On a method to construct analytic actions of non-compact Lie groups on a sphere (Q1090992)

A square matrix \(M=(m_{ij})\) of degree n with real coefficients \(m_{ij}\) satisfies the outward transversality condition if \(\frac{d}{dt}\| \exp (tM)x\| >0\) for each \(x\in {\mathbb{R}}^ n-\{0\}\) and \(t\in {\mathbb{R}}\), or, equivalently, the quadratic form \(x\cdot Mx=\sum_{i,j}m_{ij} x_ ix_ j\) is positive definite. For example, \(M=\left( \begin{matrix} a\\ c\end{matrix} \begin{matrix} b\\ d\end{matrix} \right)\) satisfies the outward transversality condition if and only if \(a>0\) and \(4ad-(b+c)^ 2>0.\) The author shows that if M satisfies the outward transversality condition, than there exists a unique real valued analytic function \(\tau\) on \({\mathbb{R}}^ n-\{0\}\) such that \(\| \exp (\tau (x)M)x\| =1\) for each \(x\in {\mathbb{R}}^ n-\{0\}\). Then, for a Lie group G and its representation \(\rho\) : \(G\to GL(n,{\mathbb{R}})\) with \(\rho (g)M=M\rho (g)\) for all \(g\in G\), the author considers an analytic action of G on \(S^{n-1}\) defined by (g,x)\(\mapsto \exp (\tau (\rho (g)x)M)\rho (g)x\), called a twisted linear action of G on \(S^{n-1}\) associated to \(\rho\). If M is the identity matrix, the action is called a linear action of G on \(S^{n-1}\) associated to \(\rho\). The author proves that if G is compact, then any twisted linear action of G on \(S^{n-1}\) associated to \(\rho\) is equivariantly analytically diffeomorphic to the linear action of G on \(S^{n-1}\) associated to \(\rho\). Finally, the author studies twisted linear actions of SL(n, \({\mathbb{R}})\) on \(S^{2n-1}\) associated to \(\rho_ n\otimes I_ 2\), where \(\rho_ n\) is the natural inclusion of SL(n, \({\mathbb{R}})\) into GL(n, \({\mathbb{R}})\) and \(I_ 2\) is the identity matrix of degree 2. The author constructs uncountably many topologically distinct twisted linear actions of SL(2, \({\mathbb{R}})\) on \(S^ 3\) associated to \(\rho_ 2\otimes I_ 2\).