MATH 7364 Lecture 11 problems

MATH 7364 Lecture 11 problems - im ι in-tersects any orbit...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 11. CM system and Hamiltonian reduction Exercise 11.1. Let X 0 be a smooth algebraic variety equipped with a free action of a finite group Γ . Show that T ( X 0 / Γ) is naturally identified with ( T X 0 ) / Γ (and that the identifica- tion intertwines the symplectic forms). Exercise 11.2. Let f 1 , . . . , f m be functions on a symplectic variety X such that { f i , f j } = 0 for all i, j . Show that the dimension of the span of d x f 1 , . . . , d x f m has dimension not exceeding 1 2 dim X . Exercise 11.3. Show that the action of G on μ 1 ( O ) Reg is free. Also check that im
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Unformatted text preview: im ι in-tersects any orbit and that elements ι ( p ) , ι ( p ′ ) are G-conjugate if and only if p and p ′ are S n-conjugate. Problem 11.1. Show that the action of G on μ − 1 ( O ) is free. Exercise 11.4. 1 Prove that a G-invariant ideal in S ( g ) is automatically Poisson. Also show that the converse is true provided G is connected. Exercise 11.5. Equip the algebra A/// I G with a natural Poisson bracket. 1 This exercise as well as the next one already appeared in PSet 10 1...
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