Gross–Hopkins duals of higher real K–theory spectra

We determine the Gross–Hopkins duals of certain higher real K–theory spectra. More specifically, let p be an odd prime, and consider the Morava E–theory spectrum of height n = p − 1. It is known, in expert circles, that for certain finite subgroups G of the Morava stabilizer group, the homotopy fixed point spectra E_n ^hG are Gross–Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups G which contain p–torsion. This generalizes previous results for n = 2 and p = 3.