Certain aspects of twisted linear actions (Q917985)

In a previous paper [Tôhoku Math. J., II. Ser. 39, 61-69 (1987; Zbl 0622.57028)], for a Lie group G and its representation \(\rho\) : \(G\to GL(n,{\mathbb{R}})\), the author has introduced the concept of a twisted linear action of G on the sphere \(S^{n-1}\) associated to \(\rho\), so that a twisted linear action is, in particular, an analytic action. The author has presented an example showing that there are uncountably many topologically distinct twisted linear actions of SL(n,\({\mathbb{R}})\) on \(S^{2n-1}\) associated to the same representation \(\rho\) : SL(n,\({\mathbb{R}})\to GL(2n,{\mathbb{R}})\). In the present paper, the author generalizes this example by showing that for \(n>k\geq 2\), there are uncountably many topologically distinct twisted linear actions of SL(n,\({\mathbb{R}})\) on \(S^{nk-1}\) associated to the same representation \(\rho\) : SL(n,\({\mathbb{R}})\to GL(nk,{\mathbb{R}})\). Moreover, he provides another examples which, in turn, shows that for \(k\geq n\geq 2\), there are uncountably many \(C^ 1\)-differentiably distinct but topologically equivalent twisted linear actions of SL(n,\({\mathbb{R}})\) on \(S^ k\) associated to the same representation. In another paper [Part II of the present one, Tôhoku Math. J., II. Ser. 41, 561-573 (1989)], the author shows that for \(n\geq 1\), there are uncountably many \(C^ 2\)- differentiably distinct but \(C^ 1\)-differentiably equivalent twisted linear actions of \({\mathbb{R}}^ n\) on \(S^ n\) associated to the same representation.