theorem (Etingof–Ginzburg; KZ functor). The Dunkl operators define a local system of differential equations on the complement of the union of the reflection hyperplanes of $V$, with values in the regular representation of $G$, and the monodromy of this local system defines the KZ functor, an exact functor from the category $\mathcal{O}$ of $H_{t,c}(V)$ to the category of finite-dimensional representations of the Hecke algebra of the orbifold. For $t \neq 0$ the functor is an equivalence on the category $\mathcal{O}$ when the group is the doubling of a complex reflection group, the case treated; in general it is the symplectic analogue of the Knizhnik–Zamolodchikov correspondence, and the precise equivalence statements are in the references.

The functor is the analogue of the Knizhnik–Zamolodchikov connection of the rational Cherednik theory and of the quantum group theory; its construction uses the local system of the Dunkl operators on the complement of the reflection hyperplanes and the monodromy of that local system, and the proofs of the equivalence are in the references.

Corollary. For $t \neq 0$ and parameters outside the hyperplanes on which the KZ functor fails to be an equivalence, the simple objects of the category $\mathcal{O}$ of $H_{t,c}(V)$ are related to the simple representations of the Hecke algebra by the KZ functor, and the Verma modules are described by the standard modules of the Hecke algebra; the dimensions of the simples are constant on the connected components of the complement of the resonant hyperplanes in the parameter space when the functor is an equivalence.

Remark. The combination of the two theorems describes the deformation completely: the level $t = 0$ is the "commutative" level at which the centre is the coordinate ring of the quotient, the level $t \neq 0$ is the "quantum" level at which the centre is trivial and the representation theory is that of the Hecke algebra, and the passage from one to the other is the generalised KZ correspondence. The Hecke algebra and the quotient singularity, which are the two objects joined by the correspondence, belong to Hecke Algebras and to the algebraic geometry of the resolutions; the symplectic reflection algebra supplies the quantised interpolation between them.

Quantisation, Symplectic Resolutions and Examples

Remark. The symplectic reflection algebra $H_{t,c}(V)$ is a quantisation of the Poisson algebra $S(V) \rtimes G$: the Poisson bracket of Symplectic Forms and Poisson Brackets extends to the smash product by the Leibniz rule, the deformation parameter $t$ multiplies the constant part of the bracket, and the coefficient $c(s)$ measures the deformation in the direction of the reflection $s$; the PBW theorem says that the quantisation is flat over the parameter space, and the centre theorem says that the Poisson centre of the orbifold is quantised at $t = 0$ and disappears at $t \neq 0$ outside a countable union of hyperplanes. This is the sense in which the article adds the symplectic form to the algebra of Part I: the smash product $S(V) \rtimes G$ is an algebra of Part I, and the form deforms it.

Example (the Hilbert scheme of points). Let $\Gamma \subseteq SL_2(\mathbb{C})$ be finite and let $G = S_n \ltimes \Gamma^n$ act on $V = (\mathbb{C}^2)^n$ as in the rank-one example. The quotient $V/G$ is the $n$-th symmetric power of the Kleinian surface $\mathbb{C}^2/\Gamma$, and it has a canonical symplectic resolution, the Hilbert scheme of points $\operatorname{Hilb}^n(\mathbb{C}^2/\Gamma)$; the symplectic reflection algebra $H_{t,c}(V)$ for the rank-one group is the quantisation of this resolution, and the Kaledin–Verbitsky theorem identifies the algebra with the quantised structure sheaf of the Hilbert scheme, so that the category of modules over $H_{t,c}$ is the category of coherent sheaves on the Hilbert scheme deformed by the parameter. The case $\Gamma = \{1\}$ gives the Hilbert scheme $\operatorname{Hilb}^n(\mathbb{C}^2)$ and the original Alday–Gaiotto–Tachikawa-style construction of the "quantum" Hilbert scheme, and the case of $\Gamma$ a binary dihedral or polyhedral group gives the Hilbert scheme of the corresponding ALE space.

Example (the Calogero–Moser space). For the rank-one group $G = S_n$ acting on $(\mathbb{C}^2)^n$ and at the level $t = 0$ with $c \neq 0$, the spherical subalgebra $e H_{0,c} e$, with $e = |G|^{-1}\sum_{g \in G} g$, is commutative and its spectrum is the Calogero–Moser space, a smooth affine variety of dimension $2n$ obtained as a hyperkähler quotient and a symplectic resolution of the quotient $(\mathbb{C}^2)^n/G = \mathbb{C}^{2n}$; for $t \neq 0$ the spherical subalgebra $e H_{t,c} e$ is a noncommutative deformation of the same space, and its representations are the quantum Calogero–Moser integrable systems of the classical theory.

