Fixed point indices of central configurations (Q497980)

Let \(E\) be \(\mathbb{R}^d\) with the euclidean product \(\cdot\), let \(\Delta:=\bigcup_{1\leq i<j\leq n}\{q\in E^n|\;q_i=q_j\}\) and \(X=E^n\setminus\Delta\) (the configuration space of \(E\)). Choose \(n\) positive reals \(m_1,\dotsc,m_n\) and define an inner product by \(\left<v,w\right>_M=\sum_{j=1}^nm_jv_j\cdot w_j\). Let \(U\) be a regular positive potential on \(X\) which is invariant under isometries of \(E\) and is homogeneous of negative degree. A configuration \(q\in X\) is said to be central if there is a nonzero real \(\lambda\) such that \(\nabla_MU(q)=\lambda q\) where of course \(\nabla_M\) is the gradient with respect to \(\left<\cdot,\cdot\right>_M\). Denote \(Y=\{q\in X|\;\sum_{j=1}^nm_jq_j=0\}\). Denote by \(G\) the symmetry group of \(U\) on \(Y\) and \(S:=\{q\in Y| \|q\|_M^2=1\}\) (the inertia ellipsoid). Define \(F:S\to S\) by \(F(q)=. -\frac{\nabla_MU(q)}{\|\nabla_MU(q)\|_M}\). Then a configuration is central iff it is a fixed point of \(F\). Moreover, it is easy to see that \(F\) is \(G\)-equivariant. The main result then reads: Assume that \(G\supset SO(E)\) and denote by \(\bar{F}\) the induced map \(\bar{F}:S/SO(E)\to S/SO(E)\) and denote the projection \(S\to S/SO(E)\) by \(\pi\). Then \(\pi(\text{Fix}(F))=\text{Fix}(\bar{F})\).