MATH 7364 Lecture 16 problems

MATH 7364 Lecture 16 problems - PROBLEMS ON SYMPLECTIC...

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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, θ ( g ) z ) . Show that x X θ ss iff G ( x, 1) doesn’t intersect X × { 0 } . Exercise 16.2. Prove that for θ = ( - 1) i Q 0 , the subset R ss consists of all elements ( x a , x a * , y i , z i ) a Q 1 ,i Q 0 (here x a Hom( V t ( a ) , V h ( a ) ) , x a * Hom( V h ( a ) , V t ( a ) ) , y i Hom( W i , V i ) , z i Hom( V i , W i ) ) such that there are no proper subspaces V i V i that are stable under all x a , x a * and such that im y i V i . Deduce that the action of GL( v ) on Rep( Q, v, w ) θ ss is free. Problem 16.1. Prove that a generic fiber μ 1 ( λ ) , λ g G , is smooth and connected. You may use the following strategy: (1) Show that all G -orbits in μ 1 ( λ ) are free. Deduce that μ 1 ( λ ) is smooth. (2) Show that μ 1 ( C λ ) is normal. (3) The affine version of Zariski’s main theorem says that the morphism μ 1 ( C λ ) C λ decomposes into a composition of a morphism μ 1 ( C λ ) X with connected general fiber and a finite morphism X C λ . Use this to prove that μ 1 ( λ ) is connected. Problem 16.2. Show that the algebra W ~ ( V )[ f 1 ] is Noetherian. Deduce from here that
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