Kevin Coulembier

Integration in superspace using distribution theory

In this paper, a new class of Cauchy integral formulae in superspace is obtained, using formal expansions of distributions. This allows us to solve five open problems in the study of harmonic and Clifford analysis in superspace.

Orthogonality of Hermite polynomials in superspace and Mehler type formulae

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview of the results for O(m) and G, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proved. Next, Hermite polynomials in a full superspace with O(m) x Sp(2n)-symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced, which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schrodinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, the new results for the Sp(2n)- and O(m) x Sp(2n)-symmetry are compared with the results in the different types of symmetry.

Orthosymplectically invariant functions in superspace

The notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced. This leads to dimensional reduction theorems for differentiation and integration in superspace. These spherically symmetric functions can be used to solve orthosymplectically invariant Schroedinger equations in superspace, such as the (an)harmonic oscillator or the Kepler problem. Finally the obtained machinery is used to prove the Funk-Hecke theorem and Bochners relations in superspace.

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 osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries 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 osp(1|2) x osp(m|2n) < 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.

Joseph Ideals and Harmonic Analysis for

The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^{m-2}. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in the tensor algebra of g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n)=spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R^{m-2|2n} and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n).

The Orthosymplectic Lie Supergroup in Harmonic Analysis

The orthosymplectic supergroup OSp(m|2n) is introduced as the supergroup of isometries of flat Riemannian superspace R^{m|2n} which stabilize the origin. It also corresponds to the supergroup of isometries of the supersphere S^{m-1|2n}. The Laplace operator and norm squared on R^{m|2n}, which generate sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual pair (osp(m|2n),sl(2)). This Howe dual pair solves the problems of the dual pair (SO(m)xSp(2n),sl(2)), considered in previous papers. In particular we characterize the invariant functions on flat Riemannian superspace and show that the integration over the supersphere is uniquely defined by its orthosymplectic invariance. The supersphere manifold is also introduced in a mathematically rigorous way. Finally we study the representations of osp(m|2n) on spherical harmonics. This corresponds to the decomposition of the supersymmetric tensor space of the m|2n-dimensional super vectorspace under the action of sl(2)xosp(m|2n). As a side result we obtain information about the irreducible osp(m|2n)-representations L_{(k,0,...,0)}^{m|2n}. In particular we find branching rules with respect to osp(m-1|2n).

q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials

We define a q-deformation of the Dirac operator, inspired by the one-dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows us to construct q-deformed Schrodinger equations in higher dimensions. The equivalence of these Schrodinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford–Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal with respect to an m-dimensional q-integration, which is related to integration on q-Euclidean space. The q-Laguerre polynomials are the eigenvectors of an suq(1|1)-representation.

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

The Primitive Spectrum of a Basic Classical Lie Superalgebra

{{{\rm SO}(4)/[{\rm SO}(2)\times {\rm SO}(2)]}}