Invariant Differential Operators Research Articles

Invariant differential operators and the generalized symmetric group

In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as a D-module, under the map π: spec OX →spec OXG(r,n), where OXG(r,n) is the ring of invariant polynomial under the action of the wreath product G(r, p):= Z/rZ ~Sn. We first describe the generators of the simple components of π+(OX) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials

Analogs of Bol operators for 𝔭𝔤𝔩(a + 1|b) ⊂ 𝔳𝔢𝔠𝔱(a|b)

Bol operators (Bols for short) are differential operators invariant under the projective action of [Formula: see text] between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: [Formula: see text]-invariant differential operators between spaces of tensor fields on [Formula: see text]-dimensional supermanifolds with irreducible, as [Formula: see text]-modules, fibers of arbitrary, even infinite, dimension for certain “key” values of [Formula: see text] and [Formula: see text] — the ones for which the solution is describable. We discovered many new operators for [Formula: see text] and for the case of [Formula: see text]-dimensional general superstring, which looks like a most natural superization of Bol’s result, additional to the cases of super analogs of Bols between spaces of weighted densities on the [Formula: see text]-dimensional superstrings with a contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension [Formula: see text], there are [Formula: see text]-parameter families of Bols, whereas there are no non-scalar nonzero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here.

Heisenberg Parabolic Subgroup of SO∗(10) and Invariant Differential Operators

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so∗(10). We use the maximal Heisenberg parabolic subalgebra P=M⊕A⊕N with M=su(3,1)⊕su(2)≅so∗(6)⊕so(3). We give the main and the reduced multiplets of indecomposable elementary representations. This includes the explicit parametrization of the intertwining differential operators between the ERS. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras so(p,q) (with p+q=10, p≥q≥2) with maximal Heisenberg parabolic subalgebra: P′=M′⊕A′⊕N′, M′=so(p−2,q−2)⊕sl(2,IR), M′C≅MC.

Some invariant differential operator properties for hyperbolic univalent convex functions

The purpose of this paper is to show some interesting properties of hyperbolic univalent convex differential functions in the case of estimating the real and imaginary parts of a hyperbolic univalent convex function using the definition of invariant differential operator of the first and second-order as stated in theorems (2.1) and (2.2). In addition, the same differential operator has been analyzed through an estimation of the differential operator when to be be a holomorphic function for each , except some poles z, and also in case of being a locally univalent function defined in the unit disk .

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

The aim of this paper is to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on Siegel domains and complex hyperbolic spaces using the method of Helgason-Fourier analysis on the complex hyperbolic spaces. Firstly, we give a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller's operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. (See Theorem 1.3.) Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group SU(1,n) and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain Un and the unit ball BCn. (See Theorems 1.4, 1.5.) Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. (See Theorems 1.9, 1.10 and 1.12.) The factorization theorem we prove on the complex hyperbolic spaces is considerably more difficult than in the case of real hyperbolic spaces and should be of its independent interest in the analysis on Heisenberg group and CR sphere and CR invariant differential operators therein.

Invariant differential operators for the Jacobi algebra 𝒢2

In this paper, we construct explicitly the intertwining differential operators for the Jacobi algebra [Formula: see text]. For the construction, we use the singular vectors of the Verma modules over [Formula: see text] which we have constructed earlier. We construct the function spaces on which the operators act. We find two versions of the left (representation) action and the right action. These actions are combined with the singular vectors to provide the intertwining differential operators.

Ellipticity of Some Higher Order Conformally Invariant Differential Operators

Ellipticity of Some Higher Order Conformally Invariant Differential Operators

$$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$

$$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$

Elementary relative tractor calculus for Legendrean contact structures

For a manifold $M$ endowed with a Legendrean (or Lagrangean) contact structure $E\oplus F \subset TM$, we give an elementary construction of an invariant partial connection on the quotient bundle $TM/F$. This permits us to develop a naïve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.

