scholarly journals A Poisson transform adapted to the Rumin complex

2020 ◽  
pp. 1-39
Author(s):  
Andreas Čap ◽  
Christoph Harrach ◽  
Pierre Julg
Keyword(s):  
Differential Forms ◽  
Compact Subgroup ◽  
General Setting ◽  
Explicit Construction ◽  
Harmonic Forms ◽  
Finite Dimensional Representation ◽  
Poisson Transform ◽  
Finite Center ◽  
Rham Complex ◽  
Rumin Complex

Let [Formula: see text] be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use [Formula: see text]-invariant differential forms on [Formula: see text] to construct [Formula: see text]-equivariant Poisson transforms mapping differential forms on [Formula: see text] to differential forms on [Formula: see text]. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on [Formula: see text] to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, [Formula: see text] is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while [Formula: see text] is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

scholarly journals POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

2020 ◽  
pp. 1-52
Author(s):  
ANNALISA BALDI ◽  
BRUNO FRANCHI ◽  
PIERRE PANSU
Keyword(s):  
Differential Form ◽  
Scale Invariance ◽  
Differential Forms ◽  
Sobolev Inequalities ◽  
Exact Form ◽  
Heisenberg Groups ◽  
Contact Manifolds ◽  
Bounded Geometry ◽  
Exterior Differential ◽  
Rumin Complex

Abstract In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$ . In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

Introduction to Kähler Manifolds

2017 ◽  
Author(s):  
Eduardo Cattani ◽  
Phillip Griffiths
Keyword(s):  
Summer School ◽  
Differential Forms ◽  
Kähler Manifolds ◽  
Theoretical Physics ◽  
Kahler Manifolds ◽  
Complex Manifolds ◽  
Linear Transformations ◽  
Harmonic Forms ◽  
Hodge Structures ◽  
Kähler Structures

This chapter provides an introduction to the basic results on the topology of compact Kähler manifolds that underlie and motivate Hodge theory. This chapter consists of five sections which correspond, roughly, to the five lectures in the course given during the Summer School at the International Centre for Theoretical Physics (ICTP). The five topics under discussion are: complex manifolds; differential forms on complex manifolds; symplectic, Hermitian, and Kähler structures; harmonic forms; and the cohomology of compact Kähler manifolds. There are also two appendices. The first collects some results on the linear algebra of complex vector spaces, Hodge structures, nilpotent linear transformations, and representations of sl(2,ℂ) and serves as an introduction to many other chapters in this volume. The second contains a new proof of the Kähler identities by reduction to the symplectic case.

scholarly journals Differential Calculus onN-Graded Manifolds

2017 ◽  
Vol 2017 ◽  
pp. 1-19
Author(s):  
G. Sardanashvily ◽  
W. Wachowski
Keyword(s):  
Differential Calculus ◽  
Differential Operators ◽  
Differential Forms ◽  
Commutative Rings ◽  
Rham Complex ◽  
Grassmann Algebras ◽  
De Rham ◽  
Linear Differential ◽  
Graded Manifold ◽  
Graded Manifolds

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.

scholarly journals RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

2017 ◽  
Vol 18 (06) ◽  
pp. 1233-1293 ◽  
Author(s):  
Federico Binda ◽  
Shuji Saito
Keyword(s):  
Cartier Divisor ◽  
Explicit Construction ◽  
Chow Groups ◽  
Chow Group ◽  
Deligne Cohomology ◽  
Fundamental Class ◽  
Normal Crossing ◽  
Rham Complex ◽  
Relative Version ◽  
Higher Chow Groups

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$ . When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$ , the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$ , we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$ . For $r=\dim \overline{X}$ , we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$ -cycles with modulus.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q I: THE DISCRETE SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 1-36 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Differential Operators ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Holomorphic Functions ◽  
Irreducible Unitary Representation ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Infinitesimal Generators ◽  
Algebraic Properties ◽  
Irreducible Unitary Representations

