Orders of elements of equivariant \(J\)-groups of complex projective spaces (Q1866469)

Let \(G\) be a finite group and let \(X\) be a compact \(G\)-space. Then the equivariant \(J\)-group \(JO_G(X)\) is defined to be the set of all stable \(G\)-fibre homotopy equivalence classes of real \(G\)-vector bundles over \(X\), so that there is a natural surjection \(J: \widetilde{KO}_G(X)\to JO_G(X)\). If \(X\) is a trivial \(G\)-space then one has \(KO_G(X)\cong KO(X)\otimes R(G; \mathbb{R})\oplus K(X)\otimes R(G;\mathbb{C})\oplus KSp(X)\otimes R(G;\mathbb{H})\), where \(R(G;\mathbb{F})\) is the free Abelian group generated by the irreducible real \(G\)-modules with their endomorphism algebra equal to \(\mathbb{F}\). Assume that \(G\) has no quaternionic type irreducible representations, that is, \(R(G;\mathbb{H})= 0\) holds and let \(\mathbb{C} P^n\) denote the complex projective \(n\)-space with trivial \(G\)-action. Then from the above decomposition one obtains expressions of elements \(w\in KO_G(\mathbb{C} P^n)\) in terms of the Hopf line bundle over \(\mathbb{C} P^n\) and real and complex type irreducible representations of \(G\). In this paper the author gives an explicit formula for computing the order of \(J(w)\) for \(w\in \widetilde{KO}_G(\mathbb{C} P^n)\) (Theorems 1.1 and 1.3). This formula is, however, too involved for brief summary here due to the complexity of the data needed to describe it. The author also gives an example of the order computation which deals with a certain element of \(JO_{\mathbb{Z}/5}(\mathbb{C} P^4)\). This sample computation shows that the formula given here is the best possible. The proof is done by division into two steps, the \(p\)-group case and the general case, and makes use of the results of \textit{T. tom Dieck} [Transformation groups and representation theory, Lecture Notes in Math. 766, Springer-Verlag (1979; Zbl 0445.57023)] and \textit{J. E. McClure} [Math. Z. 183, 229-253 (1983; Zbl 0521.55011)].