LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 10. Moment maps in algebraic setting 10.1. Symplectic algebraic varieties. An affine algebraic variety X is said to be Poisson if C[X] is equipped with a Poisson bracket. Exercise 10.1. Let A be a commutative algebra and B be a localization of A. Let A be equipped with a bracket. Show that there is a unique bracket on B such that the natural homomorphism A ! B respects the bracket. Thanks to this exercise, the sheaf OX of regular functions on X acquires a bracket (i.e., we have brackets on all algebras of sections and the restriction homomorphisms are compatible with the bracket). We say that an arbitrary (=not necessarily affine) variety X is Poisson if the sheaf OX comes equipped with a Poisson bracket. O Recall that on a variety X suchV that X is equipped with a bracket we have a bivector 2 reg 2 reg ∗ ! (=a bivector field) P Γ(X ; TX ). This gives rise to a map vx : Tx X TxX for 2 reg 7! · x X ; α Px(α; ). We say that P is nondegenerateV in x if this map is an isomorphism. 2 2 ∗ In this case, we can use this map to get a 2-form !x Tx X: !x(vx(α); vx(β)) = Px(α; β) = hα; vx(β)i = −⟨vx(α); βi. Now suppose X is smooth. Suppose that P is non-degenerate (=non-degenerate at all points). So we have a non-degenerate form ! on X. The condition that P is Poisson is equivalent to d! = 0. A non-degenerate closed form ! is called symplectic (and X is called a symplectic variety). The most important for us class of symplectic varieties is cotangent bundles. Let X0 be ∗ a smooth algebraic variety, set X := T X0. A symplectic form ! on X is introduced as follows. First, let us introduce a canonical 1-form α. We need to say how αx pairs with a tangent vector for any x 2 X. A point X can be thought as a pair (x0; β), where x0 2 X0 2 ∗ and β Tx0 X0. Consider the projection π : X X0 (defined by π(x) = x0). For x = (x0; β) we define αx by hαx; vi = hβ; dxπ(v)i. We can write α in \coordinates". If we worked in the C1- or analytic setting, we could use the usual coordinates. However, we cannot do this because we want to show that α is an algebraic form. So we will use an algebro-geometric substitute for coordinate charts: ´etale coordinates. Namely, we can introduce ´etalecoordinates in a neighborhood of each point 2 1 n 1 n x0 X0. Let us choose functions x ; : : : ; x with a property that dx0 x ; : : : dx0 x form a basis ∗ 1 n in Tx0 X0. Then dx ; : : : ; dx are linearly independent at any point from some neighborhood 0 0 ! Cn 1 n 1 n X0 of x0. So the map ' : X0 given by (x ; : : : ; x ) is ´etaleand we call x ; : : : ; x ´etale ∗ 0 coordinates. Then we can get ´etalecoordinates yP1; : : : ; yn on T X0 as follows: by definition i i n i 1 n y (x0; β) is the coefficient of dx0 x in β, i.e., β = i=1 yi(x0; β)dx0 x (andP we view x ; : : : ; x ∗ 0 ∗ 0 n i as functions on T X0 via pull-back). Then, on T X0 , α is given by i=1 yidx . There is an important remark about α: it is canonical. In particular, if we have a group ∗ action on X0, it naturally lifts to T X0: g(x0; β) = (gx0; g∗x0 β), where g∗x0 is the isomorphism 1 2 IVAN LOSEV ∗ ! ∗ Tx X0 Tgx X0 induced by g. The coordinate free definition of α implies that α is invariant 0 0 ∗ under any such group action on T X0. P − n i ^ Now set ! = dα so that, in the ´etalecoordinates, ! = i=1 dx dyi. We immediately see that ! is a symplectic form. Also let us point out that if X0 is a vector space, then ! is ⊕ ∗ a constant form (=skew-symmetric bilinear form) on the double vector space X0 X0 . The remark in the previous paragraph applies to ! as well. 10.2. Hamiltonian vector fields. Let X be a Poisson variety and f be a local section of OX . Then we can form the vector field v(f) = P (df; ·) (defined in the domain of definition of f). This is called the Hamiltonian vector field (or the skew gradient) of f. Clearly, v is linear, and satisfies the Leibniz identity v(fg) = gv(f) + fv(g). Further, we have (1) Lv(f)g = −⟨v(f); dgi = ff; gg: Here and below we