Distributions de type K-positif sur l'espace tangent

Let K be a compact subgroup of the isometry group of $\text{R}$ⁿ. A distribution T is said to be of K-positif type if it is K-invariant and if $\text{〈T, ? 1}\text{}\text{?}\text{gj〉 = ∝∝ ?(x + y)}\text{?(Y) dT(x) ? 0}$ for every K-invariant $\text{b}$^∞ function ? with compact support. We look for an integral representation of these distributions (i.e., an analog of the Bochner-Schwartz theorem). In this paper we obtain such a representation for distributions with growth of exponential type in the following case: K is the maximal compact subgroup of a semi-simple connected Lie group G with finite center, acting by the adjoint action on the tangent space of $\text{G}\text{K}$. The main step is to prove that it suffices to work with distributions of W-positif type (where W is the Weyl group associated with $\text{G}\text{K}$). This is achieved following ideas of a paper of S. Helgason [Advan. in Math.36 (1980) 297]. The end of the proof follows from the case where K is finite [N. Bopp, in “Analyse harmonique sur les groupes de Lie,” Lecture Notes in Mathematics No. 739, p. 15, Springer-Verlag, Berlin/New York 1979].     

Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by $\text{D}$(G) the space of C^∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U$\text{G}$_$\text{c}$ of the complexification of the Lie algebra of G for which the linear form $\text{? ψ(π(u 1 ?))}$ on $\text{D}$(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U$\text{G}$_$\text{c}$.     

Global solvability of the laplacians on pseudo-Riemannian symmetric spaces

Let G be a noncompact semisimple Lie group with finite center and H an open subgroup of the fixed point group of an involution of G. $\text{G}\text{H}$ becomes a pseudo-Riemannian manifold. We prove that the Laplacian P on $\text{G}\text{H}$ is globally solvable in the sense that $\text{PC}{}^{\mathrm{\infty }}\text{(}\text{G}\text{H}\text{) = C}{}^{\mathrm{\infty }}\text{(}\text{G}\text{H}\text{)}$. This generalizes the global solvability of the Casimir operators on non-compact semisimple Lie groups with finite center due to J. Rauch and D. Wigner.     

Algebraic K-theory of groups wreath product with finite groups

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example, for discrete subgroups of Lie groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], virtually poly-infinite cyclic groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], Artin braid groups [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526], a class of virtually poly-surface groups [S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press] and virtually solvable linear group [F.T. Farrell, P.A. Linnell, K-Theory of solvable groups, Proc. London Math. Soc. (3) 87 (2003) 309-336]. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G?H for H a finite group. The need for this kind of extension is already evident in [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526; S.K. Roushon, The Farrell-Jones isomorphism conjecture for 3-manifold groups, math.KT/0405211, K-Theory, in press; S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press]. We also prove the conjecture for some other classes of groups.     

On the failure of spectral synthesis for compact semisimple Lie groups

Let G be a compact connected semisimple Lie group with Lie algebra $\text{g}$. We show that the conjugacy class of a regular element of G is not a set of synthesis for the Fourier algebra of G. Similarly, the Ad(G)-orbit of a regular element of $\text{g}$ is not a set of synthesis for the algebra of Fourier transforms on $\text{g}$. In proving this latter result we demonstrate a regularity property of Ad-invariant Fourier transforms, analogous to the differentiability of radial Fourier transforms.     

Zonal measure algebras on isotropy irreducible homogeneous spaces

This paper analyzes the convolution algebra $\text{M(K}\text{G}\text{K}\text{)}$ of zonal measures on a Lie group G, with compact subgroup K, primarily for the case when $\text{M(K}\text{G}\text{K}\text{)}$ is commutative and $\text{G}\text{K}$ is isotropy irreducible. A basic result for such (G, K) is that the convolution of dim $\text{G}\text{K}$ continuous (on $\text{G}\text{K}$) zonal measures is absolutely continuous. Using this, the spectrum (maximal ideal space) of $\text{M(K}\text{G}\text{K}\text{)}$ is determined and shown to be in 1-1 correspondence with the bounded Borel spherical functions. Also, certain asymptotic results for the continuous spherical functions are derived. For the special case when G is compact, all the idempotents in $\text{M(K}\text{G}\text{K}\text{)}$ are determined.     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Analyse harmonique sur les espaces symétriques nilpotents

Let G be a connected simply connected nilpotent Lie group and H the set of fixed points of an involution of G; we give the Plancherel formula of the representation Ind_H^G(1) and infer from it the existence of an H-invariant tempered elementary solution for every nonzero G-invariant differential operator on $\text{G}\text{H}$.     

