# Gross-Hopkins Duals of Higher Real K-theory Spectra

## TL;DR

This paper determines the shifts in Gross-Hopkins duality for higher real K-theory spectra at odd primes, specifically for certain finite subgroups with p-torsion, extending known results from lower cases.

## Contribution

It identifies the specific shifts in Gross-Hopkins duality for homotopy fixed point spectra with p-torsion subgroups, generalizing previous lower-dimensional results.

## Key findings

- Determined the duality shifts for G containing p-torsion.
- Extended known duality results from n=2, p=3 to higher cases.
- Provided explicit duality shifts for these spectra.

## Abstract

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 the 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 $p=3$.