Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real $K$ -theory spectra of Hopkins and Miller at height $n=p-1$ , for $p$ an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra $E_{n}^{hG}$ , where $E_{n}$ is Lubin–Tate $E$ -theory at the prime $p$ and height $n=p-1$ , and $G$ is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.

We present a new proof of Anderson's result that the real K-theory spectrum is Anderson self-dual up to a fourfold suspension shift; more strongly, we show that the Anderson dual of the complex K-theory spectrum KU is C2-equivariantly equivalent to Σ4KU, where C2 acts by complex conjugation. We give an algebro-geometric interpretation of this result in spectrally derived algebraic geometry and apply the result to calculate 2-primary Gross-Hopkins duality at height 1. From the latter we obtain a new computation of the group of exotic elements of the K(1)-local Picard group.