Caractères des groupes de Lie

Let G be a real Lie group with Lie algebra $\text{G}$. M. Duflo has constructed irreducible unitary representations of G associated to some G-orbits Ω in the dual $\text{G}$¹ of $\text{G}$. We prove a character formula when Ω is tempered, closed, and of maximal dimension.     

Representations of certain semidirect product groups

We study a class of semidirect product groups G = N · U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincaré group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces.     

关于SL(2,R)上速降函数的反演公式

在局部紧可分群的一般理论中，分解正则表示以及获得反演公式（或 Ｐｌａｎ－ｃｈｅｒｅｌ定理的明确表示）是调和分析的基本目标之一．ＳＬ（２，　）是最简单的非交换局部紧么模半单Ｌｉｅ群．Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ在 Ｃ∞ｃ（ＳＬ（２，　））上获得了反演公式，Ｘｉａｏ和ｈｅｎｇ在文［１］中证明了Ｃ３ｃ（ＳＬ（２， ）上的反演公式．在文［２］中Ｚｈｅｎｇ引入了Ｌｉｅ群Ｇ上函数的广义微分（Ａ导数）概念．在本文中，我们利用文［２］中的微分概念来研究ＳＬ（２，　）上可微函数的Ｆｏｕｒｉｅｒ变换的阶，并获得了ＳＬ（２，　）上速降函数的反演公式．     

Global character formulae for compact Lie groups

We introduce the concept of a modulator, which leads to a family of character formulae, each generalizing the Kirillov formula. For a suitable choice of modulator, this enables one to understand the Plancherel measure of a compact Lie group as arising from a partition of the identity on the dual of its Lie algebra.

     

Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple

Our aim is to describe the semicentre of the enveloping algebra of a parabolic subalgebra p of a semisimple finite dimensional complex Lie algebra g. Whilst [F. Fauquant-Millet, A. Joseph, Transformation Groups 6 (2) (2001) 125-142] describes explicitly the semicentre of the quantized enveloping algebra associated to p, specialization at q=1 gives only part of the required classical semicentre, even when p is a Borel. Similarly the graded of a polynomial subalgebra of the Hopf dual of the enveloping algebra of g, associated to the Kostant filtration, gives a lower bound on the required semicentre. Then by a method developed from [A. Joseph, Amer. J. Math. 99 (6) (1977) 1151-1165; J. Algebra 48 (1977) 241-289] we obtain an upper bound. Finally when g is a product of simple Lie algebras of type An or Cn we show that these bounds coincide and conclude that in this case the semicentre of the enveloping algebra of p is a polynomial algebra.     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

Étude de l'algèbre de Lie double des arbres enracinés décorés

The double Lie algebra LD of rooted trees decorated by a set D is introduced, generalising the construction of Connes and Kreimer. It is shown that it is a simple Lie algebra. Its derivations and its automorphisms are described, as well as some central extensions. Finally, the category of lowest weight modules is introduced and studied.     

Abstract Plancherel theorems and a Frobenius reciprocity theorem

A method for obtaining Plancherel theorems for unitary representations of Lie groups via C^∞ vector techniques is studied. The results are used to prove the nonunimodular Plancherel theorem of Moore and to study its convergence. A C^∞ Frobenius reciprocity theorem which generalizes Gelfand's duality theorem is also proven.     

Square-integrable representations and the dual topology

We show that the square-integrable factor representations of a connected locally compact group G are precisely the normal representations whose kernels in $\text{C}{}^1\text{(G)}$ are open points of Prim_r(G) (the support of Plancherel measure). Related results hold for certain other groups. We also settle questions of Dixmier, and Duflo and Moore, by giving examples of square-integrable irreducible representations (of totally disconnected groups) which are not open in the reduced dual.     

The annihilation theorem for the completely reducible Lie superalgebras

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre. This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternativeapproach to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV) determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded component of the harmonic space in the symmetric algebra. Oblatum 1-IX-1998 & 4-XII-1998 / Published online: 10 May 1999     

On the index of nonlocal elliptic operators for compact Lie groups

We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.     

The plancherel formula for the line bundles on SL(n + 1, ℝ)/S(GL(1,ℝ) × GL(n, ℝ))

In this paper we obtain the Plancherel formula for the spaces of L²-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, ?) and H = S(GL(1, ?) × GL(n 1, ?)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ_λ of the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

On the Kashiwara–Vergne conjecture

Let G be a connected Lie group, with Lie algebra . In 1977, Duflo constructed a homomorphism of -modules , which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group G. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara–Vergne conjecture for any Lie group G. (2) We give a reformulation of the Kashiwara–Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism , as well as a more general statement for distributions.     

Lie theory and the Chern-Weil homomorphism

Let P→B be a principal G-bundle. For any connection θ on P, the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra Wg=Sg^∗⊗∧g^∗ into the algebra of differential forms A=Ω(P). Invariant polynomials inv(Sg^∗)⊂Wg map to cocycles, and the induced map in cohomology inv(Sg^∗)→H(Abasic) is independent of the choice of θ. The algebra Ω(P) is an example of a commutativeg-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map Wg→A for possibly non-commutativeg-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism “up to g-homotopy”. Hence, the induced map inv(Sg^∗)→Hbasic(A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.     

Complex contact Lie groups and generalized complex Heisenberg groups

In this paper, we use root decomposition techniques to classify the complex contact Lie groups such that the Reeb vector field action on the Lie algebra is diagonalizable. These groups turn out to be isomorphic on the Lie algebra level to a particular type of generalized Heisenberg groups, namely the semi-direct product C2nΩ_×C, where Ω is the standard symplectic 2-form on C2n.     

Determinant formula and a realization for the Lie algebra <Emphasis Type="Italic">W</Emphasis> (2, 2)

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of certain vaccum module for the algebra W(2, 2) via theWeyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).     

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E₂. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.     

Integrals of equivariant forms and a Gauss-Bonnet theorem for constructible sheaves

The Berline-Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539-541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups G_R.As an application of this generalization, we prove an analogue of the Gauss-Bonnet theorem for constructible sheaves. If F is a G_R-equivariant sheaf on a complex projective manifold M, then the Euler characteristic of M with respect to F