arXiv subjects

Jacob Bradd

Publications and source records attributed to Jacob Bradd.

5 recordsLinked to original sources

arXiv2025

Lie groupoids, the Satake compactification and the tempered dual, II: The Harish-Chandra principle

We give a geometric account of Harish-Chandra's principle that a tempered irreducible representation of a real reductive group is either square-integrable modulo center, or embeddable in a representation that is parabolically induced from such a representation. Our approach uses the Satake compactification, an associated groupoid that was constructed in the first paper of this series, and its $C^*$-algebra.

math.RT↗

arXiv2025

Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid

The (maximal) Satake compactification associated to a real reductive group $G$ is the closure of the symmetric space of all maximal compact subgroups of $G$ within the compact space of all closed subgroups of $G$. We shall present three different views of a groupoid that may be associated to the Satake compactification. To begin, we shall define our Satake groupoid, as we shall call it, as a topological groupoid, and as a special case of a general construction of Omar Mohsen. Then we shall give a Lie-theoretic account of the Satake groupoid, borrowing from work of Toshio Oshima. Finally we shall identify the Satake groupoid with the purely geometric $b$-groupoid of the Satake compactification, using the structure of the compactification as a smooth manifold with corners. In a subsequent paper we shall use the Satake groupoid to present a new proof of Harish-Chandra's principle, that all the tempered irreducible representations of $G$ may be constructed from discrete series representations using parabolic induction.

math.RT↗

arXiv2024

Operator K-Theory and Tempiric Representations

David Vogan proved that if $G$ is a real reductive group, and if $K$ is a maximal compact subgroup of $G$, then every irreducible representation of $K$ is included as a minimal $K$-type in precisely one tempered, irreducible unitary representation of $G$ with real infinitesimal character, and that moreover it is included there with multiplicity one and is the unique minimal $K$-type in that representation. We shall prove that the Connes-Kasparov isomorphism in operator $K$-theory is equivalent to a $K$-theoretic version of Vogan's result.

math.RT↗

arXiv2023

Compatible Decomposition of the Casselman Algebra and the Reduced Group C*-algebra of a Real Reductive Group

For a real reductive group $G$, we investigate the structure of the Casselman algebra $\mathcal{S}(G)$ and its similarities to the structure of the reduced group $C^*$-algebra $C_r^*(G)$. We demonstrate that the two algebras are assembled from very similar elementary components in a compatible way. In particular, we prove that the two algebras have the same $K$-theory when restricted to a finite set of $K$-types, which is a refinement of the Connes-Kasparov isomorphism.

math.OA↗

arXiv2020

Novodvorskii's theorem and the Oka principle

We give an exposition of Novodvorskii's theorem in Banach algebra K-theory, asserting that the Gelfand transform for a commutative Banach algebra induces an isomorphism in topological K-theory.

math.KT↗