On the maximal spectral type of nilsystems

TL;DR

This paper demonstrates that ergodic nilsystems have a spectral decomposition into discrete and Lebesgue spectrum parts, extending previous results to higher-step and more general nilsystems.

Contribution

It generalizes the spectral type decomposition of nilsystems to arbitrary step and broader classes, building on and extending prior foundational work.

Findings

Decomposition of $L^2(G/ Gamma)$ into discrete and Lebesgue spectrum.

Extension of spectral results from 2-step to k-step nilsystems.

Generalization to nilsystems with connected, simply connected groups.

Abstract

Let $(G/\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\geq 2$. We adapt an argument of Parry to show that $L^2(G/\Gamma)$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host, Kra and Maass for $2$-step nilsystems and a result by Stepin for nilsystems $G/\Gamma$ with connected, simply connected $G$.