Well-tempered symmetric spaces
Abstract
Let X=G/H be a real reductive symmetric space. We consider the support of the Plancherel measure for the regular representation of G on L2(X), and its Fell topology. We first recall how the Plancherel formula for X (known from work of Delorme and van den Ban--Schlicktrull, among others) yields a description of the support in terms of the discrete spectra of smaller symmetric spaces. We then point out that this description simplifies for a class of real symmetric spaces X which we call well-tempered. These are the spaces whose tangent space at the identity coset admits a Cartan subspace whose centralizer in the Lie algebra of G is abelian. Well-tempered spaces are tempered in the sense of Benoist and Kobayashi. Under mild assumptions on the disconnectedness of G, we give a description of the support of the Plancherel measure for well-tempered symmetric spaces in terms of characters of Cartan subgroups of G, and describe the Fell topology of the support. If G is a complex group and H a real form, then the support decomposes as a union of explicit quotients of relative continuous-parameter spaces by finite Weyl groups. As an application, we describe the C*-algebra associated to the regular representation of G on L2(X), and its K-theory.
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BibTeX
@misc{afgoustidis2026welltempered,
title = {{Well-tempered symmetric spaces}},
author = {Alexandre Afgoustidis and Peter Hochs and Shintaro Nishikawa and Yanli Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32916},
eprint = {2609.32916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32916},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-tempered symmetric spaces
AU - Alexandre Afgoustidis
AU - Peter Hochs
AU - Shintaro Nishikawa
AU - Yanli Song
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32916
UR - https://arxiv.org/abs/2609.32916
AB - Let X=G/H be a real reductive symmetric space. We consider the support of the Plancherel measure for the regular representation of G on L^2(X), and its Fell topology. We first recall how the Plancherel formula for X (known from work of Delorme and van den Ban--Schlicktrull, among others) yields a description of the support in terms of the discrete spectra of smaller symmetric spaces. We then point out that this description simplifies for a class of real symmetric spaces X which we call well-tempered. These are the spaces whose tangent space at the identity coset admits a Cartan subspace whose centralizer in the Lie algebra of G is abelian. Well-tempered spaces are tempered in the sense of Benoist and Kobayashi. Under mild assumptions on the disconnectedness of G, we give a description of the support of the Plancherel measure for well-tempered symmetric spaces in terms of characters of Cartan subgroups of G, and describe the Fell topology of the support. If G is a complex group and H a real form, then the support decomposes as a union of explicit quotients of relative continuous-parameter spaces by finite Weyl groups. As an application, we describe the C^*-algebra associated to the regular representation of G on L^2(X), and its K-theory.
ER -