Invariant Differential Operators Research Articles

𝔞𝔣𝔣(1|1)-Relative cohomology on ℝ1|1

Over the [Formula: see text]-dimensional real superspace [Formula: see text], we classify [Formula: see text]-invariant bilinear differential operators acting on the superspaces of weighted densities. We compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the module of [Formula: see text]-densities [Formula: see text] on [Formula: see text], where [Formula: see text] is the Lie superalgebra of contact vector fields on [Formula: see text] and [Formula: see text] is the affine Lie superalgebra. This result allows us to compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the Poisson superalgebra [Formula: see text]. We explicitly give 2-cocycles spanning these cohomology spaces.

Spherical analysis on homogeneous vector bundles of the 3-dimensional euclidean motion group

We consider $\mathbb{R}^3$ as a homogeneous manifold for the action of the motion group given by rotations and translations. For an arbitrary $\tau\in \widehat{SO(3)}$, let $E_\tau$ be the homogeneous vector bundle over $\mathbb{R}^3$ associated with $\tau$. An interesting problem consists in studying the set of bounded linear operators over the sections of $E_\tau$ that are invariant under the action of $SO(3)\ltimes \mathbb{R}^3$. Such operators are in correspondence with the $End(V_\tau)$-valued, bi-$\tau$-equivariant, integrable functions on $\mathbb{R}^3$ and they form a commutative algebra with the convolution product. We develop the spherical analysis on that algebra, explicitly computing the $\tau$-spherical functions. We first present a set of generators of the algebra of $SO(3)\ltimes \mathbb{R}^3$-invariant differential operators on $E_\tau$. We also give an explicit form for the $\tau$-spherical Fourier transform, we deduce an inversion formula and we use it to give a characterization of $End(V_\tau)$-valued, bi-$\tau$-equivariant, functions on $\mathbb{R}^3$.

Relative BGG sequences; II. BGG machinery and invariant operators

For a real or complex semisimple Lie group G and two nested parabolic subgroups Q⊂P⊂G, we study parabolic geometries of type (G,Q). Associated to the group P, we introduce the classes of relative natural bundles and of relative tractor bundles and construct some basic invariant differential operators on such bundles. We define a (rather weak) notion of “compressability” for operators acting on relative differential forms with values in a relative tractor bundle. Then we develop a general machinery which converts a compressable operator to an operator on bundles associated to completely reducible representations on relative Lie algebra homology groups.Applying this machinery to a specific compressable invariant differential operator of order one, we obtain a relative version of BGG (Bernstein–Gelfand–Gelfand) sequences. All our constructions apply in the case P=G, producing new and simpler proofs in the case of standard BGG sequences. We characterize cases in which the relative BGG sequences are complexes or even fine resolutions of certain sheaves and describe these sheaves. We show that this gives constructions of new invariant differential operators as well as of new subcomplexes in certain curved BGG sequences. The results are made explicit in the case of generalized path geometries.

The binary aff(n|1)-invariant differential operators on weighted densities on the superspace R1|n and aff(n|1)-relative cohomology

We consider the $\mathfrak{aff}(n|1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{aff}(n|1)-$invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first $\mathfrak{aff}(n|1)-$relative differential cohomology of $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities.

Editors' Note

Editors' Note

Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

<pre>Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it.</pre><pre> By the <span>Cartan</span> decomposition <span>G = KAH,</span> we must <span>compacify</span> the <span>vectorial</span> part <span>A.$</span></pre><pre> In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it.</pre><pre>Our construction is a motivation of the <span>Oshima's</span> construction and it is similar to those in N. <span>Shimeno</span>, J. <span>Sekiguchi</span> for <span>semismple</span> symmetric spaces.</pre><pre>In this note, first we will <span>inllustrate</span> the construction via the case of <span>SL (n, </span>R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification. </pre>

Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets

The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)

The Fundamental Basis Theorem of Geometry from an algebraic point of view

An algebraic analog of the Fundamental Basis Theorem of geometry is offered with a pure algebraic proof involving the famous Waring’s problem for polynomials. Unlike the geometry case the offered system of invariant differential operators is commuting, which is a new result even in the classical geometry of surfaces. Moreover the algebraic analog works in more general settings then does the Fundamental Basis Theorem of geometry.

