Pseudodifferential operators and the Connes-Kasparov isomorphism

2025/02/20 by DeBello, Peter, Higson, Nigel
#22E45 #22E46 #46L80 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2502.14985

Abstract

We compute the K-theory of the C*-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G. Our result uses the Connes-Kasparov isomorphism for G, and in fact is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan's well-known parametrization of the tempered irreducible representations of G with real infinitesimal character. When the reductive group G has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a K-theoretic version of Vogan's theorem.

Related

• On the Connes-Kasparov isomorphism, I: The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual
• On the Connes-Kasparov isomorphism, II: The Vogan classification of\n essential components in the tempered dual
• Operator K-Theory and Tempiric Representations
• The Chern-Connes Character for pseudodifferential operators on the sphere
• Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase\n factorization, and Fr "olicher Lie groups