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Xiang Tang

Publications and source records attributed to Xiang Tang.

At least 37 records · Page 2Linked to original sources
arXiv2022

Helton-Howe Trace, Connes-Chern character and Quantization

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large $t$-limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight $t$ and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces using harmonic analysis and quantization.

math.FA↗
arXiv2022

Trace Formula of Semicommutators

For weighted Bergman spaces on the unit disk, we give trace formulas of semicommutators of Toeplitz operators with $\mathscr{C}^2(\overline{\mathbb{D}})$ symbols. We generalize this formula to weighted Bergman spaces on the unit ball in higher dimensions. Applications and examples on the Hankel operators are also discussed.

math.CV↗
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↗
arXiv2021

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↗
arXiv2020

Perturbations of principal submodules in the Drury-Arveson space

We study the geometry in the perturbations of principal submodules in the Drury-Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy.

math.FA↗
arXiv2020

On the Hochschild homology of convolution algebras of proper Lie groupoids

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski's ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski's conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space.

math.KT↗
arXiv2020

A quantum Leray-Hirsch theorem for banded gerbes

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants of $\B$ twisted by various flat $U(1)$-gerbes on $\B$. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.

math.AG↗
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

Smooth Connes--Thom isomorphism, cyclic homology, and equivariant quantisation

Using a smooth version of the Connes--Thom isomorphism in Grensing's bivariant K-theory for locally convex algebras, we prove an equivariant version of the Connes--Thom isomorphism in periodic cyclic homology. As an application, we prove that periodic cyclic homology is invariant with respect to equivariant strict deformation quantization.

math.KT↗
arXiv2017

Constant Symplectic 2-groupoids

We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.

math.SG↗
arXiv2017

Resolutions of proper Riemannian Lie groupoids

In this paper we prove that every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits a desingularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov-Hausdorff distance between the quotient spaces. We construct the desingularization via a successive blow-up construction on a proper Lie groupoid. We also prove that our construction of the desingularization is invariant under Morita equivalence of groupoids, showing that it is a desingularization of the underlying differentiable stack.

math.DG↗
arXiv2017

Lagrangian Mechanics and Reduction on Fibered Manifolds

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.

math.DS↗
arXiv2016

Roe C*-algebra for groupoids and generalized Lichnerowicz Vanishing theorem for foliated manifolds

We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.

math.OA↗
arXiv2015

From double Lie groupoids to local Lie 2-groupoids

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid.

math.DG↗