Invariant Differential Operators Research Articles

The Capelli eigenvalue problem for Lie superalgebras

For a finite dimensional unital complex simple Jordan superalgebra J, the Tits–Kantor–Koecher construction yields a 3-graded Lie superalgebra $$\mathfrak {g}^\flat \cong \mathfrak {g}^\flat (-1)\oplus \mathfrak {g}^\flat (0)\oplus \mathfrak {g}^\flat (1)$$, such that $$\mathfrak {g}^\flat (-1)\cong J$$. Set $$V:=\mathfrak {g}^\flat (-1)^*$$ and $$\mathfrak {g}:=\mathfrak {g}^\flat (0)$$. In most cases, the space $$\mathscr {P}(V)$$ of superpolynomials on V is a completely reducible and multiplicity-free representation of $$\mathfrak {g}$$, and there exists a direct sum decomposition $$\mathscr {P}(V):=\bigoplus _{\lambda \in \Omega }V_\lambda $$, where $$\left( V_\lambda \right) _{\lambda \in \Omega }$$ is a family of irreducible $$\mathfrak {g}$$-modules parametrized by a set of partitions $$\Omega $$. In these cases, one can define a natural basis $$\left( D_\lambda \right) _{\lambda \in \Omega }$$ of “Capelli operators” for the algebra $$\mathscr {PD}(V)^{\mathfrak {g}}$$ of $$\mathfrak {g}$$-invariant superpolynomial differential operators on V. In this paper we complete the solution to the Capelli eigenvalue problem, which asks for the determination of the scalar $$c_\mu (\lambda )$$ by which $$D_\mu $$ acts on $$V_\lambda $$. We associate a restricted root system $$\varSigma $$ to the symmetric pair $$(\mathfrak {g},\mathfrak {k})$$ that corresponds to J, which is either a deformed root system of type $$\mathsf {A}(m,n)$$ or a root system of type $$\mathsf {Q}(n)$$. We prove a necessary and sufficient condition on the structure of $$\varSigma $$ for $$\mathscr {P}(V)$$ to be completely reducible and multiplicity-free. When $$\varSigma $$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $$c_\mu (\lambda )$$, in terms of Sergeev–Veselov’s shifted super Jack polynomials when $$\varSigma $$ is of type $$\mathsf {A}(m,n)$$, and Okounkov-Ivanov’s factorial Schur Q-polynomials when $$\varSigma $$ is of type $$\mathsf {Q}(n)$$. Along the way, we prove that the natural map from the centre of the enveloping algebra of $$\mathfrak {g}$$ into $$\mathscr {PD}(V)^{\mathfrak {g}}$$ is surjective in all cases except when $$J\cong F $$, where $$ F $$ is the 10-dimensional exceptional Jordan superalgebra.

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces

Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{\tau }$ over the unit cosphere bundle $S^{\ast }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{\tau }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{\ast }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,\sigma )$ on compatible associated vector bundles $\mathbf W_{\sigma }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $\tau$ and $\sigma$ defining the bundles $\mathbf V_{\tau }$ and $\mathbf W_{\sigma }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{\sigma }$. Our methods of proof are based on representation theory and Lie theory.

Convergence of Grid Boundary-Value Problems for Functions Defined on Grid Cells and Faces

For stationary diffusion-type equations, we study the convergence of grid inhomogeneous boundary-value problems of a version of the mimetic finite difference (MFD) technique in which grid scalars are defined inside grid cells and grid vectors are specified by their local normal coordinates on the plane faces of grid cells. Grid equations and boundary conditions are formulated in operator form using consistent grid analogs of invariant first-order differential operators and of boundary operators. Convergence is studied on the basis of the theory of operator difference schemes; i.e., a priori estimates for the norm of the solution error in terms of the norm of the approximation error are obtained that guarantee convergence of the first order under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. Grid analogs of embedding inequalities and approximation relations obtained earlier are used.

Multiplet Classification of Reducible Verma Modules over the G2 Algebra

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra G2(2) which is split real form of G2. We give the classification of reducible Verma modules G2. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over G2(2).

Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

In this paper, we establish higher order Borel-Pompeiu formulas for conformally invariant fermionic operators in higher spin theory, which is the theory of functions on m-dimensional Euclidean space taking values in arbitrary irreducible representations of the Spin group. As applications, we provide higher order Cauchy's integral formulas for those fermionic operators. This paper continues the work of building up basic integral formulas for conformally invariant differential operators in higher spin theory.

Асимптотика преобразования Радона на гиперболических пространствах

Let G/H be a hyperbolic space over R; C or H; and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K∩H-invariant functions.

Quadratic Capelli operators and Okounkov polynomials

Let $Z$ be the symmetric cone of $r \times r$ positive definite Hermitian matrices over a real division algebra $\mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_\lambda$ -- indexed by partitions $\lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y \longleftarrow X \longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $\mathbb F^n $ with $n \geq 2r$. Using this we construct a family of invariant differential operators $D_{\lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{\lambda,s}$ are given by specializations of Okounkov interpolation polynomials.

Some properties of the higher spin Laplace operator

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rarita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformal invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.

Higher-Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the sl(2)⊕u(1) invariance of the Calogero Conformal Mechanics, an osp2∣2 invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schrödinger algebra. This vector space also gives rise to a higher-spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable.

Differential Invariants and Realizations of the Deformed Smallest Galilei Algebra

Deformations of the smallest Galilei algebra are constructed and deformed algebras are realized by Lie vector fields. For these realization bases of differential invariants and operators of invariant differentiation are constructed in the case of two dependent and one independent variables.

Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain “growth-killing” invariant differential operators on H2 and those on H1×C (here Hn denotes the Siegel upper half space of degree n). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms; and as a by-product we also show that any Siegel modular form of degree 2 is determined by its ‘fundamental’ Fourier coefficients.

Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L2 on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.

Algèbres de distributions et $\mathcal D$-modules arithmétiques

Let p be a prime number, V a complete discrete valuation ring of unequal caracteristics (0,p) , G a smooth affine algebraic group over Spec V . Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level m , generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra over a finite extension K of \mathbf Q_p . We then show that these distribution algebras can be identified with invariant arithmetic differential operators over G , and prove a coherence result when the ramification index of K is < p-1 .

A Fourier transform for the quantum Toda lattice

We answer a question of V. Drinfeld by constructing an ‘algebraic Fourier transform’ for the quantum Toda lattice of a complex reductive algebraic group G, which extends the classical ‘algebraic Fourier transform’ for its subalgebra $$D(T)^W$$ of Weyl group invariant differential operators on a maximal torus. The proof is contained in Sect. 2 and relies on a result of Bezrukavnikov–Finkelberg realizing the quantum Toda lattice as the equivariant homology of the dual affine Grassmannian; the Fourier transform boils down to nothing more than the duality between homology and cohomology. In Sect. 3, we compare our result with a related result of V. Ginzburg, and explain the apparent discrepancy by showing that W-equivariant quasicoherent sheaves on $${{\mathrm{\mathfrak {t}}}}^*$$ descend to $${{\mathrm{\mathfrak {t}}}}^*//W$$ if they descend to $${{\mathrm{\mathfrak {t}}}}^*/\langle s_i\rangle $$ for every simple reflection $$s_i$$ of W.

K-Dirac Complexes

This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each $k$-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each $k$-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the $k$-Dirac operator studied in Clifford analysis.

Elliptic Complex on the Grassmannian of Oriented 2-Planes

The Grassmannian $V_2(\mathbb{R}^{n+2})$ of oriented 2-planes in $\mathbb R^{n+2}$ where $n\ge3$ carries a homogeneous parabolic contact structure of Grassmannian type. The main result of this article is that on $V_2(\mathbb{R}^{n+2})$ lives an elliptic complex of invariant differential operators of length 3 which starts with the 2-Dirac operator and that the index of the complex is zero.

Unifying relations for scattering amplitudes

We derive new amplitudes relations revealing a hidden unity among a wideranging variety of theories in arbitrary spacetime dimensions. Our results rely on a set of Lorentz invariant differential operators which transmute physical tree-level scattering amplitudes into new ones. By transmuting the amplitudes of gravity coupled to a dilaton and two-form, we generate all the amplitudes of Einstein-Yang-Mills theory, Dirac-Born-Infield theory, special Galileon, nonlinear sigma model, and biadjoint scalar theory. Transmutation also relates amplitudes in string theory and its variants. As a corollary, celebrated aspects of gluon and graviton scattering like color-kinematics duality, the KLT relations, and the CHY construction are inherited traits of the transmuted amplitudes. Transmutation recasts the Adler zero as a trivial consequence of the Weinberg soft theorem and implies new subleading soft theorems for certain scalar theories.

On osp(2|2)-relative cohomology of the Lie superalgebra of contact vector fields and deformations

Abstract We consider the 1 | 2 -dimensional real superspace R 1 | 2 endowed with its standard contact structure defined by the 1-form α . The conformal Lie superalgebra K ( 2 ) acts on R 1 | 2 as the Lie superalgebra of contact vector fields; it contains the Mobius superalgebra o s p ( 2 | 2 ) . We classify o s p ( 2 | 2 ) -invariant superskew-symmetric binary differential operators from K ( 2 ) ∧ K ( 2 ) to D λ , μ vanishing on o s p ( 2 | 2 ) , where D λ , μ is the superspace of linear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential o s p ( 2 | 2 ) -relative cohomology of K ( 2 ) with coefficients in D λ , μ . We study generic formal o s p ( 2 | 2 ) -trivial deformations of the K ( 2 ) -module structure on the direct sum of the superspaces of weighted densities. This work is the simplest superization of a result by Bouarroudj (2007).

Invariant differential operators in positive characteristic

In 1928, at the IMC, Veblen posed the problem: classify invariant differential operators between spaces of “natural objects” (in modern terms: either tensor fields, or jets) over a real manifold of any dimension. The problem was solved by Rudakov for unary operators (no nonscalar operators except the exterior differential); by Grozman for binary operators. In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case over any field of positive characteristic. Unary invariant operators are known: these are the exterior differential and analogs of the Berezin integral. We construct new binary operators from these analogs and discovered two more (up to dualizations) types of new indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators. Gordan's transvectants, aka Cohen–Rankin brackets, always invariant with respect to the simple 3-dimensional Lie algebra, are also invariant, in characteristic 2, with respect to the whole Lie algebra of vector fields on the line when the height of the indeterminate is equal to 2.

Applications of ‐invariant distributions

In this sequel to the article , a criterion for the regularity of fundamental solutions of differential operators with positive symbol is proved in analogy to HA¶rmander's criteria, see . It implies that the temperate fundamental solution of the operator [∂,∂]2+1 is not regular. Furthermore, a formula for the convolution of two O(p,q)†invariant distributions is presented, and, finally, L. Schwartz' question on the surjectivity of linear partial differential operators with constant coefficients on the space OM is completely answered in the case of O(p,q)†invariant differential operators.