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Jean-Louis Clerc

Publications and source records attributed to Jean-Louis Clerc.

At least 19 recordsLinked to original sources
arXiv2023

The source operator method: an overview

This is an overview on the {source operator method} which leads to the construction of symmetry breaking differential operators (SBDO) in the context of tensor product of two principals series representations for the conformal group of a simple real Jordan algebra. This method can be applied to other geometric contexts: in the construction of SBDO for differential forms and for spinors, and also for the construction of Juhl's operators corresponding to the restriction from the sphere $S^n$ to $S^{n-1}$.

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arXiv2021

Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\mathbb{C}\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $ρ$ and $ρ'$ be the corresponding representations of the spin group ${\rm Spin}(n)$. The group $G= {\rm Spin}(1,n+1)$ is a (twofold) covering of the conformal group of $\mathbb R^n$. For $λ, μ\in \mathbb C$, let $π_{ρ, λ}$ (resp. $π_{ρ',μ}$) be the spinorial representation of $G$ realized on a (subspace of) $C^\infty(\mathbb R^n,\mathbb S)$ (resp. $C^\infty(\mathbb R^n,\mathbb S')$). For $0\leq k\leq n$ and $m\in \mathbb N$, we construct a symmetry breaking differential operator $B_{k;λ,μ}^{(m)}$ from $C^\infty(\mathbb R^n \times \mathbb R^n,\mathbb{S}\,\otimes\, \mathbb{S}')$ into $C^\infty(\mathbb R^n, Λ^*_k(\mathbb R^n) \otimes \mathbb{C})$ which intertwines the representations $π_{ρ, λ}\otimes π_{ρ',μ} $ and $π_{τ^*_k,λ+μ+2m}$, where $τ^*_k$ is the representation of ${\rm Spin}(n)$ on the space $Λ^*_k(\mathbb R^n) \otimes \mathbb{C}$ of complex-valued alternating $k$-forms on $\mathbb{R}^n$.

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arXiv2021

Construction à la Ibukiyama of symmetry breaking differential operators, I

The construction of symmetry breaking differential operators, using invariant pluri-harmonic polynomials, due to T. Ibukiyama in the context of the Siegel upper half space, is extended for scalar representations to general Hermitian symmetric spaces of tube-type. The new context is described in terms of Euclidean Jordan algebras and their representations. As an example, new and explicit differential operators are obtained for the restriction from the tube domain over the light cone to the product of two upper half-planes.

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arXiv2020

Rankin-Cohen brackets on tube-type domains

A new formula is obtained for the holomorphic bi-differential operators on tube-type domains which are associated to the decomposition of the tensor product of two scalar holomorphic representations, thus generalizing the classical Rankin-Cohen brackets. The formula involves a family of polynomials of several variables which may be considered as a (weak) generalization of the classical Jacobi polynomials.

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arXiv2018

Conformally covariant bi-differential operators for differential forms

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.

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arXiv2018

The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type

Let $M$ be an irreducible Hermitian symmetric space of compact type, and let $ω$ be its Kähler form. For a triplet $(p_1,p_2,p_3)$ of points in $M$ we study conditions under which a geodesic triangle $\mathcal T(p_1,p_2,p_3)$ with vertices $p_1,p_2,p_3$ can be unambiguously defined. We consider the integral $A(p_1,p_2,p_3)=\int_Σω$, where $Σ$ is a surface filling the triangle $\mathcal T(p_1,p_2,p_3)$ and discuss the dependence of $A(p_1,p_2,p_3)$ on the surface $Σ$. Under mild conditions on the three points, we prove an explicit formula for $A(p_1,p_2,p_3)$ analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.

math.DG↗
arXiv2017

Conformally covariant differential operators for the diagonal action of O(p, q) on real quadrics

Let $X=G/P$ be a real projective quadric, where $G=O(p,q)$ and $P$ is a parabolic subgroup of $G$. Let $\left(π_{λ,ε}, \mathcal{H}_{λ,ε}\right)_{ (λ,ε)\in \mathbb {C}\times \{\pm\}}$ be the family of (smooth) representations of $G$ induced from the characters of $P$. For $(λ, ε), (μ, η)\in \mathbb{C} \times \{\pm\}$, a differential operator $\mathbf{D}_{(λ,ε), (μ,η)}^{reg}$ on $X\times X$, acting $G$-covariantly from $\mathcal{H}_{λ,ε} \otimes \mathcal{H}_{μ, η}$ into $\mathcal{H}_{λ+1,-ε} \otimes \mathcal{H}_{μ+1, -η}$ is constructed.

