PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
10. Moment maps in algebraic setting
Exercise 10.1. Let A be a commutative algebra and B be a localization of A. Show that
any bracket on A extends to a unique bracket on B .
Exercise 10.2. Show that the Poisson
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
13. Quantum CM systems and Rational Cherednik algebras
Exercise 13.1. Prove that wDa w1 = Dwa for all w W, a h.
Exercise 13.2. Prove an analog of Proposition on the properties of Dunkl operators for
complex reect
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
16. Symplectic resolutions and their deformations
Exercise 16.1. Consider the G-action on X C given by g (x, z ) = (gx, (g )z ). Show that
x X ss i G(x, 1) doesnt intersect X cfw_0.
Exercise 16.2. Prove that for
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
17. Procesi bundles and their deformations
Problem 17.1. Let X be an algebraic variety, F0 be a coherent sheaf on X and D be a FCS
(=at, complete and separated) deformation of OX over C[ ].
(1) Show that the cate
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
9. Commutativity and centers
Exercise 9.1. Show that cfw_, t,c = tcfw_, , where cfw_, is the standard bracket on S (V ) .
Exercise 9.2. Prove the commutativity theorem in the case when V is not necessarily sympl
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
14. Quantum Hamiltonian reduction vs SRA
Exercise 14.1. Prove that ([, ]) = 1 [( ), ( )] for any , g.
Exercise 14.2. Prove that the center of W (V ) coincides with C[ ].
Exercise 14.3. Describe the map A for A =
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
5. Symplectic quotient singularities
Problem 5.1. Consider the representation space Rep(Q, ) for cyclic quiver Q with r + 1
vertices and the corresponding morphism : Rep(Q, ) gl( ). Describe the irreducible
compo
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
2. CBH algebras (left from last time)
Exercise 2.5. Prove that there are no non-constant invariant polynomials for the action of
the one-dimensional torus C on Cn given by t.(x1 , . . . , xn ) = (tx1 , . . . , tx
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
7. Hochschild cohomology and deformations
Exercise 7.1. Let A0 be an algebra and A1 = A0 P A0 its rst order deformation with
product dened by (a, b) = ab + 1 (a, b) for a, b A0 , where 1 P C 2 (A, A). Show that
t
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
1. Kleinian singularities
Problem 1.1. Let G be a nite subgroup of SO3 (R). Consider its action on the unit sphere.
Show that any non-unit element of G xes a unique pair of opposite points and that the stabilizer
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
2. CBH algebras
Exercise 2.1. Let : A
A be an algebra epimorphism. Suppose A is ltered. Check
n
n
that A := (A ) denes an algebra ltration on A .
Problem 2.1. Show that the monomials xi y j , i, j
0, form a basis
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
3. McKay correspondence upgraded (from last time)
Exercise 3.3. A map C2 C C extends to a representation from Rep (Cx, y #, C)
if and only if it is -equivariant.
Exercise 3.4. Show that
Hom (C C, C) =
2
r
Mij Hom
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
6. SRA
Exercise 6.1. Show that the Weyl algebra W (V ) is a ltered deformation of S (V ) (the case
dim V = 2 was considered before). Moreover, check that the Poisson bracket on S (V ) induced
from W (V ) coincide
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
18. Category O for Cherednik algebras
Exercise 18.1. Use gr Hc = S (h h )#W to show that Hc is Noetherian.
Exercise 18.2. We have [h, x] = x, [h, w] = 0, [h, y ] = y for all x h , w W, y h.
Problem 18.1. Write an
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
19. KZ functor I
Problem 19.1. Let M1 , M2 be DX -modules that are coherent sheaves. Show that
dim HomDX (M1 , M2 ) < .
Exercise 19.1. 1 Let M be an Hc -module with locally nilpotent action of h. Show that M is
n
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
8. Spherical SRA
Exercise 8.1. Use the theorem that Hun H when V is symplectically irreducible to deduce
=
that H is a graded deformation of S (V )# even if V is not symplectically irreducible.
Exercise 8.2. Prov
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
11. CM system and Hamiltonian reduction
Exercise 11.1. Let X0 be a smooth algebraic variety equipped with a free action of a nite
group . Show that T (X0 /) is naturally identied with (T X0 )/ (and that the ident
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
12. CM systems and quantum mechanics
Exercise 12.1. 1 Show that the trajectories for H = 1 tr(Y 2 ) on R = T Matn (C) are of the
2
form (X tY, Y ).
Problem 12.1. Prove part (2) of the main theorem (integration of
PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS
15. Quotient singularities as quiver varieties
The purpose of this problem set is to recover results of Lecture 15 in the special case of
1 = cfw_1.
Problem 15.1. Describe the GL(n)-orbits on Matn (C) Cn . Deduce
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
1. Kleinian singularities
Kleinian singularities are remarkable singular ane surfaces (varieties of dimension 2).
They arise as quotients of C2 by nite subgroups of SL2 (C). Our main interest is not in
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
2. Algebras of Crawley-Boevey and Holland
This lecture consists of three dierent pieces. The rst two are related to the deformation
theory of Kleinian singularities and can be characterized as a genera
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
3. McKay correspondence upgraded
3.1. Categorical quotients. Let us start by reminding a few general denitions regarding
algebraic groups. By an algebraic group one means a group that is also an algebr
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
4. Deformed preprojective algebras
Recall that we have introduced the double McKay quiver Q, its representation space
Rep(Q, ) acted on by the group GL( ) and also a somewhat mysterious quadratic map
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
4. Deformed preprojective algebras, contd
4.1. Recap. Recall that in the previous lecture we have identied Cx, y # with (C2
TC
r
1
0 (CQ) . Also recall that f Cx, y #f = CQ, where f =
C), and CQ with
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
6. Symplectic reflection algebras
6.1. Denition of SRA. Let V be a nite dimensional complex vector space equipped with
a non-degenerate skew-symmetric form . Then there is a distinguished ltered deform
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
7. Hochschild cohomology and deformations
In the previous lecture we have introduced Hochschild cohomology. These were cohomology of the complex of the spaces C n (A, M ) = HomC (An , M ) with dierenti
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
8. Spherical SRA
Let V be a nite dimensional vector space equipped with a symplectic form and be
a nite subgroup in Sp(V ). Let S denote the subset of all symplectic reections s , i.e.,
all elements wi
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
9. Commutativity and centers
9.1. Commutativity theorem: statement and scheme of proof. It is a natural question to ask when the algebra eHt,c e is commutative. This happens to have a very elegant
answ
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
10. Moment maps in algebraic setting
10.1. Symplectic algebraic varieties. An ane algebraic variety X is said to be Poisson
if C[X ] is equipped with a Poisson bracket.
Exercise 10.1. Let A be a commut
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
11. Calogero-Moser system and Hamiltonian reduction
11.1. Calogero-Moser system. The Calogero-Moser system is the system of n distinct
points of the same mass, say 1, on the line (we work over C so we