A Liouville Theorem for Möbius Invariant Equations

In this paper, we classify Möbius invariant differential operators of second order in two-dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations. The equations are naturally associated with a continuous family of convex cones $$\Gamma _p$$ in $$\mathbb R^2$$ , with parameter $$p\in [1, 2]$$ , joining the half plane $$\Gamma _1:=\{ (\lambda _1, \lambda _2):\lambda _1+\lambda _2>0\}$$ and the first quadrant $$\Gamma _2:=\{ (\lambda _1, \lambda _2):\lambda _1, \lambda _2>0\}$$ . Chen and C. M. Li established in 1991 a Liouville type theorem corresponding to $$\Gamma _1$$ under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville type theorem we establish in this paper for $$\Gamma _p$$ , $$1<p\le 2$$ , does not require any additional assumption on the solution as for $$\Gamma _1$$ . This is reminiscent of the Liouville type theorems in dimensions $$n\ge 3$$ established by Caffarelli, Gidas and Spruck in 1989 and by A. B. Li and Y. Y. Li in 2003–2005, where no additional assumption was needed either. On the other hand, there is a striking new phenomena in dimension $$n=2$$ that $$\Gamma _p$$ for $$p=1$$ is a sharp dividing line for such uniqueness result to hold without any further assumption on the solution. In dimensions $$n\ge 3$$ , there is no such dividing line.

Invariant bilinear operators and the second 𝔰𝔩(2)-relative cohomology of the Lie algebra of vector fields on ℝ

Let [Formula: see text] be the Lie algebra of smooth vector fields on [Formula: see text] and [Formula: see text] be the space of bilinear differential operators acting on weighted densities. In this paper, we classify [Formula: see text]-invariant skew-symmetric binary differential operators from [Formula: see text] to [Formula: see text] vanishing on [Formula: see text]. This result allows us to compute the second differential [Formula: see text]-relative cohomology of [Formula: see text] with coefficients in [Formula: see text].

Extrinsic curvature and conformal Gauss–Bonnet for four-manifolds with corner

This paper defines two new extrinsic curvature quantities on the corner of a four-dimensional Riemannian manifold with corner. One of these is a pointwise conformal invariant, and the conformal transformation of the other is governed by a new linear second-order pointwise conformally invariant partial differential operator. The Gauss-Bonnet theorem is then stated in terms of these quantities.

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Differential operators and reection group of type Dn

In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

Differential operators and reection group of type Bn

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.

Perturbative quantization of Einstein-Hilbert gravity embedded in a higher derivative model

In a perturbative approach Einstein-Hilbert gravity is quantized about a flat background. In order to render the model power counting renormalizable, higher order curvature terms are added to the action. They serve as Pauli-Villars type regulators and require an expansion in the number of fields in addition to the standard expansion in the number of loops. Renormalization is then performed within the BPHZL scheme, which provides the action principle to construct the Slavnov-Taylor identity and invariant differential operators. The final physical state space of the Einstein-Hilbert theory is realized via the quartet mechanism of Kugo and Ojima. Renormalization group and Callan-Symanzik equation are derived for the Green functions and, formally, also for the $S$-matrix.

The λ-cosine transforms, differential operators, and Funk transforms on Stiefel and Grassmann manifolds

We introduce a new family of invariant differential operators associated with λ-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the λ-cosine transforms and yield new inversion formulas. Intermediate Funk-cosine transforms corresponding to integration over matrices of lower rank are studied. The main tools are polar decomposition and Fourier analysis on matrix space.

On the Schwartz correspondence for Gelfand pairs of polynomial growth

Let (G,K) be a Gelfand pair, with G a Lie group of polynomial growth, and let \Sigma\subset\mathbb R^\ell be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system D_1,\dots,D_\ell of G -invariant differential operators on G/K and associating to a bounded spherical function \phi the \ell -tuple of its eigenvalues under the action of the D_j 's. We say that property (S) holds for (G,K) if the spherical transform maps the bi- K -invariant Schwartz space \mathcal S(K\backslash G/K) isomorphically onto \mathcal S(\Sigma) , the space of restrictions to \Sigma of the Schwartz functions on \mathbb R^\ell . This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where G=K\ltimes N , with N nilpotent. In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs.

Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.

Differential operators and the symmetric groups

In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x 1 , …, x n ] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x 1 , …, x n ] Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.