Asymptotic expansions of matrix coefficients: The real rank one case

Let G be a real reductive Lie group of class $\text{H}$, and suppose that the split rank of G is one. We show that the asymptotic expansions of the Eisenstein integrals given in Harish-Chandra (1) give uniform approximation off of a certain naturally defined compact subset of $\text{A}\text{?}$, the unitary dual of A; G = KAN being an Iwasawa decomposition of G.     

The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by $\text{C}$^p(G)(0<p?2), (ii) the K-biinvariant elements in $\text{C}$^p(G) by $\text{I}$^p(G), (iii) the positive definite (zonal) spherical functions by $\text{P}$, and (iv) the spherical transform on $\text{C}$^p(G) by ? → $\text{}\text{?}$gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto $\text{C}$^l(G), (ii) there exists a unique measure μ of polynomial growth on $\text{P}$ such that T[ψ]=∫pψdμ for all ψ?I¹(G) (iii) all measures μ of polynomial growth on $\text{P}$ are obtained in this way, and (iv) T may be extended to a particular $\text{I}$^p(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of $\text{P}$. These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.     

Invariant convex cones and causality in semisimple Lie algebras and groups

The convex cones in a simple Lie algebra $\text{G}$ invariant under the adjoint group G of $\text{G}$ are studied. Using a earlier abstract classification of such cones, we find explicit algebraic presentations of such cones in all the classical hermitian symmetric Lie algebras. (Nontrivial such cones exist only in these cases.) The G-orbits in such cones are listed. The notion of a temporal action of a Lie group with an invariant causal orientation upon a causally oriented manifold is defined. The canonical actions of such classical groups G as above on the S?hilov boundaries of the associated (tube-type) hermitian symmetric spaces are shown to be temporal actions. Corollaries are (1) the existence of nontrivial (Lie) semigroups S in the infinite-sheeted coverings $\text{G}\text{?}$ of G, which are invariant under conjugation by $\text{G}\text{?}$ and satisfy S ∩ S^?1 = {e}, and (2) the global causality (i.e. no “closed time-like curves”) of such covering groups $\text{G}\text{?}$.     

On the existence of uncountably many matroidal families

A matroidal family is a set $\text{F}$ ≠ ? of connected finite graphs such that for every finite graph G the edge-sets of those subgraphs of G which are isomorphic to some element of $\text{F}$ are the circuits of a matroid on the edge-set of G. Simões-Pereira [5] shows the existence of four matroidal families and Andreae [1] shows the existence of a countably infinite series of matroidal families. In this paper we show that there exist uncountably many matroidal families. This is done by using an extension of Andreae's theorem, a construction theorem, and certain properties of regular graphs. Moreover we observe that all matroidal families so far known can be obtained in a unified way.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

On a conjecture of ōshima

The set of homotopy classes of self maps of a compact, connected Lie group G is a group by the pointwise multiplication which we denote by H(G), and it is known to be nilpotent. ōshima [H. ōshima, Self homotopy group of the exceptional Lie group G₂, J. Math. Kyoto Univ. 40 (1) (2000) 177-184] conjectured: if G is simple, then H(G) is nilpotent of class ?rankG. We show this is true for PU(p) which is the first high rank example.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

A Banach-Dieudonné theorem for germs of holomorphic functions

Let $\text{H}$(U) denote the space of all holomorphic functions on an open subset U of a complex Fréchet space E. Let $\text{H}$(K) denote the space of all holomorphic germs on a compact subset K of E. It is shown that $\text{H}$(K), with a natural topology, is the inductive limit of a suitable sequence of compact subsets, within the category of all topological spaces. As an application of this result it is shown that the compact-ported topology introduced by Nachbin coincides with the compact-open topology on $\text{H}$(U) whenever U is a balanced open subset of a Fréchet-Schwartz space. This last result improves earlier results of P. Boland and S. Dineen [Bull. Soc. Math. France106 (1978), 311–336], R. Meise [Proc. Roy. Irish Acad. Sect. A81 (1981), 217–223], and others.     

Factorial representations of path groups

The reduction of the energy representation of the group of mappings from I = [0, 1], S¹1, $\text{R}$₊ or $\text{R}$ into a compact semisimple Lie group G is given. For G = SU(2), the factoriality of the representation, which is of type III in the case I=$\text{R}$, is proved.