Classification of conformal representations induced from the maximal cuspidal parabolic

In the present paper we continue the project of systematic construction of invariant differential operators on the example of representations of the conformal algebra induced from the maximal cuspidal parabolic.

Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slov\'{a}k has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for arbitrary order conformally invariant differential operators, where by conformally invariant we mean equivariant with respect to the conformal group of $\mathbb{S}^m$ acting in Euclidean space $\mathbb{R}^m$. We name these the fermionic operators when the order is odd and bosonic operators when the order is even. Our approach explicitly uses convolution type operators to construct conformally invariant differential operators. These convolution type operators are examples of Knapp-Stein operators and they can be considered as the inverses of the corresponding differential operators. Intertwining operators of these convolution type operators are provided and intertwining operators of differential operators follow immediately. This reveals that our convolution type operators and differential operators are all conformally invariant. This also gives us a class of conformally invariant convolution type operators in higher spin spaces. Their inverses, when they exist, are conformally invariant pseudo-differential operators. Further we use Stein Weiss gradient operators and representation theory for the Spin group to naturally motivate the Rarita-Schwinger operators.

Projectively related metrics, Weyl nullity and metric projectively invariant equations

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries which is defined on a rank $(n+1)$-bundle. We show this connection is linked to the metrisability equations that govern the existence of metrics compatible with the structure. The fundamental 2-tensor also leads to a new class of invariant linear differential operators that are canonically associated to these geometries; included is a third equation studied by Gallot et al. We apply the results to study the metrisability equation, in the nullity setting described. We obtain strong local and global results on the nature of solutions and also on the nature of the geometries admitting such solutions, obtaining classification results in some cases. We show that closed Sasakian and K\"ahler manifold do not admit nontrivial solutions. We also prove that, on a closed manifold, two nontrivially projectively equivalent metrics cannot have the same tracefree Ricci tensor. We show that on a closed manifold a metric having a nontrivial solution of the metrisablity equation cannot have two-dimensional nullity space at every point. In these statements the meaning of trivial solution is dependent on the context. There is a function $B$ naturally appearing if a metric projective structure has nullity. We analyse in detail the case when this is not a constant, and describe all nontrivially projectively equivalent Riemannian metrics on closed manifolds with nonconstant $B$.

On 𝔬𝔰𝔭(1|2)-relative cohomology of the Lie superalgebra of contact vector fields on ℝ1|1

We consider the [Formula: see text]-dimensional real superspace [Formula: see text] endowed with its standard contact structure defined by the 1-form [Formula: see text]. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra [Formula: see text]. We classify [Formula: see text]-invariant linear differential operators from [Formula: see text] to [Formula: see text] vanishing on [Formula: see text], where [Formula: see text] is the superspace of bilinear differential operators between the superspaces of weighted densities. This result allows us to compute the first differential [Formula: see text]-relative cohomology of [Formula: see text] with coefficients in [Formula: see text]. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on [Formula: see text] acting on bilinear differential operators, Int. J. Geom. Methods Mod. Phys. 2(1) (2005) 23–40].

A functional calculus and restriction theorem on H-type groups

Let $L$ be the sublaplacian and $T$ the partial Laplacian with respect to central variables on H-type groups. We investigate a class of invariant differential operators by the joint functional calculus of $L$ and $T$. We establish Stein-Tomas type restriction theorems for these operators. In particular, the asymptotic behaviors of restriction estimates are given.

