ABSTRACT 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: M X, a coisotropic subvariety S X, and a stabilizer subgroupoid H\, \, \\ \, \, S. Our first main result is that the Poisson bracket on kM induces a Poisson bracket on the subquotient k ^-1 (S) ^ {H}. The Poisson scheme M //ₒ, {₇} G: = Spec (k ^-1 (S) ^ {H}) is then declared to be a Hamiltonian reduction of M. Other main results include sufficient conditions for M //ₒ, {₇} G 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.
Crooks et al. (Sat,) studied this question.