CORDIS - EU research results

Harmonic analysis on reductive groups

Final Report Summary - HARG (Harmonic analysis on reductive groups)

*Harmonic analysis* is a more scientific term for ''frequency analysis'' which is about decomposing a signal (function) into "harmonics"
(waves or vibrations at a single frequency). Now the frequency analysis depends in an essential way on the space on which the signal lives,
both on the local geometry (e.g. its dimension and its curvature), as well as on the global shape or boundary conditions of the space. For instance,a vibrating string admits a discrete set of frequencies determined by the length of the string, while an infinite string can vibrate as it wishes. A bounded two dimensional geometry (for instance a drum) features a discrete set of frequencies (waves) of greater complexity depending on the shape of the drum. From a mathematical perspective, the building blocks for interesting local geometries are constituted by homogeneous spaces attached to reductive groups. At the most basic level one finds the local geometry of Euclidean space. At a next step one considers homogeneous geometries such as spheres and hyperbolic spaces, and de Sitter and anti de Stitter spaces. Finally we can consider (periodic) boundary conditions, such as the two-dimensional hyperbolic geometry popularized by M.C. Escher's woodcut ''Angels and Devils''. With each step the frequency analysis does a quantum leap in sophistication compared to what happens at the base of Euclidean spaces and tori.

One major achievement of the HARG project was the discovery and development of new interesting homegeneous geometries, the *real spherical spaces*. Within the 5 years of HARG we obtained in addition a clear understanding of the continuous spectrum (frequencies) of such a space.

A second major achievement of the project was the precise characterization and classification of transfer maps between the sets of frequencies for homogeneous geometries which are related to each other by "inner twists". This yields a new way of classifying the so-called unipotent discrete frequencies of homogeneous nonarchimedean geometries, and proves a conjecture of Hiraga, Ichino and Ikeda expressing the formal degree invariant of the associated waves in terms of the Langlands dual group.

Another major achievement of HARG provides a new method to describe a part of the so-called residual discrete frequencies for spaces with periodic boundary conditions.