arXiv subjects

Nicolas Prudhon

Publications and source records attributed to Nicolas Prudhon.

5 recordsLinked to original sources

arXiv2026

Symplectic Dirac operators on homogeneous spaces

We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator decomposes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dolbeault operators can yield such a formula, in contrast with Kostant's cubic Dirac operator in the orthogonal setting. As applications, we establish an ${\mathfrak s}{\mathfrak l}_2$-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy-Dirac inequality, while the higher levels inequalities yield new constraints. For $SU(1,n)$, we show that, for representations with a specific Kraljevi\'c corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular $K$-types, precisely those satisfying an explicit highest-weight condition.

math.RT↗

arXiv2018

Exhaustive families of representations of $C^*$-algebras associated to $N$-body Hamiltonians with asymptotically homogeneous interactions

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - Δ+ \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of $X$ introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.

math.FA↗

arXiv2016

Translation of Dolbeault representations on reductive homogeneous spaces

We adapt techniques used in the study of the cubic Dirac operator on homogeneous reductive spaces to the Dolbeault operator on elliptic coadjoint orbits to prove that cohomologically induced representations have an infinitesimal character, that cohomological induction and Zuckerman translation functor commute and give a geometric interpretation of the Zuchkerman translation functor in this context.

math.RT↗

arXiv2016

Exhausting families of representations and spectra of pseudodifferential operators

Families of representations of suitable Banach algebras provide a powerful tool in the study of the spectral theory of (pseudo)differential operators and of their Fredholmness. We introduce the new concept of an exhausting family of representations of a C*-algebra A. An {\em exhausting family} of representations of a C*-algebra A is a set F of representations of A with the property that every irreducible representation of A is weakly contained in some ϕ\in F. An exhausting family F of representations of A has the property that `"a \in A is invertible if, and if, ϕ(a) is invertible for any ϕ\in F." Consequently, the spectrum of a is given by \Spec(a) = \cup_{ϕ\in F} \Spec(ϕ(a)). In other words, every exhausting family of representations is invertibility sufficient, a concept introduced by Roch in 2003. We prove several properties of exhausting families and we provide necessary and sufficient conditions for a family of representations to be exhausting. Using results of Ionescu and Williams (2009), we show that the regular representations of amenable, second countable, locally compact groupoids with a Haar system form an exhausting family of representations. If $A$ is a separable C*-algebra, we show that a family F of representations of $A$ is exhausting if, and only if, it is invertibility sufficient. However, this result is not true, in general, for non-separable C*-algebras. With an eye towards applications, we extend our results to the case of unbounded operators. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator F is invertible if, and only if, its Mellin transform is invertible. In view of possible applications, we have tried to make this paper accessible to non-specialists in C*-algebras.

math.OA↗

arXiv2010

Remarques a propos de l'operateur de Dirac cubique

Remarks on the Kostant Dirac operator In 1999, Kostant [Kos99] indroduces a Dirac operator D_g/h associated to any triple (g, h,B), where g is a complex Lie algebra provided with an ad g-invariant non degenerate nsymetric bilinear form B, and h is a Lie subalgebra of g such that the bilinear form B is non degenerate on h. Kostant then shows that the square of this operator safisties a formula that generalizes the so-called Parthasarathy formula [Par72]. We give here a new proof of this formula. First we use an induction by stage argument to reduce the proof of the formula to the particular case where h = 0. In this case we show that the vanishing of the first ordrer term in the Kostant formula for D2_g/h is a consequence of classic properties related to Lie algebra cohomology, and the fact that the square of the cubic term is a scalar follows from such considerations, together with the Jacobi identity.

math.RT↗