Invariant Differential Operators Research Articles

Global Gauge Symmetries, Risk-Free Portfolios, and the Risk-Free Rate

We define risk-free portfolios using three gauge invariant differential operators that require such portfolios to be insensitive to price changes, to be self-financing, and to produce a zero real return so there are no risk-free profits. This definition identifies the risk-free rate as the return of an infinitely diversified portfolio rather than as an arbitrary external parameter. The risk-free rate measures the rate of global price rescaling, which is a gauge symmetry of economies. We explore the properties of risk-free rates, rederive the Black Scholes equation with a new interpretation of the risk-free rate parameter as a that background gauge field, and discuss gauge invariant discounting of cash flows.

An analog of the Schwartz theorem on spectral analysis on a hyperbolic plane

Let \( \mathbb{D} \) be an open unit disk in the complex plane. It is shown that every subspace in C(\( \mathbb{D} \)) invariant under weighted conformal shifts contains a radial eigenfunction of the corresponding invariant differential operator. This function can be expressed via the Gauss hypergeometric function and is a generalization of the spherical function on the disk \( \mathbb{D} \) which is considered as a hyperbolic plane with the corresponding Riemannian structure.

On an analytic description of the α-cosine transform on real Grassmannians

The goal of this paper is to describe the [Formula: see text]-cosine transform on functions on real Grassmannian [Formula: see text] in analytic terms as explicitly as possible. We show that for all but finitely many complex [Formula: see text] the [Formula: see text]-cosine transform is a composition of the [Formula: see text]-cosine transform with an explicitly written (though complicated) [Formula: see text]-invariant differential operator. For all exceptional values of [Formula: see text] except one, we interpret the [Formula: see text]-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value [Formula: see text], which is [Formula: see text], is still an open problem.

The affine cohomology spaces and its applications

We compute the [Formula: see text] cohomology space of the affine Lie superalgebra [Formula: see text] on the (1,1)-dimensional real superspace with coefficient in a large class of [Formula: see text]-modules [Formula: see text]. We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. This work is the generalization of a result by Basdouri et al. [The linear [Formula: see text]-invariant differential operators on weighted densities on the superspace [Formula: see text] and [Formula: see text]-relative cohomology, Int. J. Geom. Meth. Mod. Phys. 10 (2013), Article ID: 1320004, 9 pp.]

Differential operators for Siegel-Jacobi forms

For any positive integers $n$ and $m$, $\mathbb{H}_{n,m}:=\mathbb{H}_n\times\mathbb{C}^{(m,n)}$ is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. We compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we construct a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for $\mathbb{H}_{n,m}$ are obtained.

On boundary value problems for some conformally invariant differential operators

ABSTRACTWe study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace, respectively, Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain Lp-spaces.The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems.

Poincaré series for non-Riemannian locally symmetric spaces

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ\\G/H, beyond the classical cases where H is compact (automorphic forms) or Γ is trivial (analysis on symmetric spaces).For any non-Riemannian reductive symmetric space X=G/H on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Γ whose action on X is “sufficiently proper”, we construct L2-eigenfunctions of the Laplacian on XΓ:=Γ\\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to XΓ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X.We prove that the Poincaré series we construct still converge, and define nonzero L2-functions, after any small deformation of Γ inside G, for a large class of groups Γ. Thus the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface.We actually construct joint L2-eigenfunctions for the whole commutative algebra of invariant differential operators on XΓ.

Branching laws for Verma modules and applications in parabolic geometry. I

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules (T. Kobayashi (2012) [22]), we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan–Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators (Juhl (2009) [16]) and its generalizations to spinor bundles. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift

AbstractGiven a measureon a locally symmetric spaceobtained as a weak-* limit of probability measures associated with eigenfunctions of the ring of invariant differential operators, we construct a measureon the homogeneous spaceX= Γ\Gthat liftsand is invariant by a connected subgroupA1⊂Aof positive dimension, whereG=NAKis an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, thenis also the limit of measures associated with Hecke eigenfunctions on X. This generalizes results of the author with A.Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

Invariants of third‐order ordinary differential equations y′′′=f(x,y,y′,y′′) via point transformations

A new systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class y′′′ = f(x, y, y′, y′′) under general point equivalence transformations in the generic case.As a result, all third‐order differential invariants, relative and absolute invariant differentiation operators are determined for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative. These relative invariant differentiation operators are used to determine the fourth‐order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex planewith four infinitesimal symmetries for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative, are provided. Copyright © 2015 John Wiley & Sons, Ltd.

