High powers of random elements of compact Lie groups

Summary. If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9]. Received: 15 October 1995 / In revised form: 7 March 1996     

Foliations invariant under Lie group transverse actions

In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups. Authors’ address: Alexandre Behague and Bruno Scárdua, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998     

A global slice theorem for proper Hamiltonian actions

An analogue of a theorem of Abels [A] reducing the proper action of a non-compact Lie group to a compact transformation group is proved for the Hamiltonian setting. Received: 26 June 1997 / Revised version: 12 October 1998     

On SS-rigid groups and A. Weil's criterion for local rigidity. I

The fundamental group Γ of a compact complete affine manifold is represented as an affine crystallographic subgroup of . L.S. Auslander conjectured that Γ is virtually solvable. Our purpose is to find the algebraic condition on Γ which leads affirmative answer to the conjecture. Received: 26 May 1997 / Revised version: 17 December 1997     

On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups

Summary. This paper presents some explicit lower bound estimates of logarithmic Sobolev constant for diffusion processes on a compact Riemannian manifold with negative Ricci curvature. Let Ric≧−K for some K>0 and d, D be respectively the dimension and the diameter of the manifold. If the boundary of the manifold is either empty or convex, then the logarithmic Sobolev constant for Brownian motion is not less than max {(d d+2) ^d 1 2(d+1)D ² exp [−1−(3d+2)D ² K],     (d−1 d+1) ^d K exp [−4D√d K]} . Next, the gradient estimates of heat semigroups (including the Neumann heat semigroup and the Dirichlet one) are studied by using coupling method together with a derivative formula modified from [11]. The resulting estimates recover or improve those given in [7, 21] for harmonic functions. Received: 19 September 1995 / In revised form 11 April 1996     

Dixmier's trace for boundary value problems

Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal for the Hilbert space of L ² sections in vector bundles E over X, F over ∂X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu=f; Tu=0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor. Received: Received: 13 January 1998     

Contributions to Real Exponential Lie Groups

 A Lie group is called exponential if its exponential map is surjective. It is called weakly exponential if the exponential image is dense, which is equivalent to the connectivity of each of the Cartan subgroups (compare [11]). In the present paper the authors study exponential Lie groups which are of mixed type, i.e., neither solvable nor semisimple. Necessary conditions and also, for special mixed Lie groups, sufficient conditions are given for exponentiality. Several counter examples are provided showing that none of the conditions which have surfaced during the course of our investigation can work as necessary and sufficient ones. All conditions considered deal with centralizers of ad-nilpotent elements of the Lie algebra. For example, it is shown that if G is exponential, there is a rather large characteristic subgroup B which contains the nilradical, all Levi factors, and all maximal compactly embedded subgroups, which is also exponential. Moreover, this subgroup is also Mal’cev splittable so that one can apply earlier results on Mal’cev splittable exponential Lie groups, which characterize exponentiality of these Lie groups (also by conditions concerning the centralizers of ad-nilpotent elements). (Received 1 June 1999; in final form 28 December 1999)     

On topologically distinct solutions of the Dirichlet problem for Yang-Mills connections

We prove that for generic Dirichlet boundary data there exist infinitely many topologically distinct solutions to the Dirichlet problem for -Yang-Mills equations over . These are absolute Yang-Mills minimizers in topologically distinct connected components of the space of connections considered. There is a special case for which only finitely many topologically distinct solutions can be found by our method. This corresponds to the simultaneous existence of self dual and anti-self dual solutions, for the given boundary data. If the boundary data is non-flat there exists always more than one solution. This paper generalizes to Yang-Mills fields an important result by Brezis and Coron, who show existence of more than one minimizing harmonic map for non-constant Dirichlet data in two dimensions. Received February 1, 1996 / Accepted March 15, 1996     

Factoring a Lie Group into a Compactly Embeddedand a Solvable Subgroup

 Let G be a real connected Lie group. A subgroup K is called compactly embedded if the closure of Ad(K) is compact in Aut(). If K is, in addition, maximal with respect to this property, then there exists a solvable subgroup S containing the nilradical such that and is the one-component of the center of G. (Received 1 June 1999; in revised form 28 December 1999)     

Maximal cofinitary groups

We discuss the cardinalities of maximal cofinitary groups under the assumption of . We also discuss various open questions in this area. Received: 24 July 1997     

Comparison of germ expansion¶on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π^′ on GL _n (D) where D is a division algebra over k of index m and the germ expansion of the representation π of GL _mn (k) associated to π^′ by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GL _mn (k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GL _n (D). Received: 30 September 1999 / Revised version: 11 February 2000     

Remarques sur les m\'ethodes de projection pour l'approximation des \'equations de Navier--Stokes

Summary. L'objet de cet article est de montrer que les estimations de convergence sur la pression pour les m\'ethodes de projection d\'ecrites dans \cite{Shen1} et \cite{Shen2} ne sont pas obtenues correctement car elles sont toutes bas\'ees sur une in\'egalit\'e fausse. Il semble qu'on ait besoin d'une convergence en de la vitesse dans pour d\'emontrer les estimations de convergence sur la pression en . La question de savoir si la m\'ethode de projection a un taux de convergence pour la pression plus \'elev\'e que reste ouverte. Received June 1, 1993