Poisson maps between character varieties: gluing and capping

Abstract

Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G = SL(2, C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic −3 surface. 1 Introduction 2 Poisson structure on character varieties: Betti point-of-view 3 Poisson structure on character varieties: De Rham point-of-view 4 Open subsets and Poisson structure 5 Capping: symplectic and Poisson extensions of Poisson character varieties 6 Gluing via symplectic quotients 7 Proof of Theorem 2.7 References 1255 1256 Biswas, Hurtubise, Jeffrey, and Lawton 2. Poisson structure on character varieties: Betti point-of-view 2.1. Reductive groups Let G be a connected reductive affine algebraic group over C. By the central isogeny theorem, G ∼ = DG × F T , where DG = [G, G] is the derived subgroup, T ∼ = (C * ) s is the maximal central torus, and F = T ∩ DG. The group G acts on itself by conjugation and the geometric invariant theoretic quotient G/ /G is isomorphic to T/W , where T ⊂ G is a maximal torus and W is the Weyl group N G (T)/T. By potentially enlarging F , we can assume DG is simply connected. With that assumption made, by results of Steinberg [St], we can say more: