Scheme-theoretic coisotropic reduction

We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over k=R or k=C, and is formulated for an affine symplectic groupoid G\,\, \\ \,\,X, an affine Hamiltonian G-scheme μ :M X, a coisotropic subvariety S⊆ X, and a stabilizer subgroupoid H\,\, \\ \,\,S. Our first main result is that the Poisson bracket on k[M] induces a Poisson bracket on the subquotient k[μ ⁻¹(S)]H. The Poisson scheme M //_S,HG:= Spec(k[μ ⁻¹(S)]H) is then declared to be a Hamiltonian reduction of M. Other main results include sufficient conditions for M //_S,HG to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to [6], where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden–Ratiu reduction [19], Mikami–Weinstein reduction [20], Śniatycki–Weinstein reduction [23], and symplectic reduction along general coisotropic submanifolds [6]. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore–Tachikawa conjecture.