The most continuous part of the Plancherel decomposition for real spherical spaces
Final Report Abstract
In recent years harmonic analysis on real spherical spaces has developed rapidly. In particular large progress has been made towards a description of the Plancherel decomposition of such spaces, up to knowledge about the discrete spectrum. The main objective of this project was to get a precise description of the most continuous part of the Plancherel decomposition. A real spherical homogeneous space is a homogeneous space Z = G/H for a real reductive group G and a closed subgroup H of G, such that each minimal parabolic subgroup of G admits an open orbit in Z. To each real spherical space Z a number of real spherical spaces ZI, called boundary degenerations of Z, can be assigned. There exists a surjective partial isometry I⊕ L2 (ZI)tds → L2 (Z), where L2 (ZI)tds is the maximal subspace of L2 (ZI) that decomposes discretely, up to the action of a center. This map is called the Bernstein morphism. The irreducible representations that occur in L2 (ZI)tds are called twisted discrete series for ZI. In view of the decomposition of L2 (Z) induced by the Bernstein morphism, an explicit determination of the Plancherel decomposition is now reduced to describing the kernel of the Bernstein morphism and the (twisted) discrete series representations for each of the boundary degenerations ZI. For the proof for the existence of the Bernstein morphism a spectral gap is needed. Together with Job Kuit, Eric Opdam and Henrik Schlichtkrull we have proven such a spectral gap to exist. To be more precise we proved that the infinitesimal characters of discrete series representations are real and integral. Together with earlier results on the tempered spectrum of real spherical spaces, this proves the existence of a spectral gap. The main goal of this project was to obtain a precise description of the most continuous part of the Plancherel decomposition for real spherical spaces. The most continuous part of the Plancherel decomposition is the image of L2 (Z∅ )tds under the Bernstein morphism, where Z∅ is the most degenerate boundary degeneration of Z. To describe the most continuous part of the Plancherel decomposition for Z = G/H explicitly, it essentially suffices to determine all H-fixed distribution vectors for principal series representations. Job Kuit and Eitan Sayag have obtained a construction of all of these H-fixed distribution vectors. At the moment they are working on a better parametrization of the functionals. Not every spherical space admits (twisted) discrete series representations. It would be very desirable to have a geometric condition that determines the existence of such representations. In an attempt to get a better understanding of discrete series representations for reductive symmetric spaces, Job Kuit and his collaborators Erik van den Ban, Henrik Schlichtkrull and Mogens Flensted-Jensen have worked on a notion of cusp form for reductive symmetric spaces. In the scope of the project two articles were published on this topic. Although cusp forms may be used to obtain a geometric condition, it is not the most promising approach at the moment. Together with Friedrich Knop, Job Kuit, Eric Opdam and Henrik Schlichtkrull, we have recently started to work on a different strategy. Together with Job Kuit and Yiannis Sakellaridis, we organized the workshop Sphericity 2019 at CIRM in Luminy, France, where amongst other topics, the results in this project were presented and discussed.
Publications
- Cuspidal integrals for SL(3)/K . Indag. Math. (N.S.) 29 (2018), no. 5, 1235–1258
M. Flensted-Jensen, and J. J. Kuit
(See online at https://doi.org/10.1016/j.indag.2018.05.005) - K-invariant cusp forms for reductive symmetric spaces of split rank one. Forum Math. 31 (2019), no. 2, 341–349
E. P. van den Ban, J. J. Kuit, and H. Schlichtkrull
(See online at https://doi.org/10.1515/forum-2018-0150) - The infinitesimal characters of discrete series for real spherical spaces
B. Krötz, J. J. Kuit, E. M. Opdam, and H. Schlichtkrull