A class of irreducible unitary representations in the discrete series of SUp,q is explicitly determined in a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square integrable representations, corresponds to a compact subgroup of SUp,q. The relevant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra [Formula: see text] in terms of differential operators is given.

scholarly journals The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)

2008 ◽  
Vol 191 ◽  
pp. 79-110
Author(s):  
Bernard Le Stum ◽  
Adolfo Quirós
Keyword(s):  
Differential Forms ◽  
Algebraic Groups ◽  
Classical Case ◽  
Group Scheme ◽  
Crystalline Cohomology ◽  
Hodge Filtration ◽  
Rham Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Invariant Differential

AbstractWe show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

scholarly journals De Rham–Hodge decomposition and vanishing of harmonic forms by derivation operators on the Poisson space

2016 ◽  
Vol 19 (02) ◽  
pp. 1650010 ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Poisson Process ◽  
Covariant Derivative ◽  
Probability Space ◽  
Differential Forms ◽  
Hodge Decomposition ◽  
Wiener Space ◽  
Harmonic Forms ◽  
Space Setting ◽  
De Rham ◽  
Poisson Space

We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham–Hodge–Kodaira decomposition as well as Weitzenböck and Clark–Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms.

scholarly journals A compact imbedding of Riemannian Symmetric Spaces

2018 ◽  
Vol 127 (1A) ◽  
pp. 55
Author(s):  
Trần Đạo Dõng
Keyword(s):  
Symmetric Space ◽  
Root System ◽  
Symmetric Spaces ◽  
Compact Subgroup ◽  
Analytic Structure ◽  
Riemannian Symmetric Space ◽  
The Real ◽  
Finite Center ◽  
Cartan Involution ◽  
Real Analytic

Let G be a connected real semisimple Lie group with finite center and θ be a Cartan involution of G. Suppose that K is the maximal compact subgroup of G corresponding to the Cartan involution θ. The coset space X = G/K is then a Riemannian symmetric space. In this paper, by choosing the reduced root system Σ0 = {α ∈ Σ | 2α /∈ Σ; α 2 ∈/ Σ} insteads of the restricted root system Σ and using the action of the Weyl group, firstly we construct a compact real analytic manifold Xb 0 in which the Riemannian symmetric space G/K is realized as an open subset and that G acts analytically on it, then we consider the real analytic structure of Xb 0 induced from the real analytic srtucture of AbIR, the compactification of the corresponding vectorial part.

scholarly journals EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES

2017 ◽  
pp. 49-53
Author(s):  
Irina Alexandrova ◽  
Irina Alexandrova ◽  
Sergey Stepanov ◽  
Sergey Stepanov ◽  
Irina Tsyganok ◽  
...  
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Differential Forms ◽  
Conformal Killing ◽  
Riemannian Symmetric Space ◽  
Vanishing Theorems ◽  
Harmonic Forms ◽  
Exterior Differential ◽  
Rank One ◽  
Riemannian Symmetric Spaces

In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

scholarly journals Langlands parameters of derived functor modules and Vogan diagrams

2003 ◽  
Vol 92 (1) ◽  
pp. 31 ◽  
Author(s):  
Paul D. Friedman
Keyword(s):  
Lie Group ◽  
Simple Root ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Classical Groups ◽  
Parabolic Subalgebra ◽  
Finite Center ◽  
Derived Functor ◽  
Langlands Parameters ◽  
Reductive Lie Group

Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $\mathrm{rank } G = \mathrm {rank } K$. Let $\ g= l\oplus u$ be a $\theta$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(\lambda)$. Suppose $\Lambda=\lambda+2\delta( u\cap p)$ is $K$-dominant and the infinitesimal character, $\lambda+\delta$, of $A_{g}(\lambda)$ is nondominant due to a noncompact simple root. By interpreting these conditions on the level of Vogan diagrams, a conjecture by Knapp is (essentially) settled for the groups $G=SU(p,q),\, Sp(p,q)$, and $SO^*(2n)$, thereby determining the Langlands parameters of natural irreducible subquotient of $A_{ g}(\lambda)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.

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