Adams operations in the connective K-theory of compact Lie groups (Q1822382)

Let G be a compact, 1-connected, simple Lie group of rank 2 or 3, that is, G be SU(3), Sp(2), \(G_ 2\), SU(4), Spin(7) or Sp(3). For these groups, the author has given a complete description of the Chern character [ibid. 22, 463-488 (1985; Zbl 0579.57020)]. Using this, he computes the Adams operations on \(K^*(G)\). Let p be an odd prime, \(KZ_{(p)}\) denote the ring spectrum representing complex K-theory localized at p, and \(kZ_{(p)}\) be its (-1)-connective cover. There is a ring spectrum g(p) such that \(kZ_{(p)}\simeq \bigvee^{p- 2}_{i=0}\Sigma^{2i}g(p).\) Let \(\theta_ r: g(p)\to \Sigma^{2p- 2}g(p)\) be a unique map of spectra such that \((v\cdot)\theta_ r\simeq \psi^ r-1,\) where \(v\cdot: \Sigma^{2p-2}g(p)\to g(p)\) is the multiplication by v and \(\psi^ r\) is the stable Adams operation \(((r,p)=1)\). Let j(p) be the fibre spectrum of \(\theta_ r\). The author computes the i-th reduced j(p)-homology (resp. cohomology) group \(j(p)_ i(G)\) (resp. \(j(p)^ i(g))\) of G.