Searcharxiv⌕ Search

arXiv subjects

Yanli Song

Publications and source records attributed to Yanli Song.

At least 19 recordsLinked to original sources
arXiv2025

Higher Lefschetz formulas on {\Gamma}-proper manifolds

Let $\Gamma$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $\Gamma$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $\tau$ in $HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$. Our formula takes place on the fixed point manifold $M^\gamma$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_\gamma$-invariant form on $M^\gamma$ that is naturally associated to $\tau\in HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$

math.DG↗
arXiv2025

Higher Orbital Integrals on Motion Groups and Mackey Deformation

We present an explicit construction of cyclic cocycles on Cartan motion groups, which can be viewed as generalizations of orbital integrals. We show that the higher orbital integral on a real reductive group associated with a semisimple element converges to the corresponding one on the associated Cartan motion group.

math.KT↗
arXiv2025

Heat kernel, large-time behavior, and representation theory

Given a real reductive group $G$, the purpose of this paper is to show an asymptotic formula of the large-time behavior of the $G$-trace of the heat operator on the associated symmetric spaces. Together with Carmona's proof on Vogan's lambda map, our results provide a geometric counterpart of Vogan's minimal $K$-type theory.

math.DG↗
arXiv2025

Lefschetz formula for locally symmetric spaces

Let $G$ be a semi-simple real Lie group of real rank one and $\Gamma$ be a discrete subgroup in $G$ such that $\Gamma \backslash G$ has finite volume. We introduce a new group $C^*$-algebra $C^*_r(G, \Gamma)$, which provides a natural framework for defining index classes of Dirac-type operators on the locally symmetirc space $\Gamma \backslash G /K$. We show that Dirac operators define elements in the $K$-theory of $C^*_r(G, \Gamma)$ and use Hecke correspondences to study their Lefschetz numbers. Our main result is an explicit formula for the Lefschetz number of Hecke operators.

math.KT↗
arXiv2023

Delocalized eta invariants of the signature operator on G-proper manifolds

Let $G$ be a connected, linear real reductive group and let $X$ be a cocompact $G$-proper manifold without boundary. We define delocalized eta invariants associated to a $L^2$-invertible perturbed Dirac operator $D_X+A$ with $A$ a suitable smoothing perturbation. We also investigate the case in which $D_X$ is not invertible but $0$ is isolated in the $L^2$-spectrum of $D_X$. We prove index formulas relating these delocalized eta invariants to Atiyah-Patodi-Singer delocalized indices on $G$-proper manifolds with boundary. In order to achieve this program we give a detailed account of both the large and small time behaviour of the heat-kernel of perturbed Dirac operators, as a map from the positive real line to the algebra of Lafforgue integral operators. We apply these results to the definition of rho-numbers associated to $G$-homotopy equivalences between closed $G$-proper manifolds and to the study of their bordism properties. We also define delocalized signatures of manifolds with boundary satisfying an invertibility assumption on the differential form Laplacian of the boundary in middle degree and prove an Atiyah-Patodi-Singer formula for these delocalized signatures.

math.DG↗
arXiv2022

Riemannian foliations and geometric quantization

We introduce geometric quantization for constant rank presymplectic structures with Riemannian null foliation and compact leaf closure space. We prove a quantization-commutes-with-reduction theorem in this context. Examples related to symplectic toric quasi-folds, suspensions of isometric actions of discrete groups, and K-contact manifolds are discussed.

math.SG↗
arXiv2022

Cartan Motion Group and Orbital Integrals

In this short note, we study the variation of orbital integrals, as traces on the group algebra $G$, under the deformation groupoid. We show that orbital integrals are continuous under the deformation. And we prove that the pairing between orbital integrals and $K$-theory element of $C^*_r(G)$ stays constant with respect to the deformation for regular group elements, but vary at singular elements.

math.KT↗
arXiv2022

On the Connes-Kasparov isomorphism, I: The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual

This is the first of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. These results were originally announced by Antony Wassermann in 1987. In Part I we shall give details of the C*-algebraic Morita equivalence, and then compute the Connes-Kasparov morphism subject to some results in tempered representation theory that we shall prove in Part II using tools from David Vogan's classification of the tempered dual.

math.RT↗
arXiv2022

On the Connes-Kasparov isomorphism, II: The Vogan classification of essential components in the tempered dual

This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. These results were originally announced by Antony Wassermann in 1987. In Part I we presented the Morita equivalence and the Connes-Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual.

math.RT↗
arXiv2021

Higher orbital integrals, rho numbers and index theory

Let $G$ be a connected, linear real reductive group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant $\eta_g (D_X)$ associated to a Dirac operator $D_X$ on a cocompact $G$-proper manifold $X$ and to the orbital integral $\tau_g$ defined by a semisimple element $g\in G$. Along the way, we give a detailed account of the large time behaviour of the heat kernel and of its short time bahaviour near the fixed point set of $g$. We prove that such a delocalized eta invariant enters as the boundary correction term in an index theorem computing the pairing between the index class and the 0-degree cyclic cocycle defined by $\tau_g$ on a $G$-proper manifold with boundary. More importantly, we also 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 $\Phi^P_g$ associated to a cuspidal parabolic subgroup $P<G$ with Langlands decomposition $P=MAN$ and a semisimple element $g\in M$. We employ these results in order to define (higher) rho numbers associated to $G$-invariant positive scalar curvature metrics.

math.DG↗
arXiv2020

Log symplectic manifolds and $[Q,R]=0$

We show, under an orientation hypothesis, that a log symplectic manifold with simple normal crossing singularities has a stable almost complex structure, and hence is Spin$_c$. In the compact Hamiltonian case we prove that the index of the Spin$_c$ Dirac operator twisted by a prequantum line bundle satisfies a $[Q,R]=0$ theorem.

math.SG↗
arXiv2020

An index theorem for higher orbital integrals

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions.

math.DG↗
arXiv2019

Geometric quantization of b-symplectic manifolds

We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction'' property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.

math.SG↗
arXiv2019

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

Let G be a connected real reductive group. Orbit integrals define traces on the group algebra of G. We introduce a construction of higher orbit integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbit integrals via Fourier transform by expressing them as integrals on the tempered dual of G. We obtain explicit formulas for the pairing between the higher orbit integrals and the K-theory of the reduced group C*-algebra, and discuss their applications to representation theory and K-theory.

math.KT↗
arXiv2019

A KK-theoretic perspective on deformed Dirac operators

We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$, where $\mathsf{c}(X)$ is a Clifford multiplication operator by an orbital vector field with respect to the action of a compact Lie group. Our main result is that the index class of such an operator factors as a KK-product of certain KK-theory classes defined by $\mathsf{D}$ and $X$. As a corollary we obtain the excision and cobordism-invariance properties first established by Braverman. An index theorem of Braverman relates the index of $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$ to the index of a transversally elliptic operator. We explain how to deduce this theorem using a recent index theorem for transversally elliptic operators due to Kasparov.

math.KT↗
arXiv2018

Norm-square localization and the quantization of Hamiltonian loop group spaces

In an earlier article we introduced a new definition for the `quantization' of a Hamiltonian loop group space $\mathcal{M}$, involving the equivariant $L^2$-index of a Dirac-type operator $\mathscr{D}$ on a non-compact finite dimensional submanifold $\mathcal{Y}$ of $\mathcal{M}$. In this article we study a Witten-type deformation of this operator, similar to the work of Tian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely many non-trivial contributions, indexed by the components of the critical set of the norm-square of the moment map. This is the main part of a new proof of the $[Q,R]=0$ theorem for Hamiltonian loop group spaces.

math.SG↗
arXiv2018

A geometric formula for multiplicities of $K$-types of tempered representations

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $\pi$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $\pi|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of $\mathrm{SU}(p,1)$, $\mathrm{SO}_0(p,1)$ and $\mathrm{SO}_0(2,2)$ restrict multiplicity-freely to maximal compact subgroups.

math.DG↗