Differentiable structures on symplectic reductions (Q1268125)

Let \(G\) be a complex reductive Lie group acting holomorphically on \(X\), a Stein manifold. The example of \(G\) being the non-zero complex numbers acting on \({\mathbb C}^3\) by \((z_1,z_2,z_3)\mapsto(\lambda z_1,\lambda z_2,\lambda^{-1}z_3)\) is already interesting. The \textit{categorical quotient} \(X//G\) is formed from the local \(G\)-invariant holomorphic functions on \(X\). In the example, this quotient is \({\mathbb C}^2\). Generally, the categorical quotient is a Stein space and \(\pi:X\to X//G\) is holomorphic. Let \(K\) be a maximal compact subgroup of \(G\). By averaging over \(K\), there are \(K\)-invariant plurisubharmonic exhaustion functions on \(X\). Such a function gives rise to a moment map and a symplectic reduction of \(X\). Though this reduction may be identified with \(X//G\) as a topological space, the notion of smooth function is generally weaker. Moreover, a different exhaustion function generally gives rise to a different class of smooth functions, even in the simple example above. The main result of this article is that, under these circumstances, there is a homeomorphism of \(X//G\) which reconciles matters: it takes the smooth functions defined with respect to one exhaustion function into those defined with respect to the other.