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arXiv2017

Another Approach to Juhl's Conformally Covariant Differential Operators from $S^n$ to $S^{n-1}$

A family $({\mathbf D}_λ)_{λ\in \mathbb C}$ of differential operators on the sphere $S^n$ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of $S^n$ which preserve the smaller sphere $S^{n-1}\subset S^n$. The family of conformally covariant differential operators from $S^n$ to $S^{n-1}$ introduced by A. Juhl is obtained by composing these operators on $S^n$ and taking restrictions to $S^{n-1}$.

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arXiv2017

Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$).

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arXiv2015

Singular conformally invariant trilinear forms, II The higher multiplicity cases

Let $S$ be the sphere of dimension $n-1, n\geq 4$. Let $(π_λ)_{λ\in \mathbb C}$ be the scalar principle series of representations of the conformal group $SO_0(1,n)$, realized on $\mathcal C^\infty(S)$. For $\boldsymbol λ= (λ_1,λ_2,λ_3) \in \mathbb C^3$, let $Tri(\boldsymbol λ)$ be the space of continuous trilinear forms on $\mathcal C^\infty(S) \times \mathcal C^\infty(S) \times \mathcal C^\infty(S)$ which are invariant under $π_{λ_1} \otimes π_{λ_2} \otimes π_{λ_3} $. For each value of $\boldsymbol λ$, the dimension of $Tri(\boldsymbol λ)$ is computed and a basis of $Tri(\boldsymbol λ)$ is described.

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arXiv2015

Singular conformally invariant trilinear forms, I Multiplicity one results

A normalized holomorphic family (depending on $\boldsymbol λ\in \mathbb C^3$) of conformally invariant trilinear forms on the sphere is studied. Its zero set $Z$ is described. For $\boldsymbol λ\notin Z$, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1, except perhaps for a denumerable subset.

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arXiv2014

Conformal covariance for the powers of the Dirac operator

A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The proof uses their relation with the Knapp-Stein intertwining operators for the spinorial principal series. We also treat the compact picture, i.e. the corresponding operators on the sphere, where certain polynomials of the Dirac operator appear. This gives a new representation-theoretic framework for earlier results.

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arXiv2012

Intertwining operators for the generalized principal series on symmetric R-spaces

Three questions about the intertwining operators for the generalized principal series on a symmetric R-space are solved : description of the functional kernel, both in the compact and the non-compact picture, domain of convergence, meromorphic continuation. A large use is made of the theory of positive Jordan triple systems. The meromorphic continuation of the intertwining integral is achieved via a Bernstein-Sato identity, and a precise description of the poles is obtained.

math.RT↗
arXiv2011

Singular conformally invariant trilinear forms and covariant differential operators on the sphere

Let $G=SO_0(1,n)$ be the conformal group acting on the $(n-1)$ dimensional sphere $S$, and let $(π_λ)_{λ\in \mathbb C}$ be the spherical principal series. For generic values of $\boldsymbol λ=(λ_1,λ_2,λ_3)$ in $\mathbb C^3$, there exits a (essentially unique) trilinear form on $\mathcal C^\infty(S)\times \mathcal C^\infty(S)\times \mathcal C^\infty(S)$ which is invariant under $π_{λ_1}\otimes π_{λ_2}\otimes π_{λ_3}$. Using differential operators on the sphere $S$ which are covariant under the conformal group $SO_0(1,n)$, we construct new invariant trilinear forms corresponding to singular values of $\boldsymbol λ$. The family of generic invariant trilinear forms depend meromorphically on the parameter $\boldsymbol λ$ and the new forms are shown to be residues of this family.

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arXiv2010

Generalized Bernstein--Reznikov integrals

We find a closed formula for the triple integral on spheres in $\mathbb{R}^{2n}\times\mathbb{R}^{2n}\times\mathbb{R}^{2n}$ whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein--Reznikov integral formula in the $n=1$ case. Our method also applies for linear and conformal structures.

math.CA↗
arXiv2010

Conformally invariant trilinear forms on the sphere

To each complex number $λ$ is associated a representation $π_λ$ of the conformal group $SO_0(1,n)$ on $\mathcal C^\infty(S^{n-1})$ (spherical principal series). For three values $λ_1,λ_2,λ_3$, we construct a trilinear form on $\mathcal C^\infty(S^{n-1})\times\mathcal C^\infty(S^{n-1})\times \mathcal C^\infty(S^{n-1})$, which is invariant by $π_{λ_1}\otimes π_{λ_2}\otimes π_{λ_3}$. The trilinear form, first defined for $(λ_1, λ_2,λ_3)$ in an open set of $\mathbb C^3$ is extended meromorphically, with simple poles located in an explicit family of hyperplanes. For generic values of the parameters, we prove uniqueness of trilinear invariant forms.

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