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Pseudodifferential operators and the Connes-Kasparov isomorphism

Peter DeBello, Nigel Higson

TL;DR

This work computes the $K$-theory of the $C^*$-category generated by equivariant, properly supported, order-zero classical pseudodifferential operators on homogeneous bundles over the symmetric space $G/K$ of a real reductive group $G$, and shows it is governed by the Connes-Kasparov isomorphism. It reframes CK as a statement about $C^*$-categories, proving $K_0(\mathsf{P}^*_{G,K}) \cong R(K)$ and $K_1(\mathsf{P}^*_{G,K})=0$, with a second viewpoint linking to tempiric representations and Vogan’s minimal $K$-types. A Fourier-transform construction is developed for real rank-one groups, yielding a direct isomorphism between the psdo category and a symbol-valued category built from tempered representations, and enabling a K-theoretic version of Vogan’s theorem. The deformation-to-normal-cone framework and a category of operator families bridge the pseudodifferential picture with CK-type index theory, while comments on van Erp-Yuncken situate the approach within broader modern theories. Overall, the paper tightens the link between noncommutative geometry, operator K-theory, and representation theory, providing explicit, category-level realizations of the Connes-Kasparov isomorphism and its consequences for tempiric representations and minimal $K$-types.

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.

Pseudodifferential operators and the Connes-Kasparov isomorphism

TL;DR

This work computes the -theory of the -category generated by equivariant, properly supported, order-zero classical pseudodifferential operators on homogeneous bundles over the symmetric space of a real reductive group , and shows it is governed by the Connes-Kasparov isomorphism. It reframes CK as a statement about -categories, proving and , with a second viewpoint linking to tempiric representations and Vogan’s minimal -types. A Fourier-transform construction is developed for real rank-one groups, yielding a direct isomorphism between the psdo category and a symbol-valued category built from tempered representations, and enabling a K-theoretic version of Vogan’s theorem. The deformation-to-normal-cone framework and a category of operator families bridge the pseudodifferential picture with CK-type index theory, while comments on van Erp-Yuncken situate the approach within broader modern theories. Overall, the paper tightens the link between noncommutative geometry, operator K-theory, and representation theory, providing explicit, category-level realizations of the Connes-Kasparov isomorphism and its consequences for tempiric representations and minimal -types.

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.

Paper Structure

This paper contains 29 sections, 40 theorems, 223 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Pseudodifferential Operators
  3. Pseudodifferential operators on Euclidean space
  4. Pseudodifferential operators on smooth manifolds
  5. The principal symbol
  6. The pseudodifferential extension of C*-algebras
  7. Operators on Proper Homogeneous Spaces
  8. Sections of equivariant vector bundles
  9. Properly supported operators
  10. Smoothing operators
  11. Pseudodifferential operators on proper homogeneous spaces
  12. Pseudodifferential extension in the equivariant context
  13. C*-Categories
  14. C*-categories from pseudodifferential operators
  15. K-Theory for the C*-category generated by smoothing operators
  16. ...and 14 more sections

Key Result

Theorem 1

Let $G$ be a real reductive group and let $K$ be a maximal compact subgroup of $G$. Denote by $R(K)$ the representation ring of the compact group $K$. There is an isomorphism of abelian groups under which the class in $R(K)$ of a finite-dimensional unitary representation $V$ of $K$ corresponds to the class of the identity operator on the bundle over $G/K$ with fiber $V$. In addition, $K_1(\mathsf

Figures (2)

  • Figure 1: The Connes-Kasparov labeling of most of the components of the tempered dual of $G=Sp(1,1)$ by indecomposable Dirac operators on the symmetric space of $G$. In this case $K \cong SU(2)\times SU(2)$, and the indecomposable Dirac operators are in bijection with irreducible representations of $K$, which are the nodes in the diagram. The circles indicate that the index of the given Dirac-type operator is a discrete series representation; each discrete series occurs exactly once. The squares indicate that the index of the given Dirac-type operator is supported on a principal series component; the components in question are precisely those that possess two minimal $K$-types (compare Figure \ref{['fig:vogan']}), and each occurs precisely once. The principal series components with unique minimal $K$-types (compare Figure \ref{['fig:vogan']} again) do not occur at all in index theory.
  • Figure 2: Vogan's labeling of the tempered dual by minimal $K$-types in the case of $G=Sp(1,1)$, where $K = SU(2)\times SU(2)$. The nodes in the diagram are the irreducible representations of $K$. The circles indicate that a given irreducible representation of $K$ occurs as the minimal $K$-type of a discrete series representation. The paired squares indicate pairs of irreducible representations of $K$ that occur as minimal $K$-types in the same principal series component of the tempered dual, while the triangles indicate irreducible representations of $K$ that occur as the unique minimal $K$-type in a principal series component of the tempered dual. Every component of the tempered dual is listed in this way exactly once.

Theorems & Definitions (84)

  • Theorem : Theorem \ref{['thm-k-theory-of-the-c-star-category']} below
  • Definition
  • Theorem : Compare \ref{['eq-k-theory-and-tempirics']} below
  • Theorem : See Theorem \ref{['thm-fourier-isomorphism-in-real-rank-one']} below
  • Definition 2.3.2
  • Theorem 2.4.1: See for instance HormanderVolIII
  • Remark 2.4.2
  • Theorem 2.4.3
  • proof
  • Theorem 2.4.4: Seeley65; see also KohnNirenberg65
  • ...and 74 more