Year: 2026
Type: preprint
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 $L^2(X)$, 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.
Source: arXiv (Cornell University)
Institution: Washington University in St. Louis
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Symmetry and analytic torsion, FRG: Collaborative Research: The Hypoelliptic Laplacian, Noncommutative Geometry, and Applications to Representations and Singular Spaces, Operator algebras and representation theory of Lie groups and groupoids, G7080048889, Higher invariants of finite-volume spaces, G913291176
OpenAlex ID: W7214733856