On geometry of the (generalized) G2-manifolds

In this paper, we first understand the classical [Formula: see text]-structure and [Formula: see text]-geometry from the viewpoint of spinor, which is a more familiar framework for physicists. Explicit construction of invariant spinor is given via the Dirac gamma matrices. We introduce a notion of multispinor bundle associated with invariant spinor and differential operator on the section of this bundle. Then we study the vector fields satisfy some additional properties on [Formula: see text]-manifold, more precisely, we prove some no-go theorems corresponding to the vector field preserving the associated 4-form on [Formula: see text]-manifold, and we also consider the nowhere-vanishing vector field which induces an integrable complex structure on the vertical direction of tangent bundle. Next we discuss the relation between the variation of metric and that of effective action on the moduli space of integrable [Formula: see text]-structures. In the last section, we deal with the structure operators on generalized [Formula: see text]-manifold after describing the integrability of generalized [Formula: see text]-structure, some identities of structure operators are derived, which are analogues of Kähler-type and Weitzenböck-type identities under the classical case. And finally, we introduce a flow of which a generalized [Formula: see text]-manifold can be realized as the fixed point.

Conformal symmetries of the super Dirac operator

In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy–Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries \mathfrak{osp}(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries \mathfrak{osp}(m+1,1|2n) . The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair \mathfrak{osp}(1|2)\times\mathfrak{osp}(m|2n)\subset \mathfrak{osp}(m+4n|2m+2n) . Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries.

Pizzetti Formulae for Stiefel Manifolds and Applications

Pizzetti's formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson-Zuber integral for the coset SO(4)/[SO(2)xSO(2)]. This integral naturally appears in the calculation of the two-point correlation function in the transition of the statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in random matrix theory.

Differential invariants and operators of invariant differentiation of the projectable action of Lie groups

We describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples.

Laplace type invariants for variable coefficient mKdV equations

We consider a class of variable-coefficient mKdV equations. We derive the equivalence transformations in the infinitesimal form and we employ them to construct differential invariants of the respective equivalence algebra. Operators of invariant differentiation are also constructed. Applications, similar to Laplace invariants, are presented.

Consistent grid analogs of invariant differential and boundary operators on an irregular triangular grid in the case of a grid nodal approximation

Grid analogs of invariant first-order differential operators and boundary operators are constructed on an irregular triangular grid. They are consistent in the sense of satisfying the grid analogs of integral relations that are consequences of the Gauss-Ostrogradskii formula for the divergence of a vector field as the product of a scalar and a vector, the vector product of vectors, or the inner product of a vector and a dyad. Construction is done using a grid-operator interpretation of integral relations written for piecewise linear interpolations of grid functions defined at the grid nodes. It is shown how to use consistent grid operators in constructing correct grid approximations of inhomogeneous boundary value problems at the operator level.

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl–Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.

Leading weight vectors and homomorphisms between generalized Verma modules

Kostant introduced leading weight vectors in his paper on Verma modules and quasi-invariant differential operators. We extend his idea to study homomorphisms between generalized Verma modules. We classify all first order leading weight vectors and thus determine corresponding Hom spaces between generalized Verma modules. Moreover, in light of the correspondence between leading weight vectors and invariant differential operators, we obtain a new and practical approach to the existence of first order invariant differential operators.

Invariant differential operators for non-compact Lie groups: the SO* (12) case

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so* (12). We give the main multiplets of indecomposable elementary representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebra so(6, 6) with suitably chosen maximal parabolic subalgebra.

The transverse index theorem for proper cocompact actions of Lie groupoids

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.