MATH 7364 Lecture 6 problems

MATH 7364 Lecture 6 problems - PROBLEMS ON SYMPLECTIC...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 6. SRA Exercise 6.1. Show that the Weyl algebra W ( V ) is a filtered 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 ) coincides with the initial bracket. Exercise 6.2. Show that if gr H κ = S ( V )#Γ , then κ : 2 V C Γ is a Γ -equivariant map (where Γ acts on C Γ via the adjoint representation). Furthermore, show that if 1 V Γ , then the image of κ lies in C Γ . Exercise 6.3. For κ = γ Γ κ γ γ , we have [ κ ( u, v ) , w ] = γ Γ κ γ ( u, v )( γ ( w ) w ) γ . Exercise 6.4. Show that im( γ 1 V ) ker( γ 1 V ) = V for any γ Γ . Further, show that the summands are orthogonal with respect to ω and, in particular, the restrictions of ω to these subspaces are non-degenerate. Exercise 6.5. Show that ω γ ( u, v )( γ ( w ) w ) + ω γ ( v, w )( γ ( u ) u ) + ω γ ( w, u )( γ ( v ) v ) = 0 . Exercise 6.6. Show that a symplectically irreducible Γ -module V is either irreducible, or is the sum U U , where U is irreducible and not symplectic.
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