On the First aff(1)-Relative Cohomology of the Lie Algebra of Vector Fields and Differential Operators

Let Vect(A¯ÂÂ’A¯ÂÂ1) be the Lie algebra of smooth vector fields on A¯ÂÂ’A¯ÂÂ1. In this paper, we classify aff(1) -invariant linear differential operators from Vect(A¯ÂÂ’A¯ÂÂ1) to A¯ÂÂ„λ,μ;v vanishing on aff(1), where A¯ÂÂ„λ,μ;vA¢Â‰Â”Homdiff(A¯ÂÂ†λ⊗A¯ÂÂ†μ;A¯ÂÂ†v) is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential aff(1)-relative cohomology of Vect(A¯ÂÂ’A¯ÂÂ1) with coefficients in A¯ÂÂ„λ,μ;v

𝔰𝔩(2, ℝ)-invariant 3-ary differential operators and cohomology of 𝔰𝔩(2, ℝ) acting on 3-ary differential operators

We consider the [Formula: see text]-module structure on the spaces of [Formula: see text]-ary differential operators acting on the spaces of weighted densities. We classify [Formula: see text]-invariant [Formula: see text]-ary differential operators. This result allows us to compute the first differential cohomology of the Lie algebra [Formula: see text] with coefficients in the space of trilinear differential operators.

A Cauchy Integral Formula for Inframonogenic Functions in Clifford Analysis

In this paper we derive a Cauchy integral representation formula for the solutions of the sandwich equation \(\partial _{\underline{x}}f\partial _{\underline{x}}=0\), where \(\partial _{\underline{x}}\) stands for the first-order vector-valued rotation invariant differential operator in the Euclidean space \({\mathbb R}^m\), called Dirac operator. Such a solutions are referred in the literature as inframonogenic functions and represent an extension of the monogenic functions, i.e., null solutions of \(\partial _{\underline{x}}\), which represent higher-dimensional generalizations of the classic Cauchy–Riemann operator.

The Capelli problem for [formula omitted] and the spectrum of invariant differential operators

The “Capelli problem” for the symmetric pairs (gl×gl,gl), (gl,o), and (gl,sp) is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [12,15,14,18]). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs (g,k):=(gl(m|2n),osp(m|2n)) and (gl(m|n)×gl(m|n),gl(m|n)), acting on W:=S2(Cm|2n) and Cm|n⊗(Cm|n)⁎.To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra g maps surjectively onto the algebra PD(W)g of g-invariant differential operators on the superspace W, thereby providing an affirmative answer to the “abstract” Capelli problem for W. Our proof works more generally for gl(m|n) acting on S2(Cm|n) and is new even for the “ordinary” cases (m=0 or n=0) considered by Howe and Umeda in [9].We next describe a natural basis {Dλ} of PD(W)g, that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [12,15,20] by showing that the spectrum of Dλ is given by a polynomial cλ, which is characterized uniquely by certain vanishing and symmetry properties.We further show that the top homogeneous parts of the eigenvalue polynomials cλ coincide with the spherical polynomials dλ, which arise as radial parts of k-spherical vectors of finite dimensional g-modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [14].Finally, we make a precise connection between the polynomials cλ and the shifted super Jack polynomials of Sergeev and Veselov [25] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the “Frobenius transform”.

The Capelli Identity and Radon Transform for Grassmannians

We study a family C-s,C-l of Capelli-type invariant differential operators on the space of rectangular matrices over a real division algebra. The C-s,C-l descend to invariant differential operators on the corresponding Grassmannian, which is a compact symmetric space, and we determine the image of the C-s,C-l under the Harish-Chandra homomorphism. We also obtain analogous results for corresponding operators on the non-compact duals of the Grassmannians, and for line bundles. As an application we obtain a Radon inversion formula, which generalizes a recent result of B. Rubin for real Grassmannians.

Relative BGG sequences: I. Algebra

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with lowest weight in one (regular or singular) affine Weyl orbit. In the regular case, we show how all the weights in the orbit can be realized as relative homology groups (with different coefficients). These results are motivated by applications to differential geometry and the construction of invariant differential operators.

$L^p$ norms of higher rank eigenfunctions and bounds for spherical functions

We prove almost sharp upper bounds for the L^p norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of independent interest.