Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra

Song, Yanli; Tang, Xiang

doi:10.48550/arxiv.1910.00175

Cited by 5 publications

(16 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The proof of the first statement is analogous to the considerations in [18] depending crucially on the inequality proved in [51,Thm A.5]:…”

Section: 21mentioning

confidence: 85%

“…Let g ∈ M be a semisimple element. Song and Tang in [51] have defined a higher delocalized cyclic cocycle [Φ P g ] on the Harish-Chandra Schwartz algebra C(G)). For m = dim(A), Φ P g is an m-cyclic cocycle on C(G) generalizing the orbital integral, Equation (1.5).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5 we study modified delocalized eta invariants. In Section 6 we recall the higher cyclic cocycles Φ P g introduced in [51] and we introduce the cyclic cocycles they define on G-proper manifolds; we also introduce the relative version of these cocycles on manifolds with boundary. In Section 7 we explain an approach to a general higher delocalized APS index theorem, using again the interplay between cyclic cocycles and relative cyclic cocycles associated to Φ P g .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Higher orbital integrals, rho numbers and index theory

Piazza¹,

Posthuma²,

Song³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let G be a connected, linear, reductive Lie group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant ηg(D X ) associated to a Dirac operator D X on a cocompact G-proper manifold X and to the orbital integral τg defined by a semisimple element g ∈ G. We prove that such an invariant enters in the the boundary correction term in a number of index theorems computing the pairing between the index class and the 0-degree cyclic cocycle defined by τg on a G-proper manifold with boundary. We prove a higher version of such a theorem, for the pairing of the index class and the higher cyclic cocycles defined by the higher orbital integral Φ P g associated to a cuspidal parabolic subgroup P < G with Langlands decomposition P = M AN and a semisimple element g ∈ M . We employ these results in order to define (higher) rho numbers associated to G-invariant positive scalar curvature metrics and to G-equivariant homotopy equivalences. Contents1. Introduction 2. Geometric preliminaries. 3. Delocalized traces and the APS index formula 3.1. Orbital integrals and associated cyclic 0-cocycles in the closed case 3.2. Manifolds with boundary: smooth index classes 3.3. 0-degree (relative) cyclic cocycles associated to orbital integrals 3.4. The 0-degree delocalized APS index theorem on G-proper manifolds 4. Delocalized eta invariants for G-proper manifolds 4.1. Large time behaviour 4.2. Small time behavior 4.3. Short time limits 5. Modified delocalized eta invariants 5.1. Spectrum on the slice 5.2. Delocalized eta invariants associated to perturbations on the slice 5.3. Delocalized eta invariants under a gap assumption 5.4. Delocalized eta invariants associated to smoothing perturbations 6. Higher delocalized cyclic cocycles 7. Toward a general higher APS index formula 8. Reduction 9. An index theorem for higher orbital integral through reduction 10. The non-invertible case 10.1. The gap case for 0-degree cyclic cocycles associated to orbital integrals 10.2. The gap case for cyclic cocycles associated to higher orbital integrals 10.3. Perturbations from the slice when G is equal rank 10.4. Perturbations from the slice for general G 11. Numeric rho invariants on G-proper manifolds 11.1. Rho numbers associated to delocalized 0-cocycles

show abstract

“…Proof. The proof of the first statement is analogous to the considerations in [18] depending crucially on the inequality proved in [51,Thm A.5]:…”

Section: 21mentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Higher orbital integrals, rho numbers and index theory

Piazza¹,

Posthuma²,

Song³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…One can also define tr g for continuous group, see [13] for example. There is also a higher generalization of the pairing between tr g and K-theory of geometric C * -algebras, introduced by [17], [7], [19], and [23], which is to consider cyclic cocycles. Since the metric m on M admits positive scalar curvature, it follows from the Lichnerowicz formula that the higher index of D M , ind G ( D M ), in the K-theory of the group C * -algebra C * r (G), is trivial with a specific trivialization.…”

Section: Introductionmentioning

confidence: 99%

Higher rho invariant and delocalized eta invariant at infinity

Chen¹,

Liu²,

Wang³

et al. 2020

Preprint

View full text Add to dashboard Cite

In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a higher index theorem for the Dirac operators. We apply our theory to study the secondary invariants for a manifold with corner with positive scalar curvature metric on each boundary face.

show abstract

“…The higher rho invariant is an obstruction to the inverse of the operator being local [4]. For some recent applications of the higher index and higher rho invariant to problems in geometry and topology, we refer the reader to [5,23,24,26,28,29,30].…”

Section: Introductionmentioning

confidence: 99%

Functoriality for higher rho invariants of elliptic operators

Guo

Xie

2020

Preprint

View full text Add to dashboard Cite

Let N be a closed spin manifold with positive scalar curvature and DN the Dirac operator on N . Let M1 and M2 be two Galois covers of N such that M2 is a quotient of M1. Then the quotient map from of M1 to M2 naturally induces maps between the geometric C * -algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the maximal higher rho invariants of the lifts of DN to M1 and M2 behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

show abstract

Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra

Cited by 5 publications

References 20 publications

Higher orbital integrals, rho numbers and index theory

Higher orbital integrals, rho numbers and index theory

Higher rho invariant and delocalized eta invariant at infinity

Functoriality for higher rho invariants of elliptic operators

Contact Info

Product

Resources

About