Example (the higher-rank and exceptional cases). For the groups that are not wreath products, the symplectic reflection algebras are the quantisations of the corresponding symplectic resolutions of the quotient singularities: the complex reflection groups doubled give the rational Cherednik algebras and their Calogero–Moser spaces, the finite symplectic reflection groups in dimension $4$ give the quantisations of the resolutions of the four-dimensional quotient singularities, and in the exceptional cases the resolutions are the "Nakajima quiver varieties" of the corresponding McKay quivers. The general framework is the quantisation theory of the symplectic resolutions due to Kaledin and to Bezrukavnikov–Etingof, and it belongs to the later part of this category; the symplectic reflection algebras are the basic examples it explains.

Summary

Let $(V, \omega)$ be a symplectic space over a field of characteristic zero and let $G \subseteq \operatorname{Sp}(V,\omega)$ be a finite group generated by symplectic reflections, the elements $s$ with $\operatorname{rank}(1-s) = 2$. The symplectic reflection $s$ fixes a hyperplane $H$ pointwise and acts on the $\omega$-orthogonal plane $W$ with eigenvalues $\lambda, \lambda^{-1}$, and the associated rank-two form $\omega_s(x,y) = \tfrac12 \omega((1-s)x,(1-s)y)$ has radical $H$. The symplectic reflection algebra $H_{t,c}(V)$ is the quotient of $T(V) \rtimes G$ by the relations $$ xy - yx = t\,\omega(x,y) - \sum_{s\in S} c(s)\,\omega_s(x,y)\,s , $$ with $t$ a scalar and $c$ an invariant function on the set $S$ of symplectic reflections. The PBW theorem of Etingof and Ginzburg gives the triangular decomposition $H_{t,c} = S(V)\cdot K[G]\cdot S(V^*)$ as vector spaces, and it is proved through the Dunkl embedding of $H_{1,c}$ into the algebra of reflection-differential operators on $V$; the algebra is a flat deformation of $S(V) \rtimes G$ with Gelfand–Kirillov dimension $\dim V$. The specialisations are $H_{0,0} = S(V)\rtimes G$ and $H_{t,0} = A(V)\rtimes G$, with $A(V)$ the Weyl algebra, and the parameters rescale by $(t,c) \mapsto (\lambda^2 t, \lambda^2 c)$. The centre has associated graded $S(V)^G$ at $t = 0$ for every $c$, and equals $S(V)^G$ at $c = 0$, while it reduces to the scalars outside a countable union of hyperplanes in the parameter space. The category $\mathcal{O}$ of locally finite $S(V)$-modules has the Verma modules $M_{t,c}(\tau)$ and their simple quotients, and the KZ functor relates it to the representations of the Hecke algebra, being an equivalence at $t \neq 0$, so that the representation theory at the deformed level is that of the Hecke algebra while at $t = 0$ it is the theory of the orbifold $V/G$. The algebras are the quantisations of the symplectic resolutions of the quotient singularities: the rank-one symplectic reflection algebras are the quantised Hilbert schemes of points on the Kleinian surfaces, and the spherical subalgebras are the quantum Calogero–Moser spaces.

Summary of Notation

Further Reading

• Pavel Etingof and Victor Ginzburg, "Symplectic Reflection Algebras, Calogero–Moser Space, and Deformed Harish-Chandra Homomorphism", Inventiones Mathematicae 147 (2002), 243–348, for the definition, the PBW theorem, the centre and the representation theory.
• Victor Ginzburg, "On the Representation Theory of Symplectic Reflection Algebras" (lecture notes, 2003), for the category $\mathcal{O}$ and the KZ functor.
• Wee Liang Gan and Victor Ginzburg, "Deformed Preprojective Algebras and Symplectic Reflection Algebras", Transactions of the American Mathematical Society 354 (2002), 3943–3959, for the relation to the preprojective algebras and the quantised resolutions.
• Arnaud Beauville, "Symplectic Singularities", Inventiones Mathematicae 139 (2000), 541–549, for the symplectic singularities and their resolutions that the algebras quantise.
• D. Kaledin and M. Verbitsky, "Non-Hermitian Yang–Mills Connections", Selecta Mathematica 4 (1998), 279–320, for the quantisation of the Hilbert schemes of the Kleinian surfaces.
• A. M. Cohen, "Finite Quaternionic Reflection Groups", Journal of Algebra 64 (1980), 293–324, for the classification of the symplectic reflection groups.