Super formal Darboux-Weinstein theorem and finite W-superalgebras

Let v=v_0¯+v_1¯ be a Z₂-graded (super) vector space with an even C^×-action and χ∈v_0¯ ^⁎ be a fixed point of the induced action. In this paper we prove an equivariant Darboux-Weinstein theorem for the formal polynomial algebras Aˆ=S[v_0¯]^∧_χ ⊗⋀(v_1¯). We also give a quantum version of it. Let g=g_0¯+g_1¯ be a basic Lie superalgebra and e∈g_0¯ be a nilpotent element. We use the equivariant quantum Darboux-Weinstein theorem to give a Poisson geometric realization of the finite W-superalgebra U(g,e) in the sense of Losev. An indirect relation between finite W-(super)algebras U(g,e) and U(g_0¯,e) is presented. Finally we use such a realization to study the finite-dimensional irreducible modules over U(g,e).