The Chromatic Fourier Transform

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height n = 0, as well as a certain duality for the E_n-(co)homology of π-finite spectra, established by Hopkins and Lurie, at heights n ≥ 1. We use this theory to generalize said duality in three different directions. First, we extend it from Z-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of E_n. Second, we lift it to the telescopic setting by replacing E_n with T_(n)-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg–Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal ∞-categories of local systems of K(n)-local E_n-modules and relate it to (semiadditive) redshift phenomena.