write Lξ for the Lie derivative of ξ so that Lξf = −@ξf. Recall that in the C1-situation, the Lie derivative is defined as follows. We pick a flow g(t) produced d j by the vector field ξ and then for a tensor field τ define Lξτ = dt g(t)∗τ t=0. In particular, d − j − if τ is a function f, then we get Lξ(f) = dt f(g( t)) t=0 = @ξf. If τ is a vector field, then Lξτ = [ξ; τ], where, by convention, the bracket on the vector fields is introduced by L[ξ,η]f = [Lξ;Lη]f. Finally, if τ is a form, then we have the Cartan formula: (2) Lξτ = −dιξτ − ιξdτ; where ιξ stands for the contraction with ξ (as the first argument): ιξτ(:::) = τ(ξ; : : :). In particular, if both ξ and τ are algebraic, then so is Lξτ, and we can define Lξτ using the formulas above. Using (1) and the Jacobi identity for {·; ·}, we deduce that the map f 7! v(f) is a Lie algebra homomorphism. Also we remark that every Hamiltonian vector field is Poisson, i.e., (3) Lv(f)P = 0 (this is yet another way to state the Jaconi identity for {·; ·}). If X is symplectic, we can rewrite the definition of the Hamiltonian vector field as (4) ιv(f)! = df: Also we have (5) !(v(f); v(g)) = ff; gg and (6) Lv(f)! = 0: So in this case f 7! v(f) is a Lie algebra homomorphism between C[X] and the algebra SVect(X) of symplectic vector fields on X. ∗ Consider the case of X = T X0, where, for simplicity, we assume that X0 is affine. Then C C [X] = SC[X0](Vect(X0)). As a function on [X] the vector field ξ is given by h i (7) ξ(x0; β) = β; ξx0 : Let us compute the vector fields v(f); f 2 C[X0]; and v(ξ); ξ 2 Vect(X0). We claim that − ∗ ∗ v(f) = df, viewed as a vertical vector field on T X0, its value on the fiber Tx X0 is constant − 0 dx0 f. To avoid confusion below we will write Df for the vector field df. Further, to a vector ~ field ξ on X0 we can assign a vector field ξ on X by requiring Lξ~η = [ξ; η];Lξ~g = Lξg for ~ ~ η 2 Vect(X0); g 2 C[X0]. We claim that v(ξ) = ξ. The vector field ξ has the following LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS 3 meaning. Assume that we are in the C1-setting. Then to ξ we can assign its flow g(t) (of ∗ ~ diffeomorphisms of X0). Then we can canonically lift this flow to T X0. The vector field ξ is ~ associated to the lifted flow. In particular, from this description one sees that d(x0,β)ξ = ξx0 . ∗ Applying (2) to τ = α and a vector field η on T X0, and using −dα = !, we get Lηα = −dιηα + ιη! and so (8) ιη! = Lηα + ιη!: If η = −Df, then ιηα = 0 (α vanishes on all vertical vector fields by the coordinate free construction). So we get ι−Df ! = L−Df α. Again, theP construction ofPα implies that L−Df α = n i n i @Df α = df (in local coordinates we have @Df α = i=1 @Df yidx = i=1 @xi fdx = df). So ι−Df ! = df = ιv(f)! so v(f) = −Df. ~ 1 Now let us check that v(ξ) = ξ. We claim that Lξ~α = 0. In the C -setting, this follows ∗ from the observation that α is preserved by any diffeomorphism of T X0 lifted from X0. Since all formulas in the algebraic setting are the same as in the C1 one, we get our claim. Also we remark that by the construction of ξ~, we have dπ(ξ~) = ξ and therefore, thanks to ∗ (7), ιξ~α = ξ (as functions on T X0). So we have ιξ~! = dιξ~α. But ιξ~α is ξ, by the description of the function ξ above. Exercise 10.2. Show that the Poisson bracket on C[X] can be characterized as follows: we have ff; gg = 0; fξ; fg = Lξf; fξ; ηg = [ξ; η] for f; g 2 C[X0]; ξ; η 2 Vect(X0). Deduce that, C − with respect to the standard grading on [X] = SC[X0](Vect(X0)), the bracket has degree 1. The construction of Hamiltonian vector fields is of importance in Classical Mechanics. Namely, we can consider a mechanical system on a Poisson variety X whose velocity vector − d is v(H) so that dt f(x(t)) = (Lv(H)f)(x(t)). In this case, the function H is interpreted as the Hamiltonian (=the full, i.e., \kinetic + potential", energy) of this system. The condition on a function F to be a first integral (=preserved quantity) of this system is Lv(H)F = 0, i.e., fH; F g = 0.