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Research paper thumbnail of Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions

Comptes Rendus Mathematique

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in orde... more 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 ⊂ X 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 = −∆ + Y ∈S v Y , where the subspaces Y ⊂ X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y ⊂ X, and Nistor and the authors considered the case when S is a finite semi lattice and Georgescu generalized these results to any families. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, 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.

Research paper thumbnail of K-theory for Sp(n,1

Research paper thumbnail of Métriques positives sur les espaces homogènes réductifs

Comptes Rendus Mathematique, 2007

Positive metrics on reductive spaces Let G be a real semi-simple Lie group and L be a reductive s... more Positive metrics on reductive spaces Let G be a real semi-simple Lie group and L be a reductive subgroup of G stable by the Cartan involution. We define a familly of positive metrics on G/L parametrized by the points of G/K, where K is a maximal compact subgroup of G. We then use these metrics to generalize a lemma of Rawnsley Schmid and Wolf from representation theory. We then show that the representation of G by left translation on the space of L de variétés de drapeaux (au sens de Wolf [6]) aété effectuée par H. W. Wong . La construction d'une structure unitaire dans ces espaces de cohomologie conduità considérer simultanément deux métriques sur ces variétés de drapeaux [5,1] : une première métrique, positive, nonéquivariante par rapportà l'action du groupe de Lie semi-simple G, et dont la définition dépend du choix d'une involution de Cartan ; et une seconde, equivariante mais seulement non-dégénérée, qui est définie intrinsèquement. Nous nous intéressons icì a la métrique positive. Nous donnons une nouvelle définition plus géométrique de cette métrique, ce qui permet d'une part de comprendre comment celle-ci dépend du choix d'une involution de Cartan, et d'autre part de montrer en toute généralité que l'action du groupe sur les formes différentielles de carré

Research paper thumbnail of Hypoellipticity and cohomological induction

Let Y = G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define... more Let Y = G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define here an equivariant differential operator on G/L ∩ K playing the role of the Dolbeault Laplacian for the complex manifold G/L, using a distrbution transverse to the fibers of G/L∩K → G/L and satisfying the Hörmander condition. We prove here that this operator is (surprisingly but hopefully) not maximal hypoelliptic.

Research paper thumbnail of Maximal hypoellipticity and Dolbeault cohomology representations for U(p,q

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define a... more Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this operator is not maximal hypoelliptic when G=U(p,q).

Research paper thumbnail of Remarques à propos de l'opérateur de Dirac cubique

Computer Physics Communications, 2010

En 1999, Kostant introduit un opérateur de Dirac cubique D g/h associéà tout triplet (g, h, B), o... more En 1999, Kostant introduit un opérateur de Dirac cubique D g/h associéà tout triplet (g, h, B), où g est une algèbre de Lie complexe munie de la forme bilinéaire symétrique ad ginvariante non dégénérée B, et h est une sous-algèbre de Lie de g sur laquelle B est non dégénérée. Kostant montre alors que le carré de D g/h vérifie une formule qui généralise la formule de Parthasarathy. Nous donnons ici une nouvelle démonstration de cette formule. Tout d'abord, au moyen d'une induction parétage, nous montrons qu'il suffit d'établir la formule dans le cas particulier où h = 0. Il apparaît alors que, dans ce cas, l'annulation du terme d'ordre 1 dans la formule de Kostant pour D 2 g/h est une conséquence de propriétés classiques en cohomologie des algèbres de Lie, tandis que le fait que le carré du terme cubique soit scalaire résulte de telles considérations, ainsi que de l'identité de Jacobi.

Research paper thumbnail of Remarques a propos de l'operateur de Dirac cubique

Comptes Rendus Mathematique, 2010

Remarks on the Kostant Dirac operator In 1999, Kostant [Kos99] indroduces a Dirac operator D_g/h ... more 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.

Research paper thumbnail of Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions

Comptes Rendus Mathematique

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in orde... more 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 ⊂ X 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 = −∆ + Y ∈S v Y , where the subspaces Y ⊂ X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y ⊂ X, and Nistor and the authors considered the case when S is a finite semi lattice and Georgescu generalized these results to any families. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, 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.

Research paper thumbnail of K-theory for Sp(n,1

Research paper thumbnail of Métriques positives sur les espaces homogènes réductifs

Comptes Rendus Mathematique, 2007

Positive metrics on reductive spaces Let G be a real semi-simple Lie group and L be a reductive s... more Positive metrics on reductive spaces Let G be a real semi-simple Lie group and L be a reductive subgroup of G stable by the Cartan involution. We define a familly of positive metrics on G/L parametrized by the points of G/K, where K is a maximal compact subgroup of G. We then use these metrics to generalize a lemma of Rawnsley Schmid and Wolf from representation theory. We then show that the representation of G by left translation on the space of L de variétés de drapeaux (au sens de Wolf [6]) aété effectuée par H. W. Wong . La construction d'une structure unitaire dans ces espaces de cohomologie conduità considérer simultanément deux métriques sur ces variétés de drapeaux [5,1] : une première métrique, positive, nonéquivariante par rapportà l'action du groupe de Lie semi-simple G, et dont la définition dépend du choix d'une involution de Cartan ; et une seconde, equivariante mais seulement non-dégénérée, qui est définie intrinsèquement. Nous nous intéressons icì a la métrique positive. Nous donnons une nouvelle définition plus géométrique de cette métrique, ce qui permet d'une part de comprendre comment celle-ci dépend du choix d'une involution de Cartan, et d'autre part de montrer en toute généralité que l'action du groupe sur les formes différentielles de carré

Research paper thumbnail of Hypoellipticity and cohomological induction

Let Y = G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define... more Let Y = G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define here an equivariant differential operator on G/L ∩ K playing the role of the Dolbeault Laplacian for the complex manifold G/L, using a distrbution transverse to the fibers of G/L∩K → G/L and satisfying the Hörmander condition. We prove here that this operator is (surprisingly but hopefully) not maximal hypoelliptic.

Research paper thumbnail of Maximal hypoellipticity and Dolbeault cohomology representations for U(p,q

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define a... more Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this operator is not maximal hypoelliptic when G=U(p,q).

Research paper thumbnail of Remarques à propos de l'opérateur de Dirac cubique

Computer Physics Communications, 2010

En 1999, Kostant introduit un opérateur de Dirac cubique D g/h associéà tout triplet (g, h, B), o... more En 1999, Kostant introduit un opérateur de Dirac cubique D g/h associéà tout triplet (g, h, B), où g est une algèbre de Lie complexe munie de la forme bilinéaire symétrique ad ginvariante non dégénérée B, et h est une sous-algèbre de Lie de g sur laquelle B est non dégénérée. Kostant montre alors que le carré de D g/h vérifie une formule qui généralise la formule de Parthasarathy. Nous donnons ici une nouvelle démonstration de cette formule. Tout d'abord, au moyen d'une induction parétage, nous montrons qu'il suffit d'établir la formule dans le cas particulier où h = 0. Il apparaît alors que, dans ce cas, l'annulation du terme d'ordre 1 dans la formule de Kostant pour D 2 g/h est une conséquence de propriétés classiques en cohomologie des algèbres de Lie, tandis que le fait que le carré du terme cubique soit scalaire résulte de telles considérations, ainsi que de l'identité de Jacobi.

Research paper thumbnail of Remarques a propos de l'operateur de Dirac cubique

Comptes Rendus Mathematique, 2010

Remarks on the Kostant Dirac operator In 1999, Kostant [Kos99] indroduces a Dirac operator D_g